Symmetric Ciphers and Asymmetric Ciphers Based on Magic Cube with 6-D

Tv~ÇÉãÄxwzÅxÇàá I would like to extend my sincere thanks and gratitude to my supervisors Prof. Dr. Abdul Monem S. Rahma and Asst. Prof. Abdul Mohssen J. Abdul Hossen for their invaluable guidance, constructive suggestions, advice and assistance during the writing of this book. Their regular comments, remarks and insightful guidance are highly appreciated. Thanks are also due to all staff members at computer science Department in the University of Technology for their encouragement and support. I am also indebted to Dr. Hazim H. Muarich for revising and editing the chapters of this study. Special thanks are extended to my mother and wife for their patience and moral support that helped me finish this study. Also many thanks are extended to my brothers and sisters for their support. Finally, I would like to thank all my friends for their support and willingness to listen and comment on different aspects of this work.

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ABSTRACT Information security cryptographic protocols are considered very important in the modern era due to the vast development and the advanced technology in the internet applications and the networks communications. The whole applications in our daily lives have become part of the electronic organization. So, the E-business, banks’ transfer money, financial marketing, remittances, distant reserving tickets for accommodation or air travel, E-Voting, E-testing, E-Marketing, E-Mailing and several other applications require secure environment. The present book includes proposing three types of cryptographic algorithms in symmetric models, asymmetric models and the magic square & magic cube mathematical problems. The symmetric type involves proposing three of distinct block cipher models that encrypt and decrypt the plain text with different key lengths and different rounds. The proposed models use the substitution and Permutation Network (SPN) structure and the reversible of operations stages in addition to the duplicated compact ciphering key. The proposed models involve (TIGRIS cipher, EUPHRATES cipher and FAROQ cipher) those are characterized with several fascinating features and render good insight in the design processes. The proposed symmetric models gave an accepted results and reasonable implementation compared with original AES model in compliance with algebraic characteristics, time of implementation, randomness tests deviation in addition to the security and efficiency factors. Although the EUPHRATES and TIGRIS models introduced some slowness in some cases especially with large data size for sixteen rounds structure but they appeared good in general with reduced rounds. FAROQ model introduced good implications in all aspects and applied a satisfied security level and conservative structure. The second part of the book comprises the developing of a secure method for public key encryption and the key exchange with digital signature, by depending upon the combining Diffie-Hellman with magic square and I

magic cube notations. The magic square and magic cube comprehensions adopted as an alternative mathematical problem in encryption and decryption process instead of the Integer Factoring Problem (IFP) and the Discrete of Logarithm Problem (DLP). The third part consists of the developing of generalized technique for the construction of magic cube based on the folded magic square. The proposed magic cube constructed with sequential or non-sequential numbers of N order by six dimensions or six faces based on the folding eleven distinct flat shapes with six surfaces from all directions using the magic square constructions’ techniques. The proposed magic cube considers a new variant of mathematical problem that is devoted for the public key development to provide a flexible solution to generate cipher keys easily without a need for complex computations. The proposed cube adds multiple of security levels represented by the start number selection, cubes’ order, difference value between the numbers and the cubes’ dimension usage. The proposed asymmetric models provide complex and untractable mathematical comprehensive that are characterized with faster implementation compared with the RSA and El-GAMAL algorithms. The proposed cryptosystem is implemented with Visual Studio 2013 C# programming language under Windows 7 Ultimate version of 32-bit operating system on CPU Intel (R) Core 2 duo 2.00 GHz and RAM 2-GB.

II

LIST OF ABBREVIATIONS Item

Descriptions

AES

Advance Encryption Standard

ATM

Automata Teller Machine

CA

Certification Authority

CRYPRREC

Cryptography Research and Evaluation Committees

DC

Differential Cryptanalysis

DES

Data Encryption Standard

DH

Diffie-Hellman

DLP

Discrete Logarithm Problem

DPA

Differential Power Analysis

DSA

Digital Signature Algorithm

ECDLP

Elliptic-Curve of Discrete Logarithm Problem

FN

Feistel Network

FS

Feistel Structure

GCD

Greatest Common Divisor

GF

Galois Field

GUI

Graphical User Interface

HECDLP

Hyperelliptic Curve Discrete Logarithm Problem

IFP

Integer Factorization Problem

IPs

Intellectual Property

LC

Linear Cryptanalysis

LC

Linear Cryptanalysis

MAC

Message Authentication Code

MC

Magic Constant

MDS

Maximum Distance Separable III

LIST OF ABBREVIATIONS Item

Descriptions

MG

Hypercube

Ms

Milliseconds

MS

Magic Square

MT

Magic Tesseract

NIST

National Institute Standard and Technology

NTT

Nippon Telegraph and Telephone

PDA

Personal Digital Assistant

PGP

Pretty Good Privacy

RCON

Round Constant

RSA

Rivest, Shamir and Adelman

SAC

Strict Avalanche Effect

SHA

Secure Hash Algorithm

SPN

Substitution Permutation Network

SSL

Secure Socket Layer

IV

LIST OF FIGURES

List of Figures Figures

Page No.

CHAPTER TWO Fig. 2.1 Block Diagram for the AES Structure

13

Fig. 2.2 ShiftRows Operation

17

Fig. 2.3 Inverse ShiftRows Operation

17

Fig. 2.4 Round Key Addition

19

Fig. 2.5 Magic Square of Order 3

22

Fig. 2.6 Three of Magic Squares with Different Pivot Element

23

Fig. 2.7 Magic Square of Odd Order Constructions’ Steps

28

Fig. 2.8 Pyramid Magical Square with order Three

29

Fig. 2.9 Pyramid Magical Square with Order Five

29

Fig. 2.10 Example of Dürer approach

30

Fig. 2.11 Other Example of Dürer approach

31

Fig. 2.12 Some Reflections and Rotations for the Dürer approach

31

Fig. 2.13 de la Hire Technique of Order Six

33

Fig. 2.14 Example of de la Hire with Order Six

34

Fig. 2.15 Semi-Magic Square with Order Three

35

Fig. 2.16 The Nasik Square Technique

35

Fig. 2.17 The Associated Squares Technique

36

Fig. 2.18 The Pan diagonal Square

36

Fig. 2.19 The Ultramagic Squares Technique

37

V

LIST OF FIGURES

Fig. 2.20 The Border Square Technique

37

Fig. 2.21 Rotations and Reflections of Magic Square

38

Fig. 2.22 Magic Cube of Order Three

39

Fig. 2.23 Magic Cube of Three Layers

40

Fig. 2.24 Eleven Distinct Shapes for the Magic Cubes

40

CHAPTER THREE Fig. 3.1 General Structure of TIGRIS Cipher

45

Fig. 3.2 TIGRIS-ReversibleShiftRows

48

Fig. 3.3 TIGRIS-Add Round Key

51

Fig. 3.4 TIGRIS-Key Expansion

52

Fig. 3.5 The EUPHRATES Structure

54

Fig. 3.6 Euphrates-ReversibleShiftColumns

58

Fig. 3.7 FAROQ Structure

62

Fig. 3.8 FAROQ-ReversibleShiftColumns

65

Fig. 3.9 Cube’s Dimension of Order Three

68

Fig. 3.10 The Magic Constant of the Six Squares

69

Fig. 3.11 The Magic Sum of the Six Squares

69

Fig. 3.12 Cube’s Dimensions of Order Four

70

Fig. 3.13 Different Shapes of Magic Cube

72

Fig. 3.14 Unfolded Cross Figure of Magic Cube

73

Fig. 3.15 The Vertical and Horizontal Track of Main Diagonals

73

Fig. 3.16 The Vertical and Horizontal Track of Secondary Diagonals

73

Fig. 3.17 The Vertical Track of Cube Rows Values

74

Fig. 3.18 The Horizontal Track of Cube Columns Values

74

VI

LIST OF FIGURES

Fig. 3.19 Diffie-Hellman Key Exchange Protocol

78

CHAPTER FOUR Fig. 4.1 Comparison of Encryption and Decryption Chart’s Time

102

Fig. 4.2 Comparison of Signature and Verification Chart’s Time

102

Fig. 4.3 Cube with Three Dimensions

103

Fig. 4.4 The Layers of Magic Cube

103

Fig. 4.5 The Main Window for the Symmetric Models

109

VII

LIST OF TABLES

List of Tables Tables

Page No.

CHAPTER TWO Table 2.1 AES-Forward S-Box

15

Table 2.2 AES-Backward S-Box

16

CHAPTER THREE Table 3.1 TIGRIS-Forward S-Box

47

Table 3.2 TIGRIS-Backward S-Box

48

Table 3.3 EUPHRATES-Forward S-Box

56

Table 3.4 EUPHRATES-Backward S-Box

57

Table 3.5 FAROQ-Forward S-Box

63

Table 3.6 FAROQ-Backward S-Box

64

Table 3.7 Magic Sum (MS) and Magic Constant (MC) Computations

70

CHAPTER FOUR Table 4.1 Analysis and Comparison Speed for the Proposed Models

89

Table 4.2 Algebraic Comparison between the AES and the Proposed

Ciphers

90 VIII

LIST OF TABLES

Table 4.3 Comparison Speed between TIGRIS Cipher and the AES (1)

91

Table 4.4 Comparison Speed between TIGRIS Cipher and the AES (2)

92

Table 4.5 Comparison Speed between EUPRATIS Cipher and the AES(1)

93

Table 4.6 Comparison Speed between EUPRATIS Cipher and the AES(2)

93

Table 4.7 Comparison Speed between FAROQ Cipher and The AES

94

Table 4.8 The Statistical Tests for the TIGRIS Cipher

95

Table 4.9 The Statistical Tests for the EUPHRATES Cipher

96

Table 4.10 The Statistical Tests for the FAROQ Cipher

97

Table 4.11 Comparison between the Proposed Cube and other Distinct Cubes 104

IX

List of Contents CHAPTER ONE INTRODUCTION

1.1 General Background …………………………..………………………..

1

1.2 Literature Survey.………..…….……………..………………………….

1

1.2.1 Related Work for the Proposed Symmetric Methods ……………

2

1.2.2 Related Work for the Proposed Asymmetric Method with Magic Square ………..…………………………………………..............

3

1.2.3 Related Work for the Proposed Magic Cube… …………………..

4

1.3 Magic Square Comprehension………….……...………………..............

5

1.4 The Block Ciphers Design ……………..……..………………...............

6

1.5 The Computational Problems of the Public Key Cipher……………......

7

1.6 Aim of Work…………………………………………………………….

8

1.7 The Book Outline…..…………………….....................................

9

CHAPTER TWO SYMMETRIC CTYPTOGRAPHY AND MAGIC SQUARE

2.1 Introduction ……………….…………………….………….….............

11

2.2 Selection the Universal Standard Cipher…………….………................

12

X

2.3 AES Structure and Characteristics………………………………………

13

2.3.1 The S-Box Construction…………………………………………...

14

2.3.2 InvSubBytes Transformation………………………………………

15

2.3.3 ShiftRows Transformations………………………………………..

16

2.3.4 Inverse ShiftRows Transformations……………………………….

17

2.3.5 Forward MixColumns Transformation…………………………….

18

2.3.6 Inverse MixColumns Transformation……………………………...

18

2.3.7 AddRoundKey Transformation……………………………………

19

2.3.8 AES Key Expansion……………………………………………….

20

2.4 Diffie-Hellman Key Exchange..…...…………………………………….

20

2.5 Introduction to Magic Square………………………………….………..

21

2.6 Magic Square from the Mathematical View Point…...….……………...

22

2.7 Magic Square Properties………………………………………………...

25

2.8 Magic Square Construction………………………..…………………….

26

2.8.1 Magic Square of Odd Order…………….…………………………

27

2.8.2 Magic Squares with Doubly Even Order.…...………………….....

29

2.8.3 Magic Squares with Singly Even Order…..…..…………………...

32

Types of Magic Square…..…………………………………………….

34

2.10 The Probability of Magic Square………………………………………

38

2.11 What is the Magic Cube?…………..…………………………………..

39

2.9

CHAPTER THREE THE PROPOSED SYSTEM

3.1 Introduction ……………………………………………………………... XI

41

3.2 The Proposed Symmetric Cipher……............…………………...............

42

3.2.1 Design Criteria……..……………………………………………...

42

3.2.2 The TIGRIS Cipher …………………..…………………………..

42

3.2.3 The EUPHRATES Cipher…………………..…………………….

52

3.2.4

Fast Algorithm Revertible Operation Queen (FAROQ)………….

60

3.3 The Proposed Magic Cube…...…………………………………………..

67

3.4 The Construction of Magic cube…………………………………………

68

3.5 The Cross Figure of Folded Cube…………………………...……….......

72

3.6 The Proposed Public Key (Asymmetric Cipher)……………………...…

74

3.7 The Proposed Signature Algorithm……………………………………...

77

CHAPTER FOUR ANALYSIS AND RESULTS WITH IMPLEMENTATION

4.1 Introduction ……….……..........................................................................

88

4.2 The Analysis for the Symmetric Cryptographic Models………………...

88

4.3 TIGRIS-Security Analysis……………………………………………….

91

4.4 EUPHRATES-Security Analysis………………………………………...

92

4.5 FAROQ Security Analysis……………………………………………….

93

4.6 The Randomness Tests…………………………………………………...

94

4.7 Applications of Proposed Symmetric Models………………....................

98

4.8 Advantages and Limitations of Proposed ciphers………………………..

99

4.8.1 The Advantages……………………………………………………

100

4.8.2 The Limitations…………………………………………………….

100

XII

4.9

Security Analysis of the Proposed Public Key…...…….…………….....

101

4.10 The Analysis of Magic Cube…………………..……….……………......

103

4.11 Advantages and Limitations of Magic Cube……………..…...…………

104

4.11.1 Advantages of the Proposed Magic cube………………………….

104

4.11.2 Limitations of the Proposed Magic cube………………………….

105

4.12 Applications of the Proposed Models……………..…………………….

106

4.13 Motivations and Limitations for Proposed Public Key……….…………

106

4.14 Graphical User Interface (GUI)………………………………................

108

CHAPTER FIVE CONCLUSIONS AND RECOMMANDATIONS

5.1 Conclusions …………..….……………………………….......................

110

5.2 Recommendations ………………………………………………………

112

5.2.1 The Symmetric Cipher Recommendations………………………...

112

5.2.2 The Asymmetric Cipher Recommendations…………….................

113

5.2.3 Magic Cube and Magic Square Recommendations………………..

113

XIII

Chapter One

Introduction

CHAPTER ONE INTRODUCTION 1.1. General Background Resent progress and development in the digital communications information technologies have turned life into a digital organization. So there is an urgent need for the handling, saving, and sharing of sensitive information under confident and secure environments which have become as an important demand for people all over the world. Hence, information security represents the nerve of digital technology. So for these reasons, more attention has been devoted to the cryptographic algorithms [1]. In recent years there has been a huge development in cryptographic algorithms of different types and with different strategies. Everyone knows that the AES cipher is a good choice and provides an acceptable performance for several applications that do not need a high level of security, but from the other perspective the AES cipher should have a limited time, and inevitably it will have a specific date of life, since the development in real effective attacks and the cryptanalysis is continuous [2]. The replacement cipher should be submitted to change the orientation thoughts for the researchers from both of designers and cryptanalysts according to the modern cryptanalysis methods and the rapid improvement in the power of computers. It is worth mentioning that the asymmetric cipher or the public key cipher also needs a dramatic change in the designing styles and the mathematical nature on the one hand of the internal operations, algebraic pattern and the general structure as well as the mathematical bases to satisfy the current need and the foreseeable future requirements, on the other [3].

1.2

Literature Survey There are several referred works and gorgeous ideas related to the

submission work of this book, so the presented contributions will be subdivided -1-

Chapter One

Introduction

into three sections that comprise those that are related to symmetric cipher, magic square and magic cube and the asymmetric cipher.

1.2.1 Related Work for the Proposed Symmetric Methods The symmetric cipher is the most important cryptographic protocols that have been developed significantly in the later years. So many researchers and countries have developed their own ciphers. Here, some of the related works will be reviewed in the following sections: In 2007, Sufyan T. Faraj has submitted a new variant of the AES symmetric cipher that is called Nahrainfish cipher. The proposed cipher has been mainly based on the Twofish cipher’s characteristics with some modifications and developments. It works with 128-bit of block size and variable cipher key that reaches to the 1024-bits. The developed algorithm has a classical Feistel network structure that is based on a key-dependent S-boxes and data-dependent rotations. The proposed cipher has applied a high level of security and an acceptable performance but with some sacrifice for efficiency [4]. In 2013, Nada H. Ali, Abdul Monem Rahma and Abdul Mohsen Jaber have applied some improvements on the AES cipher for the S-Box stage, that includes the generation of several random S-boxes by depending upon the uses of multi ciphering keys technique in order to generate several dependent S-boxes that include each one with its inverse as well as with various constant vectors. The second level is to configure a random indexing for the S-boxes that are constructed to obtain more complexity with large scale for the probability of estimation and guessing. The developed method suffers from time consuming in implementation as well as high cost in hardware [5].

-2-

Chapter One

Introduction

1.2.2 Related Work for the Proposed Asymmetric Methods with Magic Square The embedding of magic square techniques through the public key cipher still under development and research and it is still evolving, since there is no real encryption and decryption method with full dependence on magic square principles and comprehensions, because it uses the magic square just as the additional security layer and as an assistant tool for the real asymmetric algorithms. In 2012, Nitin Pandey and D.B.Ojha have proposed a new method of encryption and decryption process that is based on the RSA cipher and magic rectangle’s construction. The proposed method constructs different of singly even magic of rectangles with an even order that don’t accept the divisibility by four, where the sum of each rows and columns values is the same. The main purpose for the rectangle square is to address the numeral values to the corresponding positions with the magical rectangle in different quadrants. So the numbers then encrypted and decrypted using the RSA public cipher. The two researchers have proposed that the developed method increases the complexity and the randomness for the ciphertext and at the same time it requires an extra time for the implementation process [6]. In 2014, A. Dharini, R.M. Saranya Devi, and I. Chandrasekar have introduced a new approach for secure data transmission through the cloud environment and sharing networks as well as during the Secure Socket Layer (SSL) by the RSA combined with magic square, to provide additional security layer to the cryptosystem. The proposed model submits the confidentiality and the integrity of data over the communication to and from the cloud providers. So it discusses and combines magic square algorithm with the RSA cipher when implemented on data security in cloud computing [7]. -3-

Chapter One

Introduction

In 2009, Gopinanath, Ganapathy, and K. Mani have proposed a new approach for the public key cipher with magic square which is based on generating the magical square with order of doubly even for multiples of sixteen. The developed method is not much different from the previous work except it has taken various methods of the public key cipher. Therefore; the proposed method treated with the positions of the magic square that are corresponding to the ASCII values as an alternative. So the encryption and decryption will be with different numbers that represent the placements of the ASCII elements in the magic square. The ciphering and deciphering process is performed also by the RSA algorithm [8]. In 2014, D.I. George, J.Sai Geetha and K.Mani are three researchers from India who have proposed another technique for combining the RSA cipher with the magic rectangle of singly even order. The construction of magical rectangle is based on several initial parameters that involve magical rectangle’s seed, constant vector for the amount of column values and the beginning number for the rectangle construction. The main idea for this method is to construct a specific magic rectangle and then the encrypted text or the cipher text which is encrypted by the RSA cipher and then mapping to the positions of numeric values in the magic rectangle and after that will change the ciphertext with those corresponding positions in order to increase the time complexity and to add another security level. The proposed method requires more time of implementation and hardware cost [9].

1.2.3 Related Work for the Proposed Magic Cube For a long time, several researchers and mathematician specialists have created many related works about the magic square and magic cube. The

-4-

Chapter One

Introduction

following are some studies and discussions to the prior labor, which so far an associated work to the proposed method: In 1970, Richard Meyers has invented a perfect eighth-order magic cube that is known by Meyers cube. The Meyers cube is interested in several symmetries properties which assume that the cube is associative and every orthogonal and diagonal line sum to the same specific number. The corners values in the inner small cube as well as the corners values of each rectangular in the Meyers cube also sum to a constant certain number. The prominent feature for the symmetries properties makes that is possible for a tantalizing number and for rearrangements of the cube [10]. In 1981 J. Barkley Rosser and Robert J. Walker are two researchers who have introduced a new approach for constructing a perfect eighthorder magic cube. They have also explained and proved that the perfect pan diagonal cubes are found for whole orders with multiples of 8 and also for all the odd orders that are more than 8 order [11]. In 1988, John Hendricks submitted new ideas and published many refereed related papers. He developed a simple and clear technique in constructing of an odd order magic cube with N order. In addition, he published an extended dimension of the hyper cubes with four, five and six dimensions; also he applied an elegant work and great share in the magic square area and in the methods development for the magical constructions approaches [12].

1.3

Magic Square Comprehension The magic square is considered one of the matrices categories that

constitutes a big challenge for the mathematician researchers and for the -5-

Chapter One

Introduction

amateurs of intelligent games. The core notation for the magic square is how to arrange a set of numbers into the matrix form where each row summation values or column summation values even the two diagonals values should give the same summation result [13]. So there are several types of magic square with different shapes and different construction techniques. The regular or the ordinary magic square is the standard square and also is considered as the most well-known square that has the regular properties. Other irregular magical squares with additional conditions can be stated here like: Latin square, semimagic square, Bordered magic square, magic hexagon, magic spheres, magic series, magic spider, Nasik square, magic circle and pan-diagonal square etc. [14].

1.4

The Block Ciphers Design The block ciphers are the symmetric key algorithms that encrypt the clear

text as a fixed size of block message at each time to output the block of cipher text mixed by the secret ciphering key. The block cipher has partitioned the clear-text into the seperated blocks of the same lengths of 64-bit, 128-bits and may reach to 1024-bits [15]. Block ciphers methods build by two elementary structures that involve the Feistel Structure (FS) and the Substitution Permutation Network Structure (SPN). The algorithms which are constructed by feistel structure require only one algorithm for encryption and decryption process, because it is the same, in opposite to the SPN that requires two algorithms one for encryption process and the other for decryption process. The AES, square and serpent ciphers represent good examples for the SPN structure and the Data Encryption Standard DES, IDEA, Blowfish, Twofish, MARS, and RC6 cipher and several other algorithms act good examples for the Feistel structure [16].

-6-

Chapter One

1.5

Introduction

The Computational Problems of the Public Key Cipher There are numerous public-key cryptosystems that have been proposed and

developed along the past years until the present time. Most of them have built their mathematical foundation on trained, famous and intractable mathematical problems as shown below: A. The Integer Factorization Problem (IFP): The IFP focuses on the factoring the number to its factors prime numbers. The RSA public key is considered a good paradigm that is based on factoring problem of analyzing the composite N number to its factors of two distinct large prime’s numbers p and q, in order to find the eth root. Thus, the main difficult waylays if the factoring of N is known, so its computational mathematics will be easy to solve the RSA problem [17]. B. Discrete Logarithm Problem (DLP): The core problem here is how to compute the logarithms in a large finite field. The Digital Signature Algorithm (DSA) is a good example which is used in authentication and integrity techniques. Several algorithms use the DLP such as the DiffieHellman public key, El-Gamal cipher and the digital signature in addition to the Schnorr signature etc. [18]. C. Elliptic-Curve of Discrete Logarithm Problem (ECDLP): it is another mathematical problem that is also based on the DLP that has been defined over the group of points with an elliptic curve. The elliptic curve is viable for ciphering, digital signatures, pseudo-random generators and other assignments. The examples include the elliptic curve analogues of: DSA (so-called ECDSA), Diffie-Hellman algorithm, El-Gamal cipher and the digital signature mode etc [19].

-7-

Chapter One

Introduction

D. Hyperelliptic Curve Discrete Logarithm Problem (HECDLP): This type of problem also uses the DLP that is defined under an algebraic curve over the group of Jacobian method, which is called the group of divisors of degree 0 module principal divisors on a hyperelliptic curve. Examples comprise the hyper elliptic curve homologue of: DSA, DiffieHellman public key model, El-Gamal ciphering and deciphering in addition to the digital signature methods etc. [20]. E. Discrete Logarithm Problem on Algebraic Tori: it is another problem that is also related to the DLP in the finite field of order Fq. Which is mainly associated with DLP on an algebraic torus Tn(Fq) that is equivalent to the DLP in the subgroup of order ⱷn(q). Where ⱷn(q) equivalent to the n-th cyclotomic polynomial. The main goal of Tori is to find short representation for elements of certain subgroup of F*qn. [21].

1.6

Aim of Work There are several elementary objectives the present study is looking for: the

major goals are to design a set of secure symmetric algorithms with different models and different techniques to give the user freedom in selection that any one is suitable for his application. Hence, the choice of algorithm is selected according to the speed, efficiency and the performance or other factors. The proposed system is based, not being only on the symmetric cipher, but also on a developed method of asymmetric cipher in order to implement completed protocol with diverse usage in the cryptographic field. The main issue for the proposed public key method is to provide a secure communication over unprotected network and trusted transmission for data through the public channel by using the encryption methods. The proposed public key is to design new methods of asymmetric cipher based on the magic square and magic cube techniques to speed up the encryption and decryption process and to speed the key generation process in addition to increase the computational complexity. -8-

Chapter One

Introduction

1.7 The Book Outline This book has been divided into five chapters: Chapter One begins with explaining the whole aspects of the work and introducing the basic concepts of the cryptography. The other four chapters are organized as follows: Chapter Two covers the basic concepts of Advance Encryption Standard (AES) in addition to the designing steps of the Rijndael cipher. This chapter also submits a comprehensive study for the magic squares’ principles; properties, types and techniques in addition to the magic square construction methods and the magic cube definition. Chapter Three produces the proposing of three symmetric ciphers with distinct models and developing new method in the magic cube construction based on the folded procedure for the magic squares approaches, as well as improving new methods of public key and signature algorithms by depending upon the magic square and magic cube features combined with the Diffie-Hellman key exchange method. Chapter Four gives the analysis and the experimental results for each proposed model in addition to the results for some tests on these models. Chapter Five includes the elementary conclusions of the work and recommended notations as well as the basic suggestions for future works.

-9-

Chapter Two

Symmetric Cryptography and Magic Square

CHAPTER TWO SYMMETRIC CRYPTOGRAPHY AND MAGIC SQUARE 2.1

C

Introduction ryptography is the art and science that deals with studies, methods and

techniques of mathematical methodologies. These methods work to convert the meaningful clear-text to rubbish text. So the cryptography enables to transmit the sensitive data across insecure channels and prevents any one to figure out the intent notation except the target destination [22].The main motivations for the cryptography are to provide privacy of communication between two parties and to provide authentication of one party to another. This methodology formulated under the comprehension that each pair of communicating parties needs to have a shared secret key which should be transmitted securely. Consequently, this leads to the difficulty in obtaining signatures with non-repudiation [23]. There are two basic categories of cryptosystem algorithms: Symmetric algorithms (also called "secret key") which uses the same key for both encryption and decryption; both parties share the same key in encryption and decryption in order to provide confidentiality which needs to keep secret. Symmetric cipher includes many well-known concrete algorithms such as DES, Triple-DES, BLOWFISH, TOWFISH, RC6, Serpent, MARS, IDEA and AES. The second category is the asymmetric algorithms: (also called "public key") that uses different keys for encryption and decryption, one is used for encryption and the other one for decryption. The decryption key is typically kept secretly, therefore called the "private key'' or "secret key''. There are many famous asymmetric algorithms such as RSA, DSA, Elliptic Curve and ELGAMAL etc. [24].

- 11 -

Chapter Two

2.2

Symmetric Cryptography and Magic Square

Selection of the Universal Standard Cipher In 1997, National Institute of Standard and Technologies (NIST) invited

the cryptographers, researchers and data security specialists all over the world to participate in the contest of selecting a standard algorithm, and to apply their own algorithms in order to select the best one to be a good alternative for the DES and the Triple-DES, since the last one has encountered many problems represented by short key and non-resistance against Linear and differential attack as well as the brute force attacks. That notwithstanding with the rapid development in the information technologies [25]. Several distinct algorithms have been applied which, vary widely in their structures and implementation natures those originally presented by researchers from different nations. Fifteen algorithms have been selected from the first round of selection. After deep study and analysis processes were reduced to five algorithms in the second round. The five algorithms selected were MARS, RC6, RIJNDAEL, SERPENT and TWOFISH [26]. On October 2nd 2000, NIST announced that the Rijndael algorithm was the winner of the contest that appeared to be a very good performer in both hardware and software across a wide range of computing environments [27]. Several developed countries such as the United States and Japan submitted extensive researches to develop their own standard block cipher algorithm. The AES is American standard and it is a SPN block cipher standardized by the NIST, which supports three key lengths of 128, 192, and 256 bits [28]. Camellia is a Japanese standard cipher which is a Feistel-type block cipher jointly developed by NTT (Nippon Telegraph and Telephone Corp.) and Mitsubishi Electric. Camellia is similar to the AES cipher it also supports 128-bit, 192-bit, and 256-bit keys. An advantage of the feistel cipher is that the same data path and structure can be used for encryption and decryption process [29].

- 12 -

Chapter Two

2.3

Symmetric Cryptography and Magic Square

AES Structure and Characteristics AES cipher or what is known by (Rijndael) cipher is a symmetric and

iterated block cipher that supports the input entries with a fixed block size of 16 byte of square state array which is corresponding to128-bits and three of distinct keys length of 128-bits, 192-bits and 256-bits accompanied with 10, 12 and 14 rounds respectively. AES cipher interested with highly algebraic structure that is combined several good features in its construction instantiated by security, performance, efficiency, ease of implementation and flexibility. These characteristics make it a good implemented algorithm and an efficient cipher in both hardware and software. The Figure 2.1 states the general structure of the AES cipher [30]. State Matrix

Plaintext

AddRoundK( ) Stage

Cipher Key

SubByte ( ) Stage ShiftRow ( ) Stage

0 - 9 Rounds

MixColumn ( ) Stage AddRoundK( ) Stage

Round Key 0

SubByte ( ) Stage ShiftRow ( ) Stage AddRoundK( ) Stage

Round Key 9

Ciphertext

Figure 2.1 Block Diagram for the AES Structure - 13 -

Chapter Two

Symmetric Cryptography and Magic Square

The internal structure of the AES is designed to work with byte (8-bits) operations. AES cipher was not designed from zero, but it can be considered as a smart step in a long development process that eventually paved the way to the design of Rijndael algorithm with inherited good properties form the predecessors and successors block ciphers such as (Shark, BKSQ, Square, Crypton, Hierocrypt, GrandCru, Anubis and Khazad) [31].

2.3.1 The S-Box Construction The SubBytes operation is the first step of the data transformation process in the Rijndael cipher where each block is replaced by its substitution in S-Box table. The S-Box is table with non-linear substitution operation used in most block ciphers that can be constructed either randomly or mathematically that is defined as a square matrix 16×6 values, which contains permutation of all possible 256 values. The construction of the forward S-Box basically composed of two mappings: The first map treats the 8-bits value as an element of GF(28), and takes the multiplicative inverse if the element is non-zero, and otherwise just maps to 00[32]. The second map, treats with the resulting GF(28) element as an 8-bits vector and performs the following affine transformation f(x) over GF(2). Regarding 8-bit or bytes as an elements in GF(28) as stated in Equation (2.1).

b '0  1  b '1  1     b '2   1     b '3  = 1  b '4   1     b '5  0 b '6   0    b '7   0

0 0 1 0 1 1 1 1 1 0 0

0 1 1 1 1  b 0  1  0 0 1 1 1   b1  1  0 0 0 1 1  b 2   0      1 1 0 0 0 1   b3 0  + 1 1 1 0 0 0  b 4   0      1 1 1 1 0 0   b 5  1  1 1 1 1 1 0  b 6  1      0 1 1 1 1 1  b 7   0 

- 14 -

... ( 2.1)

Chapter Two

Symmetric Cryptography and Magic Square

The multiplicative inverse has been taken modulo the irreducible polynomial x8 + x4 + x3 + x + 1 and the result Xored with the constant vector that is represented by the value (63). Table 2.1 is the resultant table [33] Table 2.1 AES-Forward S-Box 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 63 ca b7 04 09 53 d0 51 cd 60 e0 e7 ba 70 e1 8c

1 7c 82 fd c7 83 d1 ef a3 0c 81 32 c8 78 3e f8 a1

2 77 c9 93 23 2c 00 aa 40 13 4f 3a 37 25 b5 98 89

3 7b 7d 26 c3 1a ed fb 8f ec dc 0a 6d 2e 66 11 0d

4 f2 fa 36 18 1b 20 43 92 5f 22 49 8d 1c 48 69 bf

5 6b 59 3f 96 6e fc 4d 9d 97 2a 06 d5 a6 03 d9 e6

6 6f 47 f7 05 5a b1 33 38 44 90 24 4e b4 f6 8e 42

7 c5 f0 cc 9a a0 5b 85 f5 17 88 5c a9 c6 0e 94 68

8 30 ad 34 07 52 6a 45 bc c4 46 c2 6c e8 61 9b 41

9 01 d4 a5 12 3b cb f9 b6 a7 ee d3 56 dd 35 1e 99

A 67 a2 e5 80 d6 be 02 da 7e b8 ac f4 74 57 87 2d

B 2b af f1 e2 b3 39 7f 21 3d 14 62 ea 1f b9 e9 0f

C fe 9c 71 eb 29 4a 50 10 64 de 91 65 4b 86 ce b0

D d7 a4 d8 27 e3 4c 3c ff 5d 5e 95 7a bd c1 55 54

E ab 72 31 b2 2f 58 9f f3 19 0b e4 ae 8b 1d 28 bb

F 76 c0 15 75 84 cf a8 d2 73 db 79 08 8a 9e df 16

2.3.2 InvSubBytes Transformations InvSubBytes, The inverse of the proposed S-box is constructed by applying the inverse of the affine transformation that can be shown in Equation (2.2) followed by applying the multiplicative inverse in GF(28) and the result Xored with the constant vector represented by the value (05) of hexadecimal representation, so the resultant table can be stated in Table 2.2. The rational reason for using the constant vector is to increase the complexity of the S-box and to remove fixed point respectively that gives the S-Box construction more resistant against the attacks [34].

- 15 -

Chapter Two

 b '0   0  b '1   1     b '2   0     b '3  =  1  b '4   0     b '5   0  b '6   1    b '7   0

Symmetric Cryptography and Magic Square

0 0

1 0

0 1

0 0

1 0

0 1

1 0 1

0 1 0

0 0 1

1 0 0

0 1 0

0 0 1

0 0 1

1 0 0

0 1 0

1 0 1

0 1 0

0 0 1

1   b 0  1  0   b1   0  1  b 2  1      0   b3  0  + 0  b 4   0      1   b5  0  0  b 6  0      0  b 7   0 

... ( 2 .2 )

Table 2.2 AES-Backward S-Box 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 52 7c 54 08 72 6c 90 d0 3a 96 47 fc 1f 60 a0 17

1 09 e3 7b 2e f8 70 d8 2c 91 ac f1 56 dd 51 e0 2b

2 6a 39 94 a1 f6 48 ab 1e 11 74 1a 3e a8 7f 3b 04

3 d5 82 32 66 64 50 00 8f 41 22 71 4b 33 a9 4d 7e

4 30 9b a6 28 86 fd 8c ca 4f e7 1d c6 88 19 ae ba

5 36 2f c2 d9 68 ed bc 3f 67 ad 29 d2 07 b5 2a 77

6 a5 ff ff 24 98 b9 d3 0f dc 35 c5 79 c7 4a f5 d6

7 38 87 3d b2 16 da 0a 02 ea 85 89 20 31 0d b0 26

8 bf 34 ee 76 d4 5e f7 c1 97 e2 6f 9a b1 2d c8 e1

9 40 8e 4c 5b a4 15 e4 af f2 f9 b7 db 12 e5 eb 69

A a3 43 95 a2 5c 46 58 Bd cf 37 62 c0 10 7a bb 14

B 9e 44 0b 49 cc 57 05 03 ce e8 0e fe 59 9f 9f 63

C 81 c4 42 6d 5d a7 b8 01 f0 1c aa 78 27 93 83 55

D f3 de fa 8b 65 8d b3 13 b4 75 18 cd 80 c9 53 21

E d7 e9 c3 d1 b6 9d 45 8a e6 df be 5a ec 9c 99 0c

F fb cb 4e 25 92 84 06 6b 73 6e 1b f4 5f ef 61 7d

2.3.3 ShiftRows Transformations The purpose of the ShiftRows is to distribute the bytes of each input column to different output columns. This step is a linear diffusion process that operates on individual rows and each row of the array is rotated by a certain number of byte positions. The forward shift row transformation, works by rotating the 2nd, 3rd and 4th row of the matrix. The first row is not shifted, and the remaining rows proceed as follows, for the second row 1-bytecircular left shift is - 16 -

Chapter Two

Symmetric Cryptography and Magic Square

performed for the third row 2-byte circular left shift is performed and for the fourth row 3-byte circular left shift as stated in the following Figure 2.2 [35]. S

S'

S0,0

S0,1

S0,2

S0,3

S1,0

S1,1

S1,2

S1,3

S2,0

S2,1

S2,2

S2,3

S3,0

S3,1

S3,2

S3,3

S0,0

S0,1

S0,2

S0,3

S1,1

S1,2

S1,3

S1,0

S2,2

S2,3

S2,0

S2,1

S3,3

S3,0

S3,1

S3,2

Figure 2.2 ShiftRows Operation

2.3.4 Inverse ShiftRows Transformations The decryption process is implemented by the corresponding steps of shiftrows but with an opposite direction. The first row is unchanged; the second row is shifted to the right by one byte, the third row to the right by two bytes, and the last row to the right by three bytes, all shifts being circular, as stated in Figure 2.3 [36]. S'

S S0,0

S0,1

S0,2

S0,3

S0,0

S0,1

S0,2

S0,3

S1,1

S1,2

S1,3

S1,0

S1,0

S1,1

S1,2

S1,3

S2,2

S2,3

S2,0

S2,1

S2,0

S2,1

S2,2

S2,3

S3,3

S3,0

S3,1

S3,2

S3,0

S3,1

S3,2

S3,3

Figure 2.3 Inverse ShiftRows Operation - 17 -

Chapter Two

Symmetric Cryptography and Magic Square

2.3.5 Forward Mixcolumn Transformation This operation provides both confusion and diffusion. The mixcolumn transformation is based on multiplication of the state with a certain matrix. This matrix is derived from the property that multiplication of a polynomial with a fixed polynomial over GF(28) results in a constant matrix multiplication. In GF(28) columns of the state that can be considered as polynomials. These polynomials are multiplied over GF(28) with a fixed polynomial c(x), which is defined as a(x) = {03} x3 + {01} x2 + {01} x + {02}

… (2.3)

This polynomial is coprime to (x4+1). Therefore; the transformation is invertible. This transformation can be written under the form of a matrix multiplication that is for all the 4 columns in the state matrix. As a result of this multiplication are the 4 bytes in a column of the matrix multiplication of s'(x) as follows: [37].

 S '0 ,c   S '1,c     S '2 ,c     S '3,c 

02  01   01  03

03 02 01 01

01 03 02 01

01 01 03  02

 S 0 ,c   S1,c     S 2 ,c     S 3,c 

2.3.6 Inverse Mixcolumns Transformations The InvMixColumns is similar to MixColumns process but with inverse matrix coefficients, so every column on the state treating as a four-term polynomial with coefficient in GF(28) and multiplied modulo the co-prime of x4 + 1 by the fixed polynomial a-1(x). If the inverse states'(x) is written as a matrix multiplication, then as the following formula:

s'(x) = a-1 (x)

S (x)

Where, a-1(x) ={0b} x3 + {0d} x2 + {09} x + {0e} - 18 -

… (2.4)

Chapter Two

Symmetric Cryptography and Magic Square

As a result of this multiplication the four bytes in column are replaced by the others as in the following [38]. a( x ) = a-1( x ) mod ( x4 + 1 )

 S '0 ,c   S '1,c     S '2 ,c     S '3,c 

0E 0B 0D  09 0 E 0 B  0 D 09 0 E   0 B 0 D 09

09  0 D  0B   0E 

… (2.5)

 S 0 ,c   S1,c     S 2 ,c     S 3,c 

2.3.7 AddRoundKey Transformation

The key addition operation is referred to by AddRoundKey that includes a number of words (Nb). This operation is viewed as a columnwise operation between the 4-bytes of a state column and 4-bytes of round key column. The addroundkey transformation is as simple as possible and affects every bit of the state. The forward addroundkey transformation is identical to the inverse addroundkey transformation, because the XOR operation has its own inverse [39]. As stated in the following Figure 2.4.

Figure 2.4 Round Key Addition - 19 -

Chapter Two

Symmetric Cryptography and Magic Square

2.3.8 AES Key Expansion The key generation, key scheduling or what is known by the key expansion is the method which is responsible for generating the ciphering key that should be kept secret. The AES algorithm takes the cipher key. and performs the key expansion technique to generate a key schedule. The key expansion determines how the expanded key generated from the cipher key by depending upon a few sequence steps in Rijndael's algorithm that includes the following steps: [40]: 1. Achieving one byte circular rotate on the word of (32-bit). 2. SubByte transformation. 3. The result xored with round constant table to eliminate the symmetries. 4. Key scheduling procedures.

2.4

Diffie-Hellman Key Exchange The Diffie-Hellman is one of the first and earliest public-key protocols that

allows two parties to share a secret key without any predated acquaintance over insecure or untrusted channel. The resultant key can be used later to encrypt as a symmetric key cipher. The Diffie-Hellman protocol is based mainly on the Discrete Logarithm Problem (DLP) and implemented within the finite field of cryptography. Suppose Alice and Bob want to share a secret key over public communication media and to use then in a symmetric cipher, so the information and the key that they exchange will be observed by their adversary Eve [41]. At the beginning Alice and Bob have to agree on a finite field (Fq) and the primitive root (g) in Fq, then each one secretly chooses a random positive number n and m to compute the following: Alice computes X ≡ / (mod p) Bob computes Y ≡ / - 20 -

(mod p)

Chapter Two

Symmetric Cryptography and Magic Square

The two parties will calculate the share key, where Bob raises the primitive root to the selected value that has been received from Alice, also the Alice raises the primitive root to the value that has been received from Bob to complete the agreement on the shared secret key as stated below: Alice computes X≡ ͯ

(mod p)

Bob computes Y ≡ ∿ (mod p)

Then, the two parties shall have

(

) =(

) =

∈

∗

.

The Diffie-Hellman protocol suffers from the man-in-the-middle attack which is considered a form of eavesdropping attack that happens when the malicious attacker or eavesdropper monitors, modifies and retransmits the intercepted information across the communication session between the two users by impersonating the personality of the authorized author. The Diffie-Hellman concepts paved the way to the invention the RSA public cipher that is considered the most well-known asymmetric cipher which was developed by three brilliant scientists Rivest, Shamir and Adelman [42].

2.5

Introduction to Magic Square Mathematics is the art and the study of methods that treats the logical and

quantity forms as well as arranging them. Mathematics is found everywhere and in everything that we work. It is indispensable science and the cornerstone for several tasks in daily life [43]. The variety in mathematic computational problems makes work with different branches of mathematics fields such as group theories, lattices, Latin squares, determinants, coding theory, matrices, arithmetic and logic and the magic of square [44]. Magic squares have been flourished and grown rapidly with arising of mathematics-based games like puzzles, Rubik and Sudoku games, which have been especially attractive to the amateur mathematicians. The magic square is the order of the numbers in the form of square matrix with equal dimensions that the amount of each row numbers, column numbers, and diagonal numbers is a constant number, that is known by the "magic constant" [45]. Magic square may be divided into parts - 21 -

Chapter Two

Symmetric Cryptography and Magic Square

with letters, symbols, talisman, discrete words or numbers that are engraved or written in it, whether gathered vertically, horizontally or diagonally, forming the same amount or may be added to spell out the same words that are called alpha magic squares. In out dated times it was believed that such squares have magical properties, and perhaps in touch with the orbit and the stars [46]. Figuration of magic square is computationally sensitive, and the indicator of rows and columns are influenced by restrictions, and also affect some diagonals constraints. Moving one indicator affects at least one row, one column value and the diagonal values. Thus, swapping two values will change all possible solutions for all the other indices [47].

2.6

Magic Square from the Mathematical View Point Magical squares remain an interesting phenomenon to be studied, both

mathematically and historically. It is equivalent to a square matrix as a painting full of numbers or letters in certain arrangements. Mathematics is the most interesting subject in computational squares consisting of n2 boxes, called cells or boxes, filled with different integers [48]. This array is called magic square of nxn numbers containing the numbers with consecutive order as 1; 2… n2. The total elements in any row, column, or diagonals should be the same [49]. The following Figure 2.5 is a magic square of order 3 with 9 values consecutively arranged in the magical arrangement. So, suppose Magic Constant (MC) is a fixed number that represents the amount of each row’s values, column’s values and diagonal’s values have to be added up to a vector numbers. 8

1

6

3

5

7

4

9

2

Figure 2.5 Magic Square of Order 3 - 22 -

Chapter Two

Symmetric Cryptography and Magic Square

The pivot element in the magic square represents the center element in the middle square as it is explained in the Figure 2.5. The pivot element at any magic square of odd order with sequential numbers can be calculated in the following formula: n2 +1 p= 2

... (2.6)

⩹⎆ + 1 3⎆ + 1 ⩹⎆ + 1 5⎆ + 1 P= = =, P= = ᴨ 2 2 2 2

and so on 50".

Here, is another formula to determine the pivot element in non-sequential

odd order numbers that might begin with indeterminate integer number, or have a period, another word that has a difference between the numbers is greater than one. As stated below in Equation (3.2). Where N= square order, A=starting number and D=difference number that represents the difference between the successive and the previous numbers. Figure 2.6 shows three examples a, b and c respectively that explain the notation. P = (

2 A + D ( N 2 − 1) ) 2

... ( 2.7)

75

54

69

52

17

42

17

3

13

60

66

72

27

37

47

7

11

15

63

78

57

32

57

22

9

19

5

(a) : N=3, A=54 and D=3

(b) : N=3, A=17 and D=5

(c) : N=3, A=3 and D=2

Figure 2.6 Three of Magic Squares with Different Pivot Element

2 * 54 + 3 (32 − 1) (a) P = ( 2

- 23 -

) = 66

Chapter Two

Symmetric Cryptography and Magic Square

2 *17 + 5 (32 − 1) (b) P = ( 2

) = 37

2 * 3 + 2 (32 −1) (c) P = ( 2

) = 11

The magic constant, magic vector or magic number these terms are synonyms can be computed by the derivative formula of $ %=

&(&' ( ) ) ⎆

. Thus,

3*3 is a regular magic square which should have that each values of row, values of columns and values of diagonals are equal to $ %=

* (* ' ( ) ) ⎆

=

*+ ⎆

= 15, 4*4 to

MC = 34, 5*5 to MC= 65 and 8*8 to MC = 260, and so on [51]. The regular magic square with an order n is a square matrix with the following representation: M = (aij)n x n, where i, j =1, 2, …, n, aij ∈ {1, 2, …, n2} with aij ≠

akl, for all i≠ k or j ≠1 that will give the MC of the magic square which involves the same summation for each row, column and diagonal, as stated in the following Equation (2.8) [52] : n (n2 +1) aij = ∑aji = ∑aii = ∑ajj = ∑ai, n − i +1 = = MC ∑ 2 j=1 i=1 i=1 j=1 i=1 n

n

n

n

n

... (2.8)

There exists a remarkable derivative formula to determine the sum of series magic constant for the magic square as it is explained in the following Equation (2.9). Recall that the sum of the sequential number of magic squares by dividing the previous formula by n, so that it will give the same amount for the rows and columns, as well as diagonals which include the following term:

The Magic Constant (MC) for a normal magic Cube (M Cube) can be calculated as follows: - 24 -

Chapter Two

Symmetric Cryptography and Magic Square

n (n3 + 1) M Cube = 2

... (2.10)

The magic constant (MC) for a Tesseract or quadrilateral that will be referred by (MT) formula can be calculated as follows:

n (n4 +1) MT = 2

... (2.11)

The generalization of magic constant of n-dimensional layout such as hypercube can be calculated as follows:

n (nn + 1) MG = 2

... (2.12)

Another type for computing normal magic star for n order will be as in the following: S = 4n + 2.

… (2.13)

The Magic Sum (MS) that includes the summation to the all numbers (rows, columns and diagonals) in a magic square uses the following formula in Equation (3.9).

n2 (n2 +1) MS = 2

... (2.14)

The MS for 3*3 =45, MS for 4*4 =136, MS for 5*5= 325, MS for 8*8=2080 and so on, another method for calculating MS is by multiplying MC * dimension of the magic square [53].

2.7

Magic Square Properties There are several features and marvellous properties for the normal magic

square’s construction that can be listed as follows [54]:

- 25 -

Chapter Two

Symmetric Cryptography and Magic Square

1. The addition of certain number to each number in the square keeps the square magically. 2. The multiplication of a certain number by each number in the square keeps the square magically. 3. The exchange of two columns or two rows from the centre of square equidistantly keeps the square magically. 4. The magic square of an even order with interchanged quadrants keeps the square magically. 5. The magic square of an odd order with rows interchange and partial quadrants maintains the magic square. 6. The fantastic properties for the square matrix make the square “magic” because it contains a distinct positive integers from 1, 2, …, n 2 and the vector sum for all directions is the same. 7. The rank or order of magic square is determined by the size or dimension of square matrix. 8. The probabilities of construction magic square are increasing speedily according to the size or the order of magic square which involves several forms of magic square except the rotations and reflections cases.

2.8

Magic Square Construction There is no specific method or limited algorithm to build or construct all

types of magic squares. So the method which does for singly even order of magic square will not work for the doubly even or odd order square without some additional work or modification, except for the trial and developing computer software by using programming languages [55]. Various methods for constructing magic squares have been evolved through the ages. When considering these methods, it is useful to categorize magic squares in three classes according to the matrix dimensions by discussing three most common

- 26 -

Chapter Two

Symmetric Cryptography and Magic Square

techniques for the construction of magic squares: those of an odd order; those of a singly even order and the doubly even order [56].

2.8.1 Magic Squares with Odd Order One of the easiest three types of magic squares where the order or degree n is of the form 2n + 1, where n may be any integer number (1, 2, 3, 4 etc.). An example of odd order includes the De la Loubère’s method. The matrix dimension will be 3*3, 5*5, and 7*7 and so on. Odd order magic squares are fairly easily constructed by using any one of these methods such as pyramid or serrated method and other method like de la Loubere's or what is known by the Staircase method [57].

De la Loubère’s Method The third odd order square is the smallest possible square in the odd integer number after number one, which has just one configuration in case of eliminating the reflections and rotations cases. The Mathematician scientist de la Loubère produced a novel method to construct any odd order integer of magic square that can be explained by the following steps [58]: 1. Put number ‘1’ or any starting number in the center cell of the above row, and arrange the other numbers by putting one row to up position as well as one column on the right of its predecessor number. 2. When the next number placement is outside the boundary of both rows and columns, put the number currently next below of the previous number. 3. The numbers arranged diagonally upward to the right, continually to the next column.

- 27 -

Chapter Two

Symmetric Cryptography and Magic Square

4. Whenever the position is reached that is really occupied or reserved, drop down one placement and go on. But if it reached to right corner cell in the upper, scroll down only one row. 5. If the cell that holds a number is filled in, put the new number immediately below the current one cell that is reserved by its predecessor. 6. Whenever the next number position is outside the boundary, put the next number at the antonym end of the row or opposite column, the numbers are ongoing at the last placement to the left of the relater highest row. 1

1

1 3 2

2 1 3

3

4

3

2

4

1

6

8

5

7

3

2

4

4

1

1 3

5

6

5

2

4

2

1

6

8

1

6

5

7

3

5

7

2

4

9

2

Figure 2.7 Magic Square of Odd Order Constructions’ Steps Another way can be adopted for construction of odd order square by using the pyramid method or diagonals. This method is so easy that it consists of the following steps [59]: 1. Build a pyramid or a serrated form on each side of the square. This makes a square fixed on apex. 2. Fill the diagonal sequentially; put the numbers 1 to n2 in the diagonals form as shown in Figure 2.8.

- 28 -

Chapter Two

Symmetric Cryptography and Magic Square

3. Fill the diagonal holes of each integer number is not in the square boundary (that arises through the pyramids you put) by enrolling to the opposite hole, inside to the corresponding empty square.

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7 Figure 2.8 Pyramid Magical Square with order Three The same approach may be extended for any odd order square as stated below for the 5x5 square matrix in Figure 2.9.

1

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5 9 13 17 21

10 14 18 22

15 19 23

20 24

25

3 20 7 24 11

16 8 25 12 4

9 21 13 5 17

22 14 1 18 10

15 2 19 6 23

Figure 2.9 Pyramid Magical Square with order Five

2.8.2 Magic Squares with Doubly Even Order The Doubly even order squares where the order n is of the form 4n, such as (4, 8, 12, 16, 32, etc.). The order of doubly even square can be divided by 2 and 4. The example includes the Albrecht Dürer’s method. The dimension of square matrix will be 4*4, 8*8, and 12*12 and so on. There is no specific or fixed - 29 -

Chapter Two

Symmetric Cryptography and Magic Square

algorithm for constructing the even-order squares. However, there are some methods that can be applied for building magic squares of even orders. So, the creation of ordinary magic square of 4x4 orders in the following method. Starting with the 4x4 square matrix in which the numbers from 1 to 16 are placed across the rows. However, the following method is invented to work for 4x4 normal magic square, and by making some modifications and changes may be worked for a higher dimension of doubly even matrix [60].

Albrecht Dürer’s Method This method belongs to the name of its inventor Albrecht Dürer who has created a public method for constructing doubly even magic square. The invented method including place the numbers from 1, 2 ...,16, sequentially at the upper left corner to the cells proceed straight forward horizontally without crossing by the two diagonals, which remain unoccupied by the numbers like the following square on the left of Figure 2.10 (a). Another method can be constructed by the same square by the opposite notation which comprises overfills the two diagonals with numbers and leaves the other cells empty as stated on the middle of the mentioned Figure 2.10 (b). Then complete the unoccupied cells in the square with opposite cells values as shown in the right Figure 2.10 (c) [61].

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Figure (b) Figure 2.10 Example of Dürer Approach

- 30 -

3 2 16 Figure (c)

Chapter Two

Symmetric Cryptography and Magic Square

The Dürer's method has extra appealing features addressed by the sums of the four corners or sum of the middle four numbers gives the same magic constant. Some people can construct the previous example of even square by only three steps as follows [62]: 1. Enter the numbers in series order. 2. Reveres the entries in the diagonals, see Figure 2.11. 3. Reverse the other values in the opposite direction or on cross format to complete the final shape as it is stated in the third step. 1 5 9 13

2 3 4 6 7 8 10 11 12 14 15 16 First Step

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Figure 2.11 Other Example of Dürer Approach As it was known by doing some reflections and rotations or in other words by interchanging rows, columns or corner groups you can find some other distinct magic square as stated in following Figure 2.12: 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

16 2 3 13 9 7 6 12 5 10 11 8 4 15 14 1

16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1

16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

6 12 9 7 15 1 4 14 3 13 16 2 10 8 5 11

16 2 9 7 5 11 4 14 3 13 6 12 10 8 15 1

16 2 3 13 9 7 6 12 5 11 10 8 4 14 15 1

Figure 2.12 Some Reflections and Rotations for the Dürer approach

- 31 -

Chapter Two

Symmetric Cryptography and Magic Square

2.8.3 Magic Squares with Singly Even Order Squares where n is of the form 2(2n+1) = 4n+2, such as (2, 6, 10, 14, 18, 22, etc.). The order of a singly even square can be divided by 2 but not 4. The example includes the Philippe de la Hire’s method. The matrix dimension will be 6*6, 10*10, and 14*14 and so on [63].

Philippe de la Hire’s Method This method is the name that belongs to a French mathematician Philippe de la Hire. This method creates a singly even magic square. The construction square from this type is so much difficult to be implemented by hand or with pen and paper, since the singly even orders are generally the most sophisticated methods of all squares to build [64]. 1. The first step is to partition the square into four equal sized squares.

An example, for constructing a 6x6 magic square, let the start with four 3x3 squares. Then construct four magic squares in a pattern as indicated in Figure 2.13. Basically, this means first square will build a magic square with the numbers 1 through 9 in the upper left-hand block, the second one of magic square will begin with number 10 and end with number 18, the third one will contain the numbers from 19 to 27 and the fourth one goes from 28 through 36, and so on, until to fill the whole squares or finish the numbers’ distribution. 2. The last square is imposed on the completed square. The computation for the magic constant for all columns sum to the same number 111. But for the rows this is a different case. Since, it has not a subject to the magic constant’s rule. So the first three rows in a square sum to 84 and the last three sum to 138.

- 32 -

Chapter Two

Symmetric Cryptography and Magic Square

3. It should make some changes on some elements in rows, and alter with other elements in the same column to keep the summation of magic constant equal to 111.

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Figure 2.13 de la Hire Technique of Order Six 4. The six numbers that will alter in the square have been highlighted in bold text or highlighted cell. Because there is still some more work to be done on this magic square. The bold numbers in Figure 2.14 state the exchanging process with their corresponding numbers.

- 33 -

Chapter Two

Symmetric Cryptography and Magic Square

26 21 22 17 12 13

19 23 27 10 14 18

24 25 20 15 16 11

35 1 6 26 30 5 7 21 4 9 2 22 8 28 33 17 3 32 34 12 31 36 29 13

19 23 27 10 14 18

24 25 20 15 16 11

35 1 6 26 30 5 7 21 31 9 2 22 8 28 33 17 3 32 34 12 4 36 29 13

19 23 27 10 14 18

24 25 20 15 16 11

35 1 6 26 19 3 32 7 21 23 31 9 2 22 27 8 28 33 17 10 30 5 34 12 14 4 36 29 13 18

24 25 20 15 16 11

35 1 6 26 19 3 32 7 21 23 31 9 2 22 27 8 28 33 17 10 30 5 34 12 14 4 36 29 13 18

24 25 20 15 16 11

35 3 4 8 30 31

1 5 9 28 32 36

6 7 2 33 34 29

Figure 2.14 Example of de la Hire with Order Six

2.9

Types of Magic Square There are many techniques and several types of magic square that can be

categorized according to the features they may own, but the simplest and commonly used method for all values of n, is to follow some formulas that generate regular patterns that will become necessary in discussing other properties of magic squares such as: Normal Magic Square: The most popular nxn square which was discussed earlier, the square with order n which represent an array M=(aij)n x n, where (i, j) =1, 2, …, n, aij ∈ {1, 2, 3 …, n2} with aij ≠ akl, for all i ≠ k or j ≠1 [65].

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Chapter Two

Symmetric Cryptography and Magic Square

The Semi-Magic Squares: is a magical square matrix with the same characteristics except that the two diagonals do not give the same sum of magic constant. Semi-Magic with some reflections and rotations give distinct semi-magic square. It is a an equal square matrix of order n that can be formulated by M=(aij)n x n, where i, j =1, 2, … , n , aij ∈ {1, 2, … , n2} with aij ≠ akl, for all i ≠ k or j ≠1, that includes the following formula

[66].

n (n2 +1) aij = ∑aji = = S ∑ 2 j =1 i=1 n

n

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... ( 2.15 )

Figure 2.15 Semi-Magic Square with Order Three

Nasik Square: The Nasik square is characterized with the feature of that the sum of two certain cells resultant to half of the magic constant. Another property of Nasik square that has all the “broken diagonals” sum to a magic constant as well as stated in Figure 2.16 [67].

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Figure 2.16 The Nasik Square Technique - 35 -

Chapter Two

Symmetric Cryptography and Magic Square

Associated Squares: This kind of square is associated with the skew properties that include the sum of each two certain cells symmetrically opposite the center of square that gives the sum to the half of magic constant which is equal to 17, in addition to have the properties of normal magic square. See Figure 2.17 [68].

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Figure 2.17 The Associated Squares Technique A Pandiagonal Magic Square: This type of magic square is the nearest to the associated square and it has other fascinating properties, which are resultant in the same magic constant with different patterns. The pandiagonal square of order four has several features represented by those of square with order two or the corners of each square with order three that give the same magic constant. The last discriminating feature includes any two adjacent cells vertically or horizontally together with the corresponding two cells gives the same magic constant of the square like 14+11+3+6 and 13+12+4+5. See Figure 2.18 [69]. 1

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Figure 2.18 The Pandiagonal Square - 36 -

Chapter Two

Symmetric Cryptography and Magic Square

Ultramagic Squares: This type of magic square combines both the properties of normal magic square and the pandiagonal properties in addition to the associative (center symmetric) properties. The Ultramagic square is found for order n > 4, the below Figure 2.19 is an Ultramagic square of 5x5 order which includes distinct prime numbers [70]. 113

1151 1229

839

521

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941

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911 101 1013 1091 449 461

311

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1289

Figure 2.19 The Ultramagic Squares Technique Border Magic Square: The Border square is another type of magic square that includes all the usual properties of regular magic square in addition to the border feature. The border feature means if one border (layer) is removed from the square, the square will maintain the magic properties even if it removes the second border and so on according to the square dimension until it reaches to the lowest dimension of third order. See the following Figure 2.20 [71]. 47 4 2 43 38 40 1

11 37 18 19 36 15 39

8 20 26 27 22 30 42

9 6 45 49 17 16 35 46 21 28 32 48 25 23 31 7 29 24 14 12 33 34 13 10 41 44 5 3

Figure 2.20 The Border Square Technique - 37 -

Chapter Two

Symmetric Cryptography and Magic Square

2.10 The Probability of the Magic Square The probability of construction the magical square increases considerably with the increasing in order of magic square as was mentioned earlier. So, there is merely one ordinary magic square of the third order, but by doing some reflections, transpositions and rotations, it will get seven other undifferenti ated cases of magic square as stated in Figure 2.21 [72]. The probability of constructing magic square of fourth order (4x4) is 880 time, and the probability of constructing fifth order is more than 13 million normal magic squares. There are (n2)! Styles to overfill the nxn square matrix with integer’s numbers between 1 and n2, without redundancy, but there are just a few of them match the magic squares features. The construction of magic square acts a great challenge for the intelligent search methods for existing various magic square [73]. 4

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Figure 2.21 Rotations and Reflections of Magic Square - 38 -

Chapter Two

Symmetric Cryptography and Magic Square

2.11 What is the Magic Cube? Magic cube is an extension to the magic square with three dimensions or more, that contains an arrangement set of integer number from 1,2, …n3. The sum of the entries elements in rows direction, columns direction, and all the diagonals direction give the same magic constant for the cube. A magic cube construction of order 3 is shown in Figure 2.22 [74]. 10

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Figure 2.22 Magic Cube of Order Three The magic cube is like the magic square from the point of probability construction that increases dramatically with the order of magic cube but with higher jumps. The magic cube in Figure 2.23 is another direction in the cube construction. The starting element in the diagonal cube begins from one corner of the cube that comprises the upper layer dimensions through the lower left corner. This is the smallest normal magic cube of 3x3x3 dimension with sequential numbers from 1 to 27 that are organized in three layers of nine numbers and the magic constant for this cube is sum to 42. These layers represent the dimensions or faces for the magic cube that arranged magically from all directions [75].

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Chapter Two

Symmetric Cryptography and Magic Square

Figure 2.23 Magic Cube of Three Layers Magic cubes are more than playing games with numbers like the chessboard or Rubik cube, but they substantially depend upon the mathematical rules in their construction. Magic cubes are embedded in several mathematical fields like the number theory, matrices, and combinatory. There exist eleven of distinct flat shapes that can be folded-up to construct a shape of cube as stated in the following Figure 2.24 [76].

1st Cube

2nd Cube

3rd Cube

4th Cube

5th Cube

6th Cube

7th Cube

8th Cube

9th Cube

10th Cube

11th Cube

Figure 2.24 Eleven Distinct Shapes for the Magic Cubes - 40 -

Chapter Three

The Proposed System

CHAPTER THREE THE PROPOSED SYSTEM 3.1

Introduction

The design and analysis of cryptosystems have become very necessary to all aspects of life and more popular topic due to the fact that they have become of vital importance for future requirements. Information security plays an important role in growing technology revolution and a rapid progress in networks communications. Hence, there is an urgent demand to protect numerous amounts of important data from malicious attacks that involve business management, conducted transfer of money, financial marketing, intellectual property, and a lot of other services. In addition, to protect a huge of digital information from tampering, modification and preventing untrusted parties from an unauthorized access. Thus, these challenges made the system designers and analysts think towards the developing of strong and secure cryptographic algorithms that have to face the current difficulties. The proposed system includes three main parts of designing processes those of (Symmetric cipher, Asymmetric cipher and the magic square & magic cube). So, it involves a set of symmetric and asymmetric ciphers designed with different techniques that involve the encryption data with proposed symmetric ciphers and sending the cipher key in a trusted form also by another proposed asymmetric cipher by depending upon the traditional public key combined with magic square and magic cube problem. The proposed symmetric cipher comprises designing and implementing three different block cipher algorithms. Therefore; the proposing of asymmetric cipher involves the development of a new variant method of public key cipher with digital signature depending upon the Diffie-Hellman of public key exchange combined with magic square and magic cube problem. - 41 -

Chapter Three

3.2

The Proposed System

The Proposed Symmetric Cipher The block ciphers algorithms are consider very important branch in the

cryptographic science that are considered the backbone in the designing protocols of shared secret key which is known by the symmetric cryptosystems, so in this part of the current chapter three proposed symmetric algorithms will be discussed in detail.

3.2.1 Design Criteria The proposed algorithms are designed according to the NIST standard criteria and especially according to the wide trail strategy that the AES cipher has been adopted through its design. So there are many standard criteria in the designing process that have been taken into account. The design process of all the proposed ciphers was subject to several factors that comprise the software and hardware properties and other important factors as in the following: 1. High margin of security 2. Design simplicity with repetitive shifting 3. Cost reduction in hardware 4. Resistance against all known attacks 5. Conservativeness 6. Suitability for secure implementations 7. Symmetry in the round transformation

3.2.2 The TIGRIS Cipher This is the first proposed symmetric algorithm that is considered a new revision of the AES cipher with high level of security and an elegant - 42 -

Chapter Three

The Proposed System

construction called TIGRIS cipher. The TIGRIS’S name is inspirited from one of the two famous rivers in Iraq. TIGRIS cipher is built carefully with coherent structure which is based on solid algebraic and well mathematical opinions. The proposed cipher uses the SPN structure and what is known by the Galois Field GF(28). It is an iterated cipher that has a conservative design which is simply performed on both mini hardware and various software platforms. It operates with the text of array with length of 128-bits and with three different ciphering key of lengths (128,192 and 256) bits. The proposed cipher uses basically 4 main stages in the round transformation. These are Xored add operation between the message block and the ciphering key (AddRoundKey), S-box tables (SubByte operation), row shifting (ReversibleShiftRows) and the mixing of each column in the matrix (ShiftingMixcolumn). These operations are repeated with certain numbers of rounds until to complete the encryption process.

A. TIGRIS Rational Design The proposed cipher is focused on the performance and the speed of the algorithm operations in addition to the security aspects with simplicity in design idea, which leads to figure out the algorithm and explains the entire aspects of the design rationale. This cipher provides a good resistance against theoretical and practical cryptanalytic attacks, since it uses a complex operation and extra bit wise operations with two constant vectors that represent a constant word in its key generation process. Thus, through the design process of TIGRIS algorithm many factors have been taken into account such as number of designing gates, the internal of memory requirements, and the performance of the cipher as well as several practical effective attacks. The TIGRIS cipher built by solid and proven algebraic foundation that represents the main obstacle against the expected attacks.

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Chapter Three

The Proposed System

B. TIGRIS Structure The structure of TIGRIS cipher is one of the most prominent changes that discriminate TIGRIS cipher from the AES and its predecessor’s ciphers, in spite of several algorithms built by depending upon the AES ideas and its common structural characteristics (see Figure 3.1). The round transformation of the proposed cipher also consists of the same four stages operations. The TIGRIS cipher has a flexible structure with high non-linearity in SubByte operation and the similarity in ShiftingRows that need no inverse. In addition to the extra shifting operations in diffusion layer (ShiftingMixcolumn) that repetitively recurs for each four rounds with shifting for the equation of mixcolumn four times to generate four different equations through the four repetitive rounds with the same coefficients. The ShiftingMixcolumn stage generate a new linear equations but in different coefficients’ arrangement that needs to different inverse for each round. The coefficient values of the linear equations should be matched with their inverses in forward and backward operations that keep the arrangement properly. The round key generator uses a compact method that generates the round key addition by two directions in column state and in row state with two complex (F) functions, as a result the two generated ciphering keys Xored together with two constant vectors that are represented by golden ratio and the base natural logarithm to produce the generated ciphering key in each round.

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Chapter Three

The Proposed System

Plaintext AddRoundKey (State, Round)

I= 1 SubByte-Stage ReversibleShiftRows-Stage Shifting MixColumn-Stage AddRoundKey (State, Round) SubByte-Stage ReversibleShiftRows-Stage Shifting MixColumn-Stage

SubByte-Stage ReversibleShiftRows-Stage Shifting MixColumn-Stage AddRoundKey (State, Round)

SubByte-Stage ReversibleShiftRows-Stage Shifting MixColumn-Stage AddRoundKey (State, Round)

I=I+1

Yes

I <= Nr

No SubByte-Stage ReversibleShiftRows-Stage AddRoundKey (State, Round)

Cipher text

Figure 3.1 General Structure of TIGRIS Cipher - 45 -

key Schedule

AddRoundKey (State, Round)

Chapter Three

The Proposed System

C. TIGRIS-SubBytes One of the most critical and expensive operations in any cipher design is the substitution, the substitution (S-box) which is used in TIGRIS cipher is so far similar to the one that is used in the AES cipher on one hand of the design method and the mathematical representation on the other. The S-box stage is the basic nonlinearity stage in the cipher that works with byte operations. The proposed S-Box constitutes from two levels. The first level of designing process takes byte by byte as single value in the GF (28) and applied the multiplicative inverse on each one separately. The multiplicative inverse computed modulo the following irreducible polynomial of x8 + x5 + x3 + x2 + 1. The second level represents the resultant hexa-value which is treated as a vector through the calculating of linear affine over GF(2) as it is stated in Equation 3.1, and the resultant number Xored with the constant vector represented by the value (59). The final outcome is the forward S-box which is stated in Table 3.1.

bit'0  0 bit'  0  1  bit'2  0    bit'3  = 1 bit'4  1    bit'5  0 bit'  1  6  bit'7  0

0 1 0 1 1 0 0 bit0  1 0 0 0 1 0 1 1 0 bit1       0 0 0 0 1 0 1 1 bit2      0 0 0 0 1 0 1 bit3  1 ⊕   1 1 0 0 0 0 1 0 bit4       1 1 0 0 0 0 1 bit5  0 1 0 1 1 0 0 0 0 bit6       1 0 1 1 0 0 0 bit7  0

- 46 -

… (3.1)

Chapter Three

The Proposed System

Table 3.1 TIGRIS-Forward S-Box 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 A4 D5 48 8B 86 D9 67 4A B5 FE 1A DA 45 08 D3 70

1 FC 99 28 CD C3 04 EC 56 22 7B A3 16 30 38 72 AF

2 DC 8F 6E C9 B6 77 44 75 43 59 A0 02 80 F1 DD 33

3 F4 9A 93 D7 3E 7C B8 FF 50 92 63 66 1E A1 D2 3B

4 98 A8 65 EF 95 64 46 0C AD C4 19 B9 D4 ED 4C B7

5 94 0F 74 29 2A 12 2E 20 00 0A C8 B4 2D D0 84 13

6 8C 57 BB 42 6B E7 1D 7A BD 14 9C C2 AA 01 09 8D

7 A7 40 A2 DB F8 C7 C6 88 CA AC 34 91 2B E5 E0 79

8 EE FA F6 D1 10 58 55 0D 3C 49 90 6C 81 37 F0 7D

9 5A 7E 03 07 F7 E9 A6 23 0E 27 E2 52 6F 5C 18 85

A BC 32 25 F5 CC 9E 36 82 E3 11 AB 96 E1 0B E6 FD

B EB 69 8A B1 C5 68 47 5E 26 97 F9 3D 3F 6A 9F 39

C B0 4E 5D 5F 7F 9D 83 BF 17 5B 05 61 2C 51 CB E4

D 89 06 78 E8 CF C0 FB BE F2 6D 54 62 CE 1C 1F 8E

E 71 53 D6 3A F3 2F 4F 24 DE 76 41 4D C1 1B B2 35

F 4B 31 B3 A5 A9 DF 73 87 EA D8 BA 15 21 60 9B AE

D. TIGRIS-InvSubByte The inverse of the proposed S-box is designed by implementing the opposite of the affine equation stated below in Equation (3.2) Xored with the corresponding constant vector which represents the value of (BD) subordinated by taking the multiplicative inverse with the same irreducible polynomial. The resultant of these mathematical steps is Table 3.2. So explaining of the incoming or the next proposed algorithms does not need to repeat the same context of explaining for the same reprised stages, just will be limited on the prominent changes for the evolvement in the essence of mathematical notations for the developed models. bit '0  0  bit '  0  1  bit ' 2  1     bit '3  = 1 bit ' 4  1    bit '  5  1 bit '  0  6  bit '7  1

1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 - 47 -

0 1  1  1 1  0 1  0 

bit 0  1   bit  0   1   bit 2  1      bit 3  ⊕ 1  bit 4  1      bit  5 1  bit  0   6   bit 1   7 

… (3.2)

Chapter Three

The Proposed System

Table 3.2 TIGRIS-Backward S-Box 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 85 48 75 C1 17 83 DF F0 C2 A8 A2 0C 5D D5 E7 E8

1 D6 9A CF 1F AE DC BC 0E C8 B7 D3 3B CE 38 CA D2

2 B2 55 81 1A 36 B9 BD E1 7A 93 27 EE B6 E3 A9 8D

3 29 F5 79 F2 82 1E A3 6F 6C 23 A1 2F 41 E0 8A 4E

4 51 96 7E A7 62 AD 54 25 E5 05 00 B5 94 C4 FC 03

5 AC BF 2A FE C0 68 24 72 F9 44 3F 80 4B 10 D7 3A

6 1D B1 8B 6A 64 71 B3 9E 40 BA 69 42 67 2E EA 28

7 39 8C 99 D8 6B 16 60 52 7F 9B 07 F4 57 33 56 49

8 D0 E9 21 D1 20 58 5B 2D 77 04 14 63 A5 9F 3D 47

9 E6 A4 35 FB 98 92 1B F7 0D 11 4F B4 32 50 59 AB

A 95 A0 45 3E 70 09 DB 76 2B 13 C6 AF 87 B0 8F 18

B DA DE C7 F3 0F 9C 46 91 30 EF AA 26 EC 37 0B 6D

C 74 DD CC 88 E4 D9 B8 53 06 A6 97 0A 4A 02 61 01

D 78 66 C5 BB BE 2C 9D F8 F6 5C 84 86 31 E2 D4 FA

E 89 C3 65 43 1C 7B 22 19 FD 5A FF 7D CD 8E 08 90

F 15 ED 5E CB 6E 3C C9 4C 12 EB F1 7C 4D 5F 34 73

E. TIGRIS-ReversibleShiftRows The purpose of the ReversibleShiftRows is to distribute the bytes of each input column to different output columns. This stage is a diffusion operation that works for each single row in the matrix which changes the array placements by other placements without altering the elements values. The first row is not shifted, and the remaining rows proceed as follows 2nd, 3rd and 4th rows 2bytecircular left shift is performed with specialized movement for each, (see Figure 3.2). For decryption process, the same corresponding steps shift the rows that are implemented in the same direction. This is a good method to obtain an optimal diffusion with low cost of hardware and high fast implementation.

Figure 3.2 TIGRIS-ReversibleShiftRows

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Chapter Three

The Proposed System

F. TIGRIS-ShiftingMixcolumns The ShiftingMixcolumn transformation in TIGRIS cipher takes 4×4 bytes of matrix which passes each 4-byte of the entry plaintext to the matrix by applying some linear transformation to the result. This linear transformation is called the "ShiftingMixcolumn". The shiftingmixcolumn in this model adopts the following linear equations that are addressed in Equation (3.3) and Equation (3.4) which are used in the forward and backward stages that make a shifting to one vector of the matrix as in the following: Z(x) = (03) x3 + (04) x2 + (01) x + (07)

... (3.3)

Y(x) = (0b) x3 + (08) x2 + (09) x + (0b)

... (3.4)

Z(x) * y(x) = I ... (3.5)

07 01  04  03

03 07 01 04

04 03 07 01

01 04 ⊗ 03  07

0b 09  08  0b

Where I is identity matrix

0b 0b 09 08

08 0b 0b 09

09 01 08 00 = 0b 00   0b 00

00 01 00 00

00 00 00 00 01 00  00 01

S '0,c  07 S'  01 1 ,c   ⊗  S '2,c  04    S ' 03  3,c 

03 04 01 S0,c  0b   09 07 03 04  S1,c  ⊗  = 08 01 07 03 S2,c      04 01 07 S3,c  0b

0b 08 09 S '0,c    0b 0b 08  S '1,c  = 09 0b 0b S '2,c     08 09 0b S '3,c 

S '0,c  03  S'  07 ,c 1   ⊗  S '2,c  01    S ' 04  3,c 

04 01 07 03 04 01 = 07 03 04  01 07 03

0b 0b 08 S '0,c    09 0b 0b  S '1,c  = 08 09 0b S '2,c     0b 08 09 S '3,c 

S0,c  09 S  08 ,c 1   ⊗  S 2,c  0b    S 0b  3,c 

- 49 -

Chapter Three

The Proposed System

S'0,c  04  S'    1,c  ⊗ 03 S '2,c  07    S ' 3 ,c 01  

01 07 03 04 01 07 =  03 04 01  07 03 04

S0,c  08 09 S    1,c  ⊗ 0b 08 S2,c  0b 0b    S 3 ,c 09 0b  

S'0,c  01  S'  04 1 ,c  ⊗  S'2,c  03    07 S'3,c 

07 03 04 01 07 03 = 04 01 07  03 04 01

S0,c  0b S  0b 1 ,c  ⊗  S2,c  09    08 S3,c 

0b 0b S'0,c    09 0b  S'1,c  =  08 09 S'2,c     0b 08 S'3,c 

08 09 0b S'0,c    0b 08 09  S'1,c  = 0b 0b 08 S'2,c     09 0b 0b S'3,c 

The forward ShiftingMixcolumn depends on the row shifting, for each, the last row of matrix becomes the first row in the next new matrix and so on for each four rounds repeat these operations until sixteen rounds. This step involves shifting left and exclusive-ORing bits with themselves.

G. TIGRIS-Inverse ShiftingMixcolumn The backward ShiftingMixcolumn (Inverse ShiftingMixcolumn) relies on the shifting column, for each last column of matrix becomes the first column in the next new matrix and so on for each four rounds also will repeat these operations, in order to match the same sequence of forward matrices that representative by shift rows.

H. TIGRIS- Ciphering Key Addition This stage involves that the cipher key is performed to the array of plaintext with Xored operation as stated in Figure 3.3. The round key is generated for each round from the ciphering key by the key scheduling technique. The transformation operation which includes XORing cipher key with the state matrix is referred to by ciphering key addition. - 50 -

Chapter Three

S0,0 S1,0 S2,0 S3,0

S0,1 S1,1 S2,1 S3,1

S0,2 S1,2 S2,2 S3,2

S0,3 S1,3 S2,3 S3,3

The Proposed System

K0,0 K1,0 K2,0 K3,0

K0,1 K1,1 K2,1 K3,1

K0,2 K1,2 K2,2 K3,2

K0,3 K1,3 K2,3 K3,3

=

C0,3 C1,3 C2,3 C3,3

C0,0 C1,0 C2,0 C3,0

C0,1 C1,1 C2,1 C3,1

C0,2 C1,2 C2,2 C3,2

Figure 3.3 TIGRIS-Add Round Key

I. TIGRIS-Function (F) The function (F) forms the core for the SubKey generator. It is a complex function that is responsible for the key expansion, the terms F1 and F2 are working with same manner and with the same three steps but the main difference between them is that the (F1) manipulates the state matrix row by row and uses the base natural logarithm as a constant vector whether (F2) manipulates the state matrix column by column and uses a golden ratio also as a constant vector in the third step, which is performed as Xored vector with previous vector. So, it consists of three sub-functions that make entry byte works during the key-dependent of SubByte operation with 8-bit permutation: it takes eight bits of input entry and gives eight bits as a result. The second sub-function applies 8-bit to the right that rotates over the vector and the third sub-function includes two constant vectors (base natural logarithm and golden ratio) that Xored with the previous result of the second sub-function that can be shown in Figure 3.4. The key generator of this cipher operates competently in environments which need advance key alters, and may be applied in devoted tiny hardware or with few gates devices. The use of two constant vectors is to derive the key material and to add a substantial resistance to related-key attacks. Note

Pw = base natural logarithm (b7e15163) Qw = golden ratio (9e377969) >> 8-bit =right rotate over the vector

- 51 -

Chapter Three

The Proposed System

Figure 3.4 TIGRIS-Key Expansion

3.2.3 The EUPHRATES Cipher The second proposed symmetric cipher is called the EUPHRATES cipher. The EUPHRATES’s name also is inspirited from one of the two famous rivers in Iraq. It is a developed revision of the TIGRIS cipher which was proposed in the previous sections. It works also with block size 128-bits and three of different key size of (128,192 and 256) bits, similar to the AES standard but also with sixteen rounds. The EUPHRATES cipher uses the Galois Field GF(28) as the mathematical representation and provides a good insight to the block cipher design with complex structure.

A. The EUPHRATES Structure The EUPHRATES cipher is composed of four main transformations that include SubBytes, ReversibleShiftColumns, ShiftingMixcolumn and a bit by - 52 -

Chapter Three

The Proposed System

bit Xored operation with a ciphering key (Round key Addition). The last round of the EUPHRATES is compounded of the similar stages as the origin round in the AES except it includes the ShiftingMixcolumn stage. The EUPHRATES cipher uses a compact S-box which is considered the source of nonlinearity in the algorithm and the equations that describe it. It will be the main hitch that prevents the system from being easily solved. It is similar to the AES S-box on the one hand of the work, but from the construction view, it uses a different affine transform with different constant vector and different irreducible polynomial. The second step of the EUPHRATES is the linearity which is provided by the shifts of the columns of the state matrix by a certain circular shifting which is reversible and symmetric forward and backward. The third step of the round transformation is the shifting for the coefficient of the linear equation to generate new equations by simple reordering arrangement for the same values; for more details (see Figure 3.5). The core notation for the design process is in the shifting step especially in the backward’s linear equation, since it is not used in the decryption process, because as a replacement it will use the forward linear equation Xored with the complement (difference) values that make the forward linear equation coefficients equal to the backward linear equation coefficients as the alternative in decryption process. This means that the ShiftingMixcolumn stage does not use the inverse linear equation in backward decryption process, but it uses the same forward linear equation in encryption and decryption except it Xored with complement (difference) values in decryption process. This means the proposed cipher can works with two equivalent structures according to the ShiftingMixcolumn stage, either it implemented straight as the traditional ciphers or by use the forward stage with difference as a replacement in backward stage.

- 53 -

Chapter Three

The Proposed System

Plaintext

Ciphertext

Add round key ( )

Add round key ( ) I =1

I =1

ReversibleShiftColumns

ReversibleShiftColumns

ShiftingMixColumn( )

ShiftingMixColumn( )

Add round key

Add round key

SubByte ( )

InvSubByte ( )

ReversibleShiftColumns

ReversibleShiftColumns

ShiftingMixcolumn ( )

ShiftingMixcolumn ( )

Add round key ( )

Add round key ( )

SubByte ( )

InvSubByte ( )

ShiftingMixcolumn ( )

Schedule

InvSubByte ( )

Add round key ( )

key

SubByte ( )

ReversibleShiftColumns

ShiftingMixcolumn ( )

ReversibleShiftColumns

ReversibleShiftColumns

ShiftingMixcolumn ( )

ShiftingMixcolumn ( )

Add round key ( )

Add round key ( ) I=I+1

I=I+1

Yes I<=Nr

I<=Nr No SubByte ( )

SubByte ( )

ReversibleShiftRows ( )

ReversibleShiftRows ( )

Add round key ( )

Add round key ( )

Ciphertext Encryption Process

Complement ShiftingMixColumn( )

Add round key ( ) InvSubByte ( )

No

Complement ShiftingMixColumn( )

ReversibleShiftColumns

SubByte ( )

Yes

Complement ShiftingMixColumn( )

Plaintext Decryption Process

Figure 3.5 The EUPHRATES Structure

- 54 -

Complement ShiftingMixColumn( )

Chapter Three

The Proposed System

Thus; the proposed algorithm can be worked with the linear equation of backward decryption as an equivalent algorithm. This modification is the prominent change and it is considered as the main difference between the TIGRIS and EUPHERATES cipher. The last step of the algorithm is the Xored the Subkey with the state matrix. The round key generator of Subkey is generated from the entry ciphering key through the process of the approach that assists in defeat the cryptanalysis.

B. EUPHRATES-Fundamental Operations The round transformation of the EUPHRATES cipher also consists of four main stages operations that are also similar to AES cipher stages which recur repeatedly with the number of rounds that change according to the key’s length that should have length of multiples four:

C. EUPHRATES-SubBytes The design of this stage is so far similar to the same procedures followed in the previous S-Box of TIGRIS cipher but with different mathematical affine transform equations in forward and backward operations, Xored with other constant vector that involves the value (3D) of hexa-representation. The multiplicative inverse has taken modulo new other irreducible polynomial equation of eight degrees that are represented by resultant of these routines is Table 3.3.

- 55 -

8

X

+ x4 + x3 + x2 + 1 and the

Chapter Three

The Proposed System

bit '0  1 bit'  1  1  bit'2  1    ' bit 3   = 0 bit'4  0    bit'5  0 bit '  1  6  bit'7  1

1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0  1 1 1 1 1 0 0 0 1 1 1 1 1 0  0 0 1 1 1 1 1 0 0 0 1 1 1 1  1 0 0 0 1 1 1

bit0  1 bit  0 1     bit2  1     bit 3   ⊕ 1 bit4  1     bit5  1 bit  0  6   bit 0  7 

… (3.6)

Table 3.3 EUPHRATES-Forward S-Box 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 3D 6D 15 F2 29 87 DA FB 37 EF 60 89 CE 3C 5E 9F

1 FA C7 31 27 D9 8D E7 94 AF 3A 51 0F 49 8F D0 80

2 70 2F 40 5D 95 0A 9E 86 E1 D3 CB 79 FE 11 47 82

3 B4 8E 77 6B 8B FF C6 AA A7 A5 BF 42 61 3B AB 66

4 9B D7 34 39 2D 19 A3 2E 69 D4 A6 A1 EC 52 4E 75

5 71 17 20 67 62 F0 30 DD 0E 7B A0 24 64 84 9D B7

6 F9 ED E4 CC B6 1D 16 E2 C8 54 5C 33 6E 04 58 1B

7 EB A4 6F 1E 22 F3 2A B1 5A 91 E5 4F 0C C3 6C D6

8 C0 A2 48 B0 B9 09 3F 25 35 4B 81 F7 DC 41 1A 43

9 44 FD 53 4A E0 E8 38 1F 7E 72 CF 2B FC 00 57 4C

A B5 68 28 46 B3 59 BE AD 92 BC DB 6A BB D8 E3 E9

B 9C DE 7D DF EF 7C 12 2C 23 BD 98 3E 01 76 45 90

C 5F 26 55 50 D1 F5 C5 C1 F8 07 83 7A 06 5B D2 21

D 97 96 A8 1C 63 93 F6 C9 CD C4 03 13 AE 8C 02 C2

E 56 BA F1 99 14 EE AC 05 B2 78 F4 88 18 B8 D5 E6

F 32 74 9A 01 EA 0B 36 8A 73 A9 0D CA 08 65 85 4D

D. EUPHRATES-InvSubByte The inverse of the proposed S-box is constructed by applying the reverse procedures that involve computing the inverse of affine transformation for each byte and the result Xored with the specific constant vector that is represented by the value (73), and then subordinated by applying the multiplicative inverse of the same irreducible polynomial as it was mentioned in TIGRIS S-Box.

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Chapter Three

The Proposed System

bit '0  1  bit '   0  1  bit '2   0     bit '3  = 1 bit '4   0    bit '  5  1 bit '   0  6  bit '7   0

0

0

1

0

1

0

1

0

0

1

0

1

0 0

1 0

0 1

0 0

1 0

0 1

1

0

0

1

0

0

0

1

0

0

1

0

1

0

1

0

0

1

0

1

0

1

0

0

0 0  1  0 1  0 0  1 

bit 0  1   bit  1  1     bit 2  0      bit 3  ⊕  0  bit 4  1      … (3.7)  bit 5  1  bit  1   6    0  bit 7 

Table 3.4 EUPHRATES-Backward S-Box 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 D9 3F 25 65 22 3C A0 02 F1 FB A5 38 08 E1 49 55

1 CB D2 FC 21 D8 A1 C3 05 A8 97 B4 77 7C 4C 82 2E

2 ED 6B 47 0F B3 29 45 99 F2 8A 18 8E FC EC 76 30

3 AD BD 8B B6 F8 96 4D 8F AC 5D 64 4A D7 92 EA 57

4 D6 4E B5 24 09 2C C5 1F D5 71 17 03 9D 94 26 AE

5 7E 20 78 88 EB 96 DF F4 EF 42 93 0A 6C EE A7 5C

6 CC 66 1C 6F 3A 0E F3 DB 72 1D A4 46 63 F7 FE 6D

7 9C 15 31 80 E2 E9 35 23 50 0D 83 F5 11 14 61 B8

8 CF CE 2A 69 28 E6 1A 9E BE AB 2D DE 86 DA 59 8C

9 58 54 40 34 C1 5A 84 B2 B0 3E 9F 48 7D 41 FA 06

A 52 E8 67 91 39 87 BA BC 7F 2F 73 1E BF 60 4F 01

B 5F F6 B9 D3 98 DC 33 95 43 04 E3 CA A2 AA 07 70

C C7 3D 7B D0 F9 A6 E7 5B DD 0B 6E 9A 36 C8 C4 C9

D AF 56 44 00 FF 32 10 2B 51 E5 7A 9B 8D 75 16 19

E 85 37 74 BB E4 E0 C6 89 13 62 CD 6A C0 1B 5E C2

F B1 79 12 68 B7 0C 27 90 D1 F0 81 A3 A9 3B 4B 53

E. EUPHRATES-ReversibleShiftColumns ReversibleShiftColumns is actually the same of transposition cipher, it only rearranges the positions of the elements without changing their identities and their values, for the encryption process the 1stcolumn is shifted by 2 byte to the down, 2nd column is also 2 byte of circular shift, the first one to the down and the second one to the above, 3rd column involves 2 byte of circular shift the first one to the last position and the second one to the third byte position and the last is the 4th column remains unchanged as shown in Figure 3.6 below.

- 57 -

Chapter Three

The Proposed System

Figure 3.6 Euphrates-ReversibleShiftColumns The decryption process involves the implementation process similar to that for encryption process. So, the corresponding step shifts the implemented rows exactly in the same direction without any change like symmetric shifting. This is a good approach to obtain an optimal diffusion with low cost on hardware and high fast implementation in addition to acquire more resistance athwart the dictionary and saturation attacks.

F. EUPHRATES-ShiftingMixColumns The ShiftingMixcolumn transformation considers the confusion and diffusion process that takes 4 × 4 matrix of bytes, which passes each byte to the linear transformation process and then to the result. The ShiftingMixcolumn provides better diffusion and better protection against differential attacks. The ShiftingMixcolumn in this model uses two elementary of different linear equations with matrices formula. The ShiftingMixcolumn makes a shifting to the one vector of the matrix. The forward ShiftingMixcolumn depends on row shifting, for each last row of matrix becomes the first row in the next new matrix and so on. Each four rounds will repeat these operations periodically until the last round is completed. This step involves shifting left and exclusive-ORing bits with themselves. The following are two linear equations that are used for forward and backward ShiftingMixcolumn as stated in Equation (3.8) and Equation (3.9). Z(x) = (05) x3 + (06) x2 + (06) x + (04) - 58 -

… (3.8)

Chapter Three

The Proposed System

Y(x) = (0a) x3 + (0c) x2 + (09) x + (0e)

… (3.9)

Z( x ) = Y( x ) mod ( x4 + 1 )

04 06  06  05

05 06 06 0e 04 05 06 09 × 06 04 05 0c   06 06 04 0a

… (3.10)

0a 0c 09 1 0e 0a 0c 0 = 09 0e 0a 0   0c 09 0e 0

0 0 0 1 0 0 0 1 0  0 0 1

 p'0,c  04  p'    1,c  ⊗ 06  p'2,c  06     p'3,c  05

05 06 06  p0,c  0a p   04 05 06 03 1,c ⊕ ⊗ p2,c  06 06 04 05     06 06 04  p3,c  05

05 06 03 0a 05 06 = 03 0a 05  06 03 0a

 p'0,c  0e  p'    1,c  ⊗ 09 p'2,c  0c     p'3,c  0a

0a 0c 09 0e 0a 0c 09 0e 0a  0c 09 0e

 p'0,c  05  p'    1,c  ⊗ 04  p'2,c  06     p'3,c  06

06 05 04 06

06 06 05 04

04  p0,c  04 p   06 0a 1,c ⊕ ⊗ p2,c  04 06     05  p3,c  08

08 04 0a 04

04 08 04 0a

0a 04 = 08  04

 p'0,c  09  p'    1,c  ⊗ 0c p'2,c  0a     p'3,c  0e

0e 09 0c 0a

0a 0e 09 0c

0c 0a 0e  09

p'0,c  06  p'    1,c  ⊗ 05 p'2,c 04     p'3,c  06

06 06 05 04

04 06 06 05

05 p0,c  06   04  p1,c  05 ⊕ ⊗ 06 p2,c 0a     06  p3,c  03

03 06 05 0a

0a 03 06 05

05 0a = 03  06

p'0,c  0c  p'    1,c  ⊗ 0a p'2,c 0e     p'3,c  09

09 0c 0a 0e

0e 09 0c 0a

0a 0e 09  0c

p'0,c  06  p'    1,c  ⊗ 06 p'2,c 05     p'3,c  04

04 06 06 05

05 04 06 06

06 p0,c  04   05  p1,c  08 ⊕ ⊗ 04 p2,c 04     06  p3,c  0a

0a 04 08 04

04 0a 04 08

08 04 = 0a  04

p'0,c  0a  p'    1,c  ⊗ 0e p'2,c 09     p'3,c  0c

0c 0a 0e 09

09 0c 0a 0e

0e 09 0c  0a

G. EUPHRATES-Inverse ShiftingMixcolumn The backward ShiftingMixcolumn (Inverse ShiftingMixcolumn) that is used in decryption process which relies basically on shifting column for each - 59 -

Chapter Three

The Proposed System

last column of matrix becomes the first column in the next new matrix and so on for each four rounds, also will repeat these operations in order to superpose the same sequence of forward matrices represented by shift rows. The same of forward ShiftingMixcolumn matrix is used backward but Xored with the complement matrix (values of differences) for the shifting process as inverse.

H. EUPHRATES-Key Schedule The key scheduling or adding round key technique is the same technique adopted in all proposed ciphers that comprise the bitwise XOR operation and does not need to be explained again. The key schedule is the main part of the key expansion that is responsible for accepting of 128-bits input key and producing 128-bits ciphering key which represents the secret key. The key generation routines determines how expanded key generated from the key scheduling which has been modified to be probable to implement the cipher key using restricted memory storage with high key agility on a wide range of processors.

3.2.4 Fast Algorithm Revertible Operation Queen (FAROQ) This is the third proposed symmetric algorithm that so far uses the same steps and the same strategy in the designing process for the previous algorithms with the same significant modifications and amendments. The proposed model is called FAROQ cipher that involves four elementary stages in the round transformation. The proposed cipher constructed with some flexibility according to the block and key lengths that can be assimilate the modifications of round’s numbers and the block size, furthermore with developed reversible structure. It has balanced internal operations that seem to have a perfect possibility to exploit the parallelism property and the real time’s speed implementation.

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The Proposed System

A. FAROQ Structural Features and Characteristics The proposed cipher is a new internal structure and it is considered as a new method to integrate the encrypter and decrypter operations with full utilization potentials. This method is designed with a simple and modern architecture that comprises hardware reduction cost and efficient in executing the ciphering and deciphering. The developed structure is suitable for the embedded gates resources of momentously applications, like smartcard, Personal Digital Assistant (PDA), and cellular phone, etc. This cipher adopts a new innovative pattern in accordance with the dependency of ciphering key with the S-box that creates an extraordinarily powerful system between these security of the algorithm and the structure of the key scheduling that makes the known attacks very hard, so among those primitives, light weight block cipher is necessary and takes a vital role as a building cornerstone for the security methodologies. The FAROQ cipher consists of three elementary layers: confusion layer, diffusion layer and key addition layer. The confusion layer involves the substitution of bytes by means of a nonlinear transformation. The diffusion layer includes that every byte is replaced by a linear combination of the bytes within the same column arrangement. The key addition layer comprises the bytes that are Xored with an n-bit round key to give the cipher text. The FAROQ cipher supports the standard input plaintext block of 128-bits that represent a s a state array of 4*4 dimensions and with three variable keys of 128-bits, 192bits and 256-bits and with three different rounds 10, 12, and 14 respectively. The main structure of the proposed cipher with SPN that has been amended and modified to be equivalent and suitable for the implementation in a wide variety of environments and platforms, as it is stated in Figure 3.7.

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Chapter Three

The Proposed System

Cipher Key

Plaintext Add Round Key ( )-Stage

Words [0,3]

SubByte() /InvSubByte( )- Stage RiversibleShiftColumn( )- Stage RiversibleMixColumn( )- Stage Add Round Key ( )- Stage

Words [4,7]

: SubByte() /Inverse SubByte( )- Stage RiversibleShiftColumn( )- Stage RiversibleMixColumn( )- Stage Add Round Key ( )- Stage

Words [36 ,39]

SubByte() /Inverse SubByte( )- Stage RiversibleShiftColumn( )- Stage Add Round Key ( )- Stage

Words [40,43]

Cipher text Encryption/Decryption

Figure 3.7 FAROQ Structure

B. FAROQ-SubBytes This stage follows the same construction routines which are adopted in the previous algorithms, by selecting another irreducible mathematical polynomial that acts as x8 + x5 + x3 + x + 1 and other affine equation as shown in matrix below of Equation (3.11) Xored with a different constant vector represented by

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Chapter Three

The Proposed System

the value (25). These steps are similar to the S-Box construction of the AES cipher but with different mathematical selected equations.

bit '0  0 bit '  1  1  bit' 2  0    bit ' 3   = 1 bit' 4  0    bit '5  0 bit '  0  6  bit'7  1

1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1  0 1 0 1 0 0 0 1 0 1 0 1 0 0  0 1 0 1 0 1 0 0 0 1 0 1 0 1  0 0 0 1 0 1 0

bit0  1 bit  0  1   bit2  1     bit 3   ⊕ 0 bit4  0     … (3.11) bit5  1 bit  0 6     0 bit7 

Table 3.5 FAROQ-Forward S-Box 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 25 8A FA BF C2 53 60 CA D6 70 16 84 87 E7 DA F8

1 AF C1 42 CD 11 9A 02 2E 88 FE E6 7C 9B D0 D4 65

2 68 DF 5F 20 9E 4E 51 91 3F 7F F2 29 BE DD A0 F4

3 AC 6F C7 79 AD 82 F7 1A ED 2C 38 A8 39 A6 04 40

4 8B FC 50 0E 18 1C A7 B8 F0 26 90 34 1F A4 77 0D

5 52 F1 19 13 14 EE 7B CF FB 30 6B 5D A2 7E C6 48

6 E1 78 00 73 54 D9 03 80 61 49 F6 F5 4C E5 BA 0C

7 1B AE 07 3D 45 A5 CC 09 4D DB 99 57 B9 98 E2 2A

8 7A 10 C9 EA 9F 76 B0 8E B3 6C B1 56 64 BC EB 12

9 C0 2F F3 5C 81 2D C4 86 D7 31 72 35 05 1D D5 55

A 96 63 47 1E 33 17 36 75 BD 46 C8 01 0A 66 58 71

B 8F 89 DC 6E FD 94 B4 67 44 E9 74 85 CB B6 D2 4A

C 4F AA 83 62 B7 C3 06 4B 9D A1 5B 6A 3E 27 FF 32

D A9 43 41 95 E3 7D A3 8D E4 EC 23 5A 97 92 B5 DE

E 3A BB E8 F9 3C AB 21 0B 15 22 6D 24 D1 D8 3B 5E

F 08 9C 28 69 2B 37 CE D3 59 8C E0 B2 C5 0F EF 93

C. FAROQ-InvSubBytes The inverse of S-Box is constructed by reverse steps of construction method with the following affine transform Xored with the constant vector represented by the value (7C). As shown in the Equation (3.12). Then take the multiplicative inverse modulo the irreducible polynomial that has been - 63 -

Chapter Three

The Proposed System

mentioned earlier. The resultant of these steps is the backward S-Box table that can be shown in Table 3.6:

bit '0  0  bit '  1  1  bit ' 2  0    bit '3  = 0 bit ' 4  0    bit '5  1 bit '  0  6  bit '7  1

1 0 1 0 0

0 1 0 1 0

1 0 1 0 1

0 1 0 1 0

0 0 1 0 1

0 0 0 1 0

1 0  0  0 1  0 0 0 1 0 1 0 1 0 0 0 1 0 1  0 1 0 0 0 1 0 

bit 0  0   bit  0  1     bit 2  1      bit 3  ⊕ 1  ...( 3.12 ) bit 4  1      bit 5  1  bit  1   6   bit 0   7 

Table 3.6 FAROQ-Backward S-Box 0 1 2 3 4 5 6 7 8 9 A B C D E F

0 26 18 32 95 F3 24 60 90 76 A4 E2 68 09 D1 AF 84

1 BA 41 6E 99 2D 62 86 FA 49 72 9C A8 11 CE 06 15

2 61 F8 9E FC 21 05 3C A9 53 DD C5 BF 40 EB E7 A2

3 66 35 AD 4A 1D 50 1A 36 2C FF 6D 88 5C 7F 4D 29

4 E3 45 BE B4 8B 46 C8 AB B0 5B D4 6B 69 E1 8D F2

5 C9 8E 00 B9 47 F9 F1 7A BB 3D 57 ED CF E9 D6 B6

6 6C A0 94 6A 9A B8 DA 58 79 0A D3 DB E5 80 A1 A6

7 27 5A DC 5F 2A B7 7B E4 C0 CD 64 4C 23 89 D0 63

8 0F 44 2F A3 F5 EA 02 16 81 D7 B3 74 AA DE 2E F0

9 77 25 B2 C3 96 8F 3F 33 1B A7 0D C7 28 56 9B 3E

A CA 73 F7 0E FB BD BC 08 10 51 1C E6 70 E0 38 20

B 7E 07 4F EE 7C AC A5 65 04 C1 5E 1E CB 97 E8 85

C F6 54 93 4E C6 39 98 B1 9F 1F 03 D8 67 2B 9D 14

D F4 D9 59 37 87 B5 AE 5D 7D 8C 43 8A 31 D2 83 4B

E 34 3A 71 CC 52 FE 3B D5 78 42 17 C2 6F FD 55 91

F DF C4 19 82 0C 22 13 92 0B 48 01 30 75 12 EF EC

D. FAROQ-ReversibleShiftColumns ReversibleShiftColumns is actually permutation process; it only rearranges the positions of the elements without changing their values. The ReversibleShiftColumns include changing the positions of elements with other placements, for the encryption process the 1st column remains unchanged. The 2nd column is shifted by 2 byte to the bottom. The 3rd column is shifted by 2 - 64 -

Chapter Three

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byte, the first one to the bottom and the last one to the top; the last column is the 4th column that is shifted by 2 byte, the first one to the last position and the second one to the third byte position. The decryption process will be implemented similarly to that for encryption process exactly. As shown in Figure 3.8.

Figure 3.8 FAROQ-ReversibleShiftColumns

E. FAROQ-ReversibleMixColumns This step involves shifting left and exclusive-ORing bits with themselves and provides both confusion and diffusion. The ReversibleMixColumns is based on multiplication of the state with a certain matrix, if one of the input elements is changed, all the output elements must be changed, which is defined in the following Equation (3.13) as a self-reversible equation. Z(x) = (05) x3 + (06) x2 + 0(5) x + (07)

... (3.13)

This equation formulated as a matrix representation with a circular shifting, for the each row analogues to

Z0; Z3; Z2; Z1.

So, the later row is gained

by a circular movement for the preceding one by one placement to the left direction in order to constitute the matrix multiplication instead of the linear equation which it handles is easier. This polynomial equation is coprime to (x4 + 1). That is considered invertible matrix which produces the identity array through the multiplication process. So the matrix multiplication of s'(x) can be stated as follows: - 65 -

Chapter Three

 p'0,c  07  p'    1,c  ⊗ 05  p'2,c  06    p ' 3 ,c 05   for 0 ≤ p
The Proposed System

05 06 05 07 05 06 = 05 07 05  06 05 07

 p0,c  07 p    1,c  ⊗ 05  p2,c  06    p 3 ,c 05  

05 06 05  p'0,c    07 05 06  p'1,c  = 05 07 05  p'2,c     06 05 07  p'3,c 

where Nb=4 is no of bytes

The consequence of this operation is a vector of bytes in the column that are multiplied by the same matrix multiplication, forward and backward, since multiplying this matrix by itself gives the identity matrix (self-inverse). Z(x) = Y(x) mod (x4 + 1)… (3.14)

07  05  06   05

05 06 05  07 05   05 07 07 05 06  ×    06 05 05 07 05   06 05 07   05 06

06 05  1 05 06  0 = 07 05  0   05 07  0

0 0 0 1 0 0 0 1 0  0 0 1

F. FAROQ-Ciphering Key Addition The key addition is denoted by ciphering key addition. In this layer 128-bits of state matrix are XORing by corresponding of 128-bit round ciphering key. The process is seen as a vector or column of bit by bit that makes exclusive-or operation between the vectors of bytes with other column’s vector of the ciphering key’s column. The Key generation method is not different from the key expansion technique which has been mentioned earlier in the previous algorithms, so there is no need to explain it again and no need to explain the inverse since the simple XOR operation has its own inverse.

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3.3

The Proposed System

The Proposed Magic Cube The proposed method includes constructing a magic cube by using the

folded magic square technique and it is considered a new step towards the magic cube construction that applies a good insight and provides an easy generalized technique. This method generalizes the design of magic cube with N order regardless of the type of magic square whether odd order or even order squares. The proposed method is fairly easy, since it depends mainly on the magic square construction methods, and all what the designer needs is just how to build six magic square sequentially or with constant difference value between each pair of the numbers in the square matrix, whereby each one of this magic square will represent the surface or dimension for the magic cube configuration. The next step for the designer will be how to arrange each square in the proper order to constitute the regular cube in order to maintain the properties of magic cube, where the sum of rows, columns and the diagonals from all directions are the same. Magic cubes are more than playing games with numbers like the chessboard or Rubik cube, but they substantially depend upon the mathematical rules in their construction. Magic cubes are embedded in several mathematical fields like the number theory, matrices, and combinatorics. There exist eleven of distinct flat shapes that can be folded-up to construct a shape of cube as they were mentioned in the previous chapter. These shapes have been coloured with three distinctive colours (green, yellow and orange) each pair of opposite sides are coloured with same colour to constitute a folded cube with six surfaces each two opposite surfaces with the same colour as it is shown with Figure 2.24 in chapter two. The purpose of these colours is to keep the arrangement of the magic square as we shall explain in the following sections.

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Chapter Three

3.4

The Proposed System

The Construction of Magic Cube The construction of magic cube is the most difficult problem that has

become interesting to researchers in mathematical sciences for a long time. Therefore; the methods that work for an odd order of magic cube will not work for doubly even or singly even methods and vice versa. The proposed method works for all types of magic cube and with any order and it depends basically on the magic square techniques. The proposed method allows to construct several magic cubes with sequential numbers or with constant differences among the series of numbers. The work for six squares (surfaces) will give one magic cube and the work for twelve squares will constitute two magic cubes and so on, so the work with cube should be multiple of six numbers to introduce several cubes regarding the need or to the task requirements. The following example explains the core notation for the magic cube construction. 1. Build six separated magic squares of any order corresponding to the six surfaces of the cube dimensions as shown in Figure 3.9 with order three. 2. Arrange the six surfaces (squares) of the cube by the following way: the first surface should be put opposite to the sixth surface and the second surface opposite to the fifth surface and finally places the third surface opposite to the forth surface with the corresponding colours respectively. 21

0

15

48

27

42

75

54

69

6

12

18

33

39

45

60

66

72

24

3

36

51

30

63

78

57

9 st

1 Square

2

nd

rd

Square

3 Square

102

81

96

129

108

123

156

135

150

87

93

99

114

120

126

141

147

153

90

105

84

117

132

111

144

159

138

4th Square

5th Square

6th Square

Figure 3.9 Cube’s Dimension of Order Three - 68 -

Chapter Three

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After constructing and coloring the six surfaces (dimensions) for the magic cube as it is illustrated in Figure 3.9 above, the magic constant and the magic sum are computed for each square consecutively. The summation for each pair of analogues colored square will give the same result as stated in Figure 3.10 and in Figure 3.11 below successively.

Figure 3.10 The Magic Constant of the Six Squares 36 + 441= 477, 117 + 360 = 477, 198 + 279 = 477

Figure 3.11 The Magic Sum of the Six Squares 108 + 1323= 1431, 351 + 1080 = 1431, 594 + 837= 1431 In the previous Figure 3.9, the magic cube is of the order three with starting value 0 and the constant difference 3 between each pair of numbers, in the 1st square like the following sequence (0, 3, 6, 9…24). The 2nd square starts and ends with the following arrangement (27, 30, 33, 36…51) and so on until to the last or the 6th square of the cube that starts with the number of 135 and ends with 159. Table 3.7 below represents numerical values for some important

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The Proposed System

properties of the magic sum and magic constant for the magic square that symbolizes the dimensions for the magic cube. Table 3.7 Magic Sum (MS) and Magic Constant (MC) Computations MC 441-360 = 81 360 – 279 = 81 279-198 = 81 198-117 = 81 117-36 = 81

MS 1323-1080 =243 1080-837 =243 837-594 =243 594-351 =243 351-108 =24

MC 36 mod 81 = 36 117 mod 81 = 36 198 mod 81 = 36 279 mod 81 = 36 360 mod 81 = 36 441 mod 81 = 36

MS 108 mod 243 = 108 351 mod 243 = 108 594 mod 243 = 108 837 mod 243 = 108 1080 mod 243 = 108 1323 mod 243 = 108

Figure 3.12 below represents the six surfaces for the magic cube with order four of doubly even order squares that contain sequential numbers from 1 to 96 distributed magically. 16

2

3

13

32

18

19

29

48

34

35

45

5

11

10

8

21

27

26

24

37

43

42

40

9

7

6

12

25

23

22

28

41

39

38

44

4

14 15

1

20

30 31

17

36

46 47

33

2

1st Square

nd

rd

Square

3 Square

64

50

51

61

80

66

67

77

96

82

83

93

53

59

58

56

69

75

74

72

85

91

90

88

57

55

54

60

73

71

70

76

89

87

86

92

52

62

63

49

68

78

79

65

84

94

95

81

4th Square

5th Square

6th Square

Figure 3.12 Cube’s Dimensions of Order Four As it has been mentioned earlier, there exist eleven forms of distinct shapes that can be folded-up to construct a cube. The proposed method exploits this feature and fills the squares with different numbers according to the magic squares’ rules to constitute the magic cube. The following Figure 3.13 explains the shapes with six dimensions, surfaces or faces painted with three different colors which construct the magic cube after they are folded from all directions. - 70 -

Chapter Three

The Proposed System

to be continued Shapes

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Chapter Three

The Proposed System

Figure 3.13 Different Shapes of Magic Cube

3.5

The Cross Figure of Folded Cube The cross shape that has been stated in Figure 3.14 acts as one of the

several shapes that construct the magic cube, selected to be traversed and tested with vertical and horizontal track for the main diagonals values and the secondary diagonals values, from all directions which will produce the same constant number of (954). There is no problem or restricted matter in the selection of any shape from the magic cube’s shapes else cross shape, where the cross shape is chosen because it is very easy, simple and clear to figure out the procedures of folding in the magic square surfaces. The cross shape will be partitioned into several basic parts in order to study its properties in detail.

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Figure 3.14 Unfolded Cross Figure of Magic Cube The first part will comprise the main and secondary diagonals for the matrices of magic cube with circular movement in both directions vertically and horizontally as explained in Figure 3.15 and Figure 3.16 consecutively.

Figure 3.15 The Vertical and Horizontal Track of Secondary Diagonals

Figure 3.16 The Vertical and Horizontal Track of Main Diagonals - 73 -

Chapter Three

The Proposed System

The second part of the cube will involve the rows and columns of magic cube with circular movement in both directions vertically and horizontally and the traversing process that comprises the tracks of values with rows and columns of the magic cube from all directions. These operations also will produce the same constant number of (954) as it is explained in Figure 3.15 and Figure 3.16 respectively. The traversing process comprises the tracks values with rows and columns of the magic cube from all directions. These operations will also produce the same constant number of (954) as it is explained in Figures 3.17 and 3.18 respectively.

Figure 3.17 The Vertical Track of Cube Rows Values

Figure 3.18 The Horizontal Track of Cube Columns Values

3.6

The Proposed Public Key (Asymmetric Cipher) The proposed cipher is a new variant of asymmetric cipher (Public Key

cipher) algorithm that is focused on the DLP uses Diffie-Hellman public key exchange and the mathematical features of the magic square. The proposed method exploits the DLP problem only in the key exchange by using the final - 74 -

Chapter Three

The Proposed System

value of agreement procedures, just as the dimension to the magic construction and through which it determines the type of magic square construction if it is (odd, singly-even or doubly-even) magic square, as well as through which determines the starting number and the difference value. From another point of view, it exploits the magic squares problem in encryption/decryption and signing/verifying operations. The developed method extremely speeds in the Encryption/Decryption process as well as in the digital signature that uses a Secure Hash Algorithm SHA-1, since the proposed cipher does not use logarithm and the factorization as the traditional algorithms in ciphering and deciphering operations just in the mutual exchanging, but it depends mainly upon the magic constant and magic sum as alternate to generate the cipher key that is deduced from the result of magic square after it multiplied by fixed value, which has been selected randomly and imposed to keep secret, as it will be explained in the following sections. The proposed cipher has formulated the problem of transfer as a secret key to both communicating parties before secure communication starts, besides; encrypts and signs the message uses a new methodology of public key based on the magic square problem. Since, the encryption and decryption process represent the difficult task, because it needs a secure channel. The proposed cipher is mainly focused on the Diffie-Hellman method and the magic square construction problem which comprises several factors for the predisposing the agreement: The dimension of magic square (type of magic square), the start number, the difference between the numbers and the fixed multiplied value that enlarges the magic sum and magic square. The generalized method for the magic square is the magic cube that involve six dependent magic squares as a six dimensions; each one has a magic constant and magic sum that can be used as a private key for the encryption and decryption procedures. This technique can be generalized for more than two entities up to six parties for each face of magic cube. So; these parties can be increased rapidly with the number of magic - 75 -

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cub construction. The algorithm (1) explains the full notation step by step with manual solution for simple example. Algorithm (1) The Proposed Public Key Cipher 1) Initialization of Algorithm

a. Alice and Bob choose a Finite Field (Fq). b. Alice and Bob choose a primitive element (p).

2) Key Generation a. Alice selects a secret random integer (a), 1< a < p-1, and she compute (Pa mod Fq). b. Bob selects a secret random integer (b), 1< b < p-1, and he compute (Pb mod Fq). c. Makes Pa and Pb public and keep a, b secret as the Diffie-Hellman. d. Alice computes D1=(Pb)a and Bob computes D2=(Pa)b, hence D1=D2 Since it exchanges key which will consider the Dimension of Magic Square (D) and it will illustrate the type of Magic Square if it is (odd magic square, even magic square or odd even magic square). e. Announce the Starting element (S) which is selected randomly and then multiplied by the secret value of D privately to give new start number. f. Announce the Difference number (DF) or added value up to user selection which is selected randomly and then subtracted or added by the secret value of D secretly to give new difference or new addition generated values. 3) Calculation The Secret Key a. Build the magic square according to the dimension (D), Starting element (S) and Difference number (DF). b. Select a random number integer as a fixed multiplied value (FV) for the magic square to regenerate the magic square with other big values. c. Compute the Init value as follow: Init = FV *D d. Compute the Magic Sum (MS) or the summation for all the magic square values and consider the result as a secret key. e. Compute the Magic Constant (MC) for the magic square that uses in the Signature computation.

4) Encryption and Decryption Encryption Process

a. Alice encrypts the message (M) as follow: C =M * MS mod p b. Alice sends (Init , C) to Bob

Decryption Process - 76 -

Chapter Three

The Proposed System

a. Bob receives (Init , C) b. Bob computes the Fixed Value (FV) as follow: FV = Init *D-1 c. Bob generates the magic square and multiplied it by FV d. Bob computes the MS and MC from the new magic square e. Bob decrypts the ciphertext (C) as follows: M = C * MS-1

3.7

The Proposed Signature Algorithm The proposed signature algorithm responsible for the signing and

verifying operations, is one of the simplest algorithms that are similar to the traditional signature algorithm, that depending upon the magic constant vector as a private key that is represented by vector summation in magic square matrix (row, column or diagonal values), in addition to the outcome message digest of Secure Hash Algorithm (SHA1). Algorithm (2) The Proposed Signature Algorithm Signature Algorithm

Singing the Message 1) Alice computes the message digest by using Hash Function SHA-512 h = H(M) mod p. 2) Alice computes the ciphertext as follow: C=MC * h module p. and sends the (C) to the Bob. Verifying the Message 1) Bob receives the ciphertext (C) and compute the message digest As follows: h'=C*MC-1 mod p 2) If h= h' then accepts else rejects the signature

Example (1):

(Diffie-Hellman with Magic Squares) Network Space

Select ( Finite field (Fq =11), Primitive Element=2) Alice announces the result of step (2) =10 publicly on Network Bob announces the result of step (2) =7 publicly on Network Announce the Staring Number (S) S=D * 1 Difference Value DF = D - 1 - 77 -

Chapter Three

The Proposed System

Alice 1) Select a random integer a=5 2) Compute Pa mod Fq 25 mod 11= 10 3) Alice takes from the net value (7) and computes D1=(Pb)a 75 mod 11= 10= Dimension. Alice

Bob

b

1

a, g, p

g, p, A

B = (gb ) mod p

a

A= (g )mod p a

K= (B )mod p

K= (Ab )mod p

2

B

Figure 3.19 Diffie-Hellman Key Exchange Protocol K=Ab module p = (ga module p)b module p = gab module p = (gb module p)a module p = Ba module p = ( D ) Dimension of Magic Square 4) Build the magic square according to the dimension (D) =10*10. 5) Suppose the result starting number is (5) and the difference value (1) to alleviate the computation burden on the reader and makes the track process easy only for this example. 6) Select a random number as a fixed multiplied value (FV) = 6 and multiplied the magic square by 6 to generate new magic square as the stated in the following magic square :

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Chapter Three

45 46 52 33 39 70 71 77 58 64

The Proposed System

63 69 75 76 57 38 44 50 51 32

56 62 68 74 80 31 37 43 49 55

79 60 61 67 73 54 35 36 42 48

72 20 13 6 104 97 78 21 19 12 85 103 59 27 25 93 86 9 65 8 26 24 92 90 66 14 7 30 98 91 47 95 88 81 29 22 53 96 94 87 10 28 34 102 100 18 11 84 40 83 101 99 17 15 41 89 82 105 23 16

Magic Constant = 555

Magic sum = 5550

240 246 282 168 204 390 396 432 318 354

90 96 132 18 45 540 546 582 468 504

348 384 420 426 312 198 234 270 276 162

306 382 378 414 450 156 192 228 264 300

444 330 336 372 408 294 180 186 222 258

402 438 324 360 366 252 288 174 210 216

New Magic Constant = 3030 The result of MS = 30300 The result of MC = 3030

48 84 120 126 12 498 534 570 576 462

6 42 528 114 150 456 492 78 564 600

595 480 486 522 558 144 30 36 72 108

552 588 24 510 516 102 138 474 60 66

New Magic sum = 30300

after multiplied by 6 after multiplied by 6

7) Compute the Magic Sum (MS) and the Magic Constant (MC) for the new Magic square MS=30300 and MC=3030 8) Compute the Init value. Init=FV*D 6*10 mod 11 60 mod 11= 5 9) Encrypt the message with plaintext m (B=2) C=m*MS 2*30300 mod 11 60600 mod 11 = 1 10) Send the Init and the ciphertext (Init=5, C=1) to Bob. Bob 1) Select a random integer b=7 2) Compute Pb mod Fq 27 mod 11= 7 3) Bob takes from the net value (10) and computes D2=(Pa)b - 79 -

Chapter Three

The Proposed System

107 mod 11= 10. D1=D2 represent the (D) Dimension of the magic square for the two parties. 4) Receive the (Init = 5, C = 1) 5) Compute the fixed value. FV =Init * D-1 mod p = 5*10-1 mod 11 = 50 mod 11 = 6. 6) Decrypt the message M=C*MS-1 mod p =1*30300 mod11= 1 * 2 mod 11= 2= B retrieved the original message. Signature Algorithm Sign the message 1) Compute the message digest h=H(message) mod p h=H(75139) mod 11 = 9 2) Compute the ciphertext C= MC * h mod 11 =3030 * 9 mod 11 = 27270 mod 11= 1 3) Send the ciphertext (C= 1) Verify the message 1) Receive the ciphertext (C=1) 2) Compute the message digest h = C * MC-1 mod 11 1*3030-1 mod 11= 1*9 mod 11 = 9 Accept the signature Example (2): (with Generalized Method- Diffie-Hellman & Magic Cube) • Two parties or more can use the proposed system to encrypt and decrypt the cleartext using one of the three proposed symmetric algorithms (TIGRIS, EUPHRATES and FAROQ) cipher and then encrypts and sends the cipher key in secure form by Asymmetric cipher using DiffieHellman algorithm with magic cube technique as well as singing and verifying the intended message.

- 80 -

Chapter Three

The Proposed System

Network Space Diffie-Hellman Key Exchange Starting... The Handshaking & Key Exchange • (Alice and Bob Handshaking) The Handshaking process is just used to ensure that the two parties are existing (live) and available on the network and to prevent the man in the middle attack by depending upon the famous authentication protocols. The string below represents a simple simulation for the designed program for the handshaking session. Alice Sends: 39ZQKRYMOZ7C5G5LZDY7STQV9G3RB7JCS0RX6KBH1US4H0S2OH |5|31PFQKPILDN92TRLC3HY2P99V58KH9QZ98505LOZJXRO68903J Bob Responds: 2DN194JXL3T4I50ICPUJDZF2UH1NRZIZOKGXH23D3ASXL7YKNC • (Alice and Bob Key Exchange) The key exchange is implemented and generated by the Diffie-Hellman algorithm. The designed program treats with several hundred of prime numbers that are generated and selected randomly to constitute the agreement key for the two parties module a specific primitive elements. The string below represents the generated key module the prime number (P =179424673). Alice key: 2010698254200173423720713182091051681507448569721132472179412461 43113212165129 Bob key: - 81 -

Chapter Three

The Proposed System

2010698254200173423720713182091051681507448569721132472179412461 43113212165129 The dimension of the constructed cube in this example will involve only the first byte from the generated key for easy calculation, since the designed program can take unlimited number of order n*n. Thus, the dimension of cube constructing will be equal to 20 Press any key... Cube or square … or print either c or s character… c Enter the dimension … Enter the dimension is generated ... 20 Enter the lower range … 20 Enter the upper range … 2567867 Enter the period … 5 Enter the multiplied value … 3 Magic Cube with Period Press any key to construct the cube dimensions consequently:

- 82 -

Chapter Three

The Proposed System

The First Dimension 425 46 66 365 345 126 146 285 265 206 226 205 185 286 306 125 105 366 386 45

27 404 384 87 107 324 304 167 187 244 224 247 267 164 144 327 347 84 64 407

28 422 421 31 32 418 417 403 49 50 400 399 53 54 383 69 70 380 379 73 74 88 362 361 91 92 358 357 108 342 341 111 112 338 337 323 129 130 320 319 133 134 303 149 150 300 299 153 154 168 282 281 171 172 278 277 188 262 261 191 192 258 257 243 209 210 240 239 213 214 223 229 230 220 219 233 234 248 202 201 251 252 198 197 268 182 181 271 272 178 177 163 289 290 160 159 293 294 143 309 310 140 139 313 314 328 122 121 331 332 118 117 348 102 101 351 352 98 97 83 369 370 80 79 373 374 63 389 390 60 59 393 394 408 42 41 411 412 38 37

35 396 376 95 115 316 296 175 195 236 216 255 275 156 136 335 355 76 56 415

36 395 375 96 116 315 295 176 196 235 215 256 276 155 135 336 356 75 55 416

414 57 77 354 334 137 157 274 254 217 237 194 174 297 317 114 94 377 397 34

Magic Constant =4510

413 39 40 410 409 43 44 58 392 391 61 62 388 387 78 372 371 81 82 368 367 353 99 100 350 349 103 104 333 119 120 330 329 123 124 138 312 311 141 142 308 307 158 292 291 161 162 288 287 273 179 180 270 269 183 184 253 199 200 250 249 203 204 218 232 231 221 222 228 227 238 212 211 241 242 208 207 193 259 260 190 189 263 264 173 279 280 170 169 283 284 298 152 151 301 302 148 147 318 132 131 321 322 128 127 113 339 340 110 109 343 344 93 359 360 90 89 363 364 378 72 71 381 382 68 67 398 52 51 401 402 48 47 33 419 420 30 29 423 424

406 65 85 346 326 145 165 266 246 225 245 186 166 305 325 106 86 385 405 26

Magic Sum =90200

The Second Dimension 430 51 71 370 350 131 151 290 270 211 231 210 190 291 311 130 110 371 391 50

32 409 389 92 112 329 309 172 192 249 229 252 272 169 149 332 352 89 69 412

33 408 388 93 113 328 308 173 193 248 228 253 273 168 148 333 353 88 68 413

427 54 74 367 347 134 154 287 267 214 234 207 187 294 314 127 107 374 394 47

426 55 75 366 346 135 155 286 266 215 235 206 186 295 315 126 106 375 395 46

36 405 385 96 116 325 305 176 196 245 225 256 276 165 145 336 356 85 65 416

37 404 384 97 117 324 304 177 197 244 224 257 277 164 144 337 357 84 64 417

423 58 78 363 343 138 158 283 263 218 238 203 183 298 318 123 103 378 398 43

422 59 79 362 342 139 159 282 262 219 239 202 182 299 319 122 102 379 399 42

40 401 381 100 120 321 301 180 200 241 221 260 280 161 141 340 360 81 61 420

41 400 380 101 121 320 300 181 201 240 220 261 281 160 140 341 361 80 60 421

419 62 82 359 339 142 162 279 259 222 242 199 179 302 322 119 99 382 402 39

Magic Constant =4610

418 44 63 397 83 377 358 104 338 124 143 317 163 297 278 184 258 204 223 237 243 217 198 264 178 284 303 157 323 137 118 344 98 364 383 77 403 57 38 424

45 396 376 105 125 316 296 185 205 236 216 265 285 156 136 345 365 76 56 425

415 66 86 355 335 146 166 275 255 226 246 195 175 306 326 115 95 386 406 35

414 67 87 354 334 147 167 274 254 227 247 194 174 307 327 114 94 387 407 34

Magic Sum =92200 - 83 -

48 393 373 108 128 313 293 188 208 233 213 268 288 153 133 348 368 73 53 428

49 392 372 109 129 312 292 189 209 232 212 269 289 152 132 349 369 72 52 429

411 70 90 351 331 150 170 271 251 230 250 191 171 310 330 111 91 390 410 31

Chapter Three

The Proposed System

The Third Dimension 435 56 76 375 355 136 156 295 275 216 236 215 195 296 316 135 115 376 396 55

37 414 394 97 117 334 314 177 197 254 234 257 277 174 154 337 357 94 74 417

38 413 393 98 118 333 313 178 198 253 233 258 278 173 153 338 358 93 73 418

432 59 79 372 352 139 159 292 272 219 239 212 192 299 319 132 112 379 399 52

431 60 80 371 351 140 160 291 271 220 240 211 191 300 320 131 111 380 400 51

41 410 390 101 121 330 310 181 201 250 230 261 281 170 150 341 361 90 70 421

42 409 389 102 122 329 309 182 202 249 229 262 282 169 149 342 362 89 69 422

428 63 83 368 348 143 163 288 268 223 243 208 188 303 323 128 108 383 403 48

427 64 84 367 347 144 164 287 267 224 244 207 187 304 324 127 107 384 404 47

45 406 386 105 125 326 306 185 205 246 226 265 285 166 146 345 365 86 66 425

46 405 385 106 126 325 305 186 206 245 225 266 286 165 145 346 366 85 65 426

Magic Constant =4710

424 423 67 68 87 88 364 363 344 343 147 148 167 168 284 283 264 263 227 228 247 248 204 203 184 183 307 308 327 328 124 123 104 103 387 388 407 408 44 43

49 402 382 109 129 322 302 189 209 242 222 269 289 162 142 349 369 82 62 429

50 401 381 110 130 321 301 190 210 241 221 270 290 161 141 350 370 81 61 430

420 71 91 360 340 151 171 280 260 231 251 200 180 311 331 120 100 391 411 40

419 72 92 359 339 152 172 279 259 232 252 199 179 312 332 119 99 392 412 39

53 398 378 113 133 318 298 193 213 238 218 273 293 158 138 353 373 78 58 433

54 397 377 114 134 317 297 194 214 237 217 274 294 157 137 354 374 77 57 434

424 77 97 364 344 157 177 284 264 237 257 204 184 317 337 124 104 397 417 44

58 403 383 118 138 323 303 198 218 243 223 278 298 163 143 358 378 83 63 438

59 402 382 119 139 322 302 199 219 242 222 279 299 162 142 359 379 82 62 439

416 75 95 356 336 155 175 276 256 235 255 196 176 315 335 116 96 395 415 36

Magic Sum =94200

The Fourth Dimension 440 61 81 380 360 141 161 300 280 221 241 220 200 301 321 140 120 381 401 60

42 419 399 102 122 339 319 182 202 259 239 262 282 179 159 342 362 99 79 422

43 418 398 103 123 338 318 183 203 258 238 263 283 178 158 343 363 98 78 423

437 64 84 377 357 144 164 297 277 224 244 217 197 304 324 137 117 384 404 57

436 65 85 376 356 145 165 296 276 225 245 216 196 305 325 136 116 385 405 56

46 415 395 106 126 335 315 186 206 255 235 266 286 175 155 346 366 95 75 426

47 414 394 107 127 334 314 187 207 254 234 267 287 174 154 347 367 94 74 427

433 68 88 373 353 148 168 293 273 228 248 213 193 308 328 133 113 388 408 53

432 69 89 372 352 149 169 292 272 229 249 212 192 309 329 132 112 389 409 52

50 411 391 110 130 331 311 190 210 251 231 270 290 171 151 350 370 91 71 430

51 429 410 72 390 92 111 369 131 349 330 152 310 172 191 289 211 269 250 232 230 252 271 209 291 189 170 312 150 332 351 129 371 109 90 392 70 412 431 49

Magic Constant =4810

428 73 93 368 348 153 173 288 268 233 253 208 188 313 333 128 108 393 413 48

54 407 387 114 134 327 307 194 214 247 227 274 294 167 147 354 374 87 67 434

55 406 386 115 135 326 306 195 215 246 226 275 295 166 146 355 375 86 66 435

425 76 96 365 345 156 176 285 265 236 256 205 185 316 336 125 105 396 416 45

Magic Sum =96200 - 84 -

421 80 100 361 341 160 180 281 261 240 260 201 181 320 340 121 101 400 420 41

Chapter Three

The Proposed System

The Fifth Dimension 445 66 86 385 365 146 166 305 285 226 246 225 205 306 326 145 125 386 406 65

47 424 404 107 127 344 324 187 207 264 244 267 287 184 164 347 367 104 84 427

48 423 403 108 128 343 323 188 208 263 243 268 288 183 163 348 368 103 83 428

442 69 89 382 362 149 169 302 282 229 249 222 202 309 329 142 122 389 409 62

441 70 90 381 361 150 170 301 281 230 250 221 201 310 330 141 121 390 410 61

51 420 400 111 131 340 320 191 211 260 240 271 291 180 160 351 371 100 80 431

52 419 399 112 132 339 319 192 212 259 239 272 292 179 159 352 372 99 79 432

438 73 93 378 358 153 173 298 278 233 253 218 198 313 333 138 118 393 413 58

437 74 94 377 357 154 174 297 277 234 254 217 197 314 334 137 117 394 414 57

55 416 396 115 135 336 316 195 215 256 236 275 295 176 156 355 375 96 76 435

56 415 395 116 136 335 315 196 216 255 235 276 296 175 155 356 376 95 75 436

434 77 97 374 354 157 177 294 274 237 257 214 194 317 337 134 114 397 417 54

Magic Constant =4910

433 78 98 373 353 158 178 293 273 238 258 213 193 318 338 133 113 398 418 53

59 412 392 119 139 332 312 199 219 252 232 279 299 172 152 359 379 92 72 439

60 411 391 120 140 331 311 200 220 251 231 280 300 171 151 360 380 91 71 440

430 81 101 370 350 161 181 290 270 241 261 210 190 321 341 130 110 401 421 50

429 82 102 369 349 162 182 289 269 242 262 209 189 322 342 129 109 402 422 49

63 408 388 123 143 328 308 203 223 248 228 283 303 168 148 363 383 88 68 443

64 407 387 124 144 327 307 204 224 247 227 284 304 167 147 364 384 87 67 444

426 85 105 366 346 165 185 286 266 245 265 206 186 325 345 126 106 405 425 46

Magic Sum =98200

The Sixth Dimension 450 71 91 390 370 151 171 310 290 231 251 230 210 311 331 150 130 391 411 70

52 429 409 112 132 349 329 192 212 269 249 272 292 189 169 352 372 109 89 432

53 428 408 113 133 348 328 193 213 268 248 273 293 188 168 353 373 108 88 433

447 74 94 387 367 154 174 307 287 234 254 227 207 314 334 147 127 394 414 67

446 75 95 386 366 155 175 306 286 235 255 226 206 315 335 146 126 395 415 66

56 425 405 116 136 345 325 196 216 265 245 276 296 185 165 356 376 105 85 436

57 424 404 117 137 344 324 197 217 264 244 277 297 184 164 357 377 104 84 437

443 78 98 383 363 158 178 303 283 238 258 223 203 318 338 143 123 398 418 63

442 79 99 382 362 159 179 302 282 239 259 222 202 319 339 142 122 399 419 62

60 421 401 120 140 341 321 200 220 261 241 280 300 181 161 360 380 101 81 440

61 420 400 121 141 340 320 201 221 260 240 281 301 180 160 361 381 100 80 441

Magic Constant =5010

439 438 64 82 83 417 102 103 397 379 378 124 359 358 144 162 163 337 182 183 317 299 298 204 279 278 224 242 243 257 262 263 237 219 218 284 199 198 304 322 323 177 342 343 157 139 138 364 119 118 384 402 403 97 422 423 77 59 58 444

65 416 396 125 145 336 316 205 225 256 236 285 305 176 156 365 385 96 76 445

435 434 86 87 106 107 375 374 355 354 166 167 186 187 295 294 275 274 246 247 266 267 215 214 195 194 326 327 346 347 135 134 115 114 406 407 426 427 55 54

Magic Sum =100200 - 85 -

68 69 413 412 393 392 128 129 148 149 333 332 313 312 208 209 228 229 253 252 233 232 288 289 308 309 173 172 153 152 368 369 388 389 93 92 73 72 448 449

31 90 110 371 351 170 190 291 271 250 270 211 191 330 350 131 111 410 430 51

Chapter Three

The Proposed System

Magic Constant of First Dimension is = 4510

Magic Constant of Sixth dimension = 5010

The Summation of Magic Constant is = 9520 Magic Sum of First Dimension is = 90200

Magic Sum of Sixth dimension = 100200

The Summation of Magic Sum is= (190400) Magic Constant of Second Dimension is = 4610

Magic Constant of Fifth dimension = 4910

The Summation of Magic Constant is= 9520 Magic Sum of Second Dimension is = 92200

Magic Sum of Fifth dimension = 98200

The Summation of Magic Sum is= (190400) Magic Constant of Third Dimension is = 4710

Magic Constant of Fourth dimension = 4810

The Summation of Magic Constant is= 9520 Magic Sum of Third Dimension is =94200

Magic Sum of Fourth dimension = 96200

The Summation of Magic Sum is= (190400)

Select the face of Cube Number 5 The Fifth Dimension: Magic Constant =4910

Magic Sum =98200

• The Encryption Process … Enter the Plaintext Message: M= “11223344” Ciphertext (C) = M * (K=MS) mod P = 11223344 * 98200 mod 179424673 = 106039234 The Encrypted message is: (C= 106039234) The Decryption Process … Plaintext = C * (K-1 =MS) mod P = 106039234 *(98200-1 = (72835089)) mod 179424673 =11223344 The Decrypted message is: 11223344 • The Signature Algorithm Message Digest 103 141 180 190

124 197 90 174 229 85 93 216 143 242 39 121 146 47 254 232 208 5 94 232 134 105 155 187 183 127 242 44 193 22 102 97 2 7 74 167 125 209 22 111 242 38 108 195 60 195 51 55 117 83 59 228 73 157 211 62 235 186 171 186 173 - 86 -

Chapter Three

The Proposed System

Message abstract for the Message Digest includes also the first byte for easy calculation in tracking and evidence. The first byte for the message digest: 103 The Signature Process … Signing = Message digest * Magic Constant mod P Signing = 103 * 4910 mod 179424673 = 505730 The Signature is: 505730 Verify = Sign * Inverse Magic Constant mod P Verify = 505730 * 4910-1 mod 179424673 = 103 The Verifying is: 103 Second Method: the second method of this model considers the MS and MC as two keys K1 & K2 respectively (MS=K1, MC=K2),

Magic Sum = 100200 = K1,

Magic Constant = 5010 = K2.

• The Encryption Process … C = K1 * M + K2 mod P. = 98200 * 11223344 + 4910 mod 179424673 = 106044144. • The Decryption Process … M = K1-1 (C-K2) mod P. = 72835089 (106044144 - 4910) mod 179424673 = 11223344 = M The Signature Algorithm … • Sing Process

C = K1 * M + K2 mod P.

= 98200 * 103 + 4910 mod 179424673 = 10119510 • Verifying Process

M = K1-1 (C - K2) mod P.

72835089 (10119510 - 4910) mod 179424673 = (103=M) - 87 -

Chapter Four

Analysis and Results with Implementation

CHAPTER FOUR ANALYSIS AND RESULTS WITH IMPLEMENTATION 4.1

Introduction The proposed ciphers algorithms do not base their security or part of it

on obscure or ambiguous interactions between arithmetic operations and algebraic notations. The confirming was based on the illustration of essential comprehensible and concentrating on finding of appropriate solutions for many sophisticated cryptographic problems related to the malicious attacks and security aspects. Therefore, the designed philosophy of all ciphers models depends on offering a high level of security and considerable efficiency. The designed strategy of the proposed ciphers with high margin security became a necessary demand, therefore; the mathematical internal operations are built by a provable security against differential and linear cryptanalysis. This part of this study will comprise the analytical sides and the randomness tests for the proposed cryptosystems that include the symmetric and asymmetric models.

4.2

The Analysis for the Symmetric Cryptographic Models

The proposed ciphers adopted the increase of the key complexity and the linearity according to the wide trail strategy as well as make the block encryption of ciphertext pass through several refereed stages in order to improve immunity and enhance the resistance against malignant actions. The designed ciphers are built by the diversities of techniques that include various modern internal

operations.

The

reversible

operation

of

ShiftingMixcolumn,

ReversibleShiftRows and ReversibleShiftColumn provide better diffusion and confusion as well as protection against differential attacks, truncated differential - 88 -

Chapter Four

Analysis and Results with Implementation

attacks and saturation attacks. The basic purpose behind the idea of reversible operation is to design an excellent algorithm that is very well suited for restricted space environments and to be implemented very efficiently on cheap smartcard processors and to use a very small amount of RAM and a reasonable amount of ROM, in addition, to make cipher execution and key setup/key changes are very fast. Table 4.1 explains some analysis and comparisons for the proposed symmetric ciphers and the AES algorithm. Table 4.1 Analysis and Comparison Speed for the Proposed Models Algorithms

AES Cipher

NO. of Rounds

(10, 12 and 14) Rounds

Reversible Operations

No Reversible Operations

SubByte

FAROQ Cipher (10, 12 and 14) Rounds

Very- High Moderate Level

High

Two Directions Very-High Moderate Level Moderate

Moderate

High

Unbalancing

Unbalancing

Unbalancing

Balancing

Key-Expansion Security Performance

High High Level

Speed Balancing Time in Encryption/Decryption

MixColumns

(16 and multiples of 4) Rounds with Reversible Operations IrreversibleStage ReversibleStage With-Shifting

EUPHRATES Cipher (16 and multiples of 4) Rounds with Reversible Operations

with Reversible Operations IrreversibleStage ReversibleStage ReversibleStage Two Directions High Very- High

IrreversibleStage IrreversibleStage IrreversibleStage One Direction

ShiftRows

TIGRIS Cipher

IrreversibleStage ReversibleStage With-Shifting Two Directions

There are several important criteria and numerous algebraic properties that have been taken in to consideration through the design steps for the proposed symmetric ciphers which are compared to the standard AES cipher. The algebraic analysis involve the round transformation stages for each proposed model according to several features as it stated in Table 4.2 - 89 -

Chapter Four

Analysis and Results with Implementation

Table 4.2 Algebraic Comparison between the AES and the Proposed Ciphers Algebraic Properties Correlation Immunity Algebraic Degree Algebraic Complexity Bijection Strict Avalanche Criteria (SAC) NonLinearity Difference Propagation Power Mapping Diffusion Invertibility Offset Positions DimensionDegree Correlation Matrices Linearity 8-Bit Process Complexity Constant Vector Symmetry Elimination Complexity S-Box Dependency

TIGRIS Cipher

EUPHRATES FAROQ Cipher Cipher S-Box Properties

AES Cipher

Zero

Zero

Zero

Zero

Seven

Seven

Seven

Seven

8*256!

8*256!

8*256!

8*256!

Yes

Yes

Yes

Yes

½

1/2

1/2

1/2

GF(28)

GF(28)

GF(28)

GF(28)

Small as Possible By-Affine

Small as Possible

Small as Possible

Small as Possible

By-Affine

By-Affine

By-Affine

ShiftRows/ShiftColumns Properties Optimal Optimal Optimal Straightforward StraightforwardStraightforward -Order Order -Order (i,j)= (J + Ri) (i,j)= (J + Ci) mod (i,j)= (J + Ci) mod mod Nb Nb Nb 4-Bytes CrossCorrelation Over GF(2) Sum of Power (02) 4*4!*4! Pw/Qw

MixColumns Properties 4-Bytes 4-Bytes

Optimal With ReverseOrder (i,j)= (J + Ci) mod Nb 4-Bytes

Cross-Correlation

Auto -Correlation

Cross-Correlation

Over GF(2) Sum of Power (02)

Over GF(2) Sum of Power (02)

Over GF(2) Sum of Power (02)

4*4!*4! 4*4! Key Expansion Properties Pw/Qw Pw/Qw

Non-weak key

Non-weak key

Non-weak key

4!*4!*16! Yes

4!*4!*16! Yes

4!*4!*10! Yes

- 90 -

4*4! Rcon-Table Non-weak key 4!*10! Yes

Chapter Four

4.3

Analysis and Results with Implementation

TIGRIS-Security Analysis

The TIGRIS cipher submits a full diffusion property to prevent the mounting of shortcut attack, since it depends on iterated ShiftingMixColumn stage for each four rounds that reduce the space through which the plaintext can be exploited. The developed ReversibleShiftRows stage provides another layer of linearity that means the absence of implementation attacks which exploit its internal structure. The TIGRIS algorithm proposed with new design of strong and efficient key-schedule algorithm to defeat different types of attacks. The compact key expansion method increases the probability of keys that leads to increase the difficulty in brute force attack and dictionary attack. TIGRIS structure increased the difficulty of Differential Power Analysis (DPA) attack by the needs to the extra time of computations to the mathematical operations and to the internal state of the processing, since it has innovation key setup and an effective mechanism to overcome the effective possible threats. The TIGRIS cipher can be performed by the encryption/decryption process in parallel and balancing way. Table 4.3 and Table 4.3 express the executive time comparison in Milliseconds MS with different number of rounds. Table 4.3 Comparison Speed between TIGRIS Cipher and the AES (1) Algorithms

No. of Rounds

Key Size

Rijndael Cipher

10-Rounds

128-bit

TIGRIS Cipher

12-Rounds

128-bit

- 91 -

Number of Bytes

Time of Encryption

1000 10000 100000 1000 10000 100000

0.68 0.76 1.19 0. 68 0.79 1.34

Chapter Four

Analysis and Results with Implementation

Table 4.4 Comparison Speed between TIGRIS Cipher and the AES (2) Algorithms

No. of Rounds

Key Size

Rijndael Cipher

10-Rounds

128-bit

TIGRIS Cipher

12-Rounds

128-bit

4.4

Number of Bytes

Time of Encryption

1000 10000 100000 1000 10000 100000

0.68 0.76 1.19 0.90 0.98 2.93

EUPHRATES-Security Analysis

The EUPHRATES cipher is also designed to defeat the differential and linear cryptanalysis attacks that require more exploiting time than the AES cipher until to flawing the proposed algorithm. The main envisioned target algorithm can be worked efficiently on a constrained device or similar. After studying the efficiency of the key avalanche effect and completeness, the conclusion was that the complexity of exhaustive search if one tries to bypass one round will be difficult to analysis. The proposed cipher focuses on increasing the algebraic complexity for S-box and its inverse to defeat algebraic attacks and interpolation attack. This cipher is slower than the Rijndael cipher especially in decryption process compared with encryption process, and the implementation requires more time for the key scheduling whereby it work with more number of rounds than the AES. The internal algebraic operations constitute a form of security margin against the known attacks by reducing the interrelation or correlation among the linear representation of input and output data to face the linear attack, and at the same moment reducing the probability of variance and difference propagation in order to defeat differential attack. The number of rounds has been increased to sixteen rounds in order to defeat the square attack and related key attack that is possible for reducing rounds. It uses wise and precise techniques in generating Subkeys with the ability to change - 92 -

Chapter Four

Analysis and Results with Implementation

keys with a minimum number of resources. Simple assessment and comparison for the time implementation in Milliseconds (Ms) between the AES and the EUPHRATES cipher according to 16-rounds and 12-rounds can be shown in Table 4.5 and Table 4.6 respectively in order to ensure the justice in time of implementations. Table 4.5 Comparison Speed between EUPHRATES Cipher and the AES (1) Algorithms

No. of Rounds

Key Size

Rijndael Cipher

10-Rounds

128-bit

EUPHRATES 12-Rounds Cipher

128-bit

Number of Bytes 1000 10000 100000 1000 10000 100000

Time of Encryption 0.68 0.76 1.19 0.68 0.82 1.38

Table 4.6 Comparison Speed between EUPHRATES Cipher and the AES (2) Algorithms

No. of Rounds

Rijndael Cipher

10-Rounds

128-bit

EUPHRATES 16-Rounds Cipher

128-bit

4.5

Key Size

Number of Bytes 1000 10000 100000 1000 10000 100000

Time of Encryption 0.68 0.76 1.19 0.93 0.101 3.30

FAROQ-Security Analysis

FAROQ cipher is a new symmetric cipher which supports key expansion unit with a simple design and high complexity that works competently in implementations with low reduced resources that have the ability to change keys quickly with a minimum number of resources (high key agility). The proposed algorithms involve more efficient combination of ReversibleShiftrows and - 93 -

Chapter Four

Analysis and Results with Implementation

ReversibleMixColumn operations that are used in the same manner in forward and backward in order to use less resources than the separated block structures’ stages. This cipher ensures high performance with the applications that require frequent change of the key scheduling with high parallelism of computations to ensure fast encryption speed. The timing and power analysing attacks and several other effective attacks have been taken into account. The designed structure takes reasonably less hardware resources as it is known and can be applied with sufficient throughput, in order to alleviate the hardware requirements and to produce high efficiency. The

S-Box is designed with constraining the linear

interrelation among the inputs and the outputs sequence of bits in order to abort the linear cryptanalysis; this design is reasonable and provides an acceptable degree of confusion. FAROQ cipher also is characterized with a high degree of balancing that makes the power analysis attack impossible for exploiting the difference in internal implementation of instructions. A simple comparison in encryption speed in Ms between the proposed cipher and the standard AES cipher can be shown in Table 4.7. Table 4.7 Comparison Speed between FAROQ Cipher and the AES Algorithms

No. of Rounds

Key Size

Rijndael Cipher

10-Rounds

128-bit

FAROQ Cipher

10-Rounds

128-bit

4.6

Number of Bytes 1000 10000 100000 1000 10000 100000

Time of Encryption 0.68 0.76 1.19 0.49 0.69 1.10

The Randomness Tests

The randomness tests and the statistical analysis are very important for these proposed ciphers neither to ensure the level of security nor to reveals the performance of efficiency but to determine if there is any deviation or biases in the correlation of input/output bits and to ensure the random mapping for the - 94 -

Chapter Four

Analysis and Results with Implementation

cipher text. The proposed symmetric ciphers give an accepted and reasonable implications according to the NIST randomness tests. Therefore a list of acquired empirical results can be stated below in Table 4.8, Table 4.9 and Table 4.10 respectively. Table 4.8 The Statistical Tests for the TIGRIS Cipher Statistical Tests 1-Block Frequency Test 2-Mono-bit Frequency Test 3-Overlapping Templates Test 4-Non- Overlapping Templates Test 5-Serial Test Serial (1) Serial (2) 6-Aproximate Entropy Test 7-Linear Complexity Test 8-Maurer’s “Universal Statistical Test” 9-Cumulative Sums Test Cumulative Sums (Forward) Cumulative Sums (Reverse) 10-Runs Test 11-Longest Run of Ones Test 12-Binary Matrix Rank Test 13-Spectral DFT Test 14-Random Excursions Test Random Excursions (1) Random Excursions (2) Random Excursions (3) Random Excursions (4) Random Excursions (5) Random Excursions (6) Random Excursions (7) Random Excursions (8) 15- Random Excursions Variant Test Random Excursions Variant (1) Random Excursions Variant (2) Random Excursions Variant (3) Random Excursions Variant (4) Random Excursions Variant (5) - 95 -

Input Size (n) 2048 128 1000000 1000000 1000000 128 1000000 1000000 128 128 128 38912 1024 1000000

1000000

P-Value >=0.01 0.586731 0.100357 0.097526 0.867537 0.752853 0.643827 0.455273 0.785291 0.592013 0.182732 0.183894 0.492101 1.000000 0.361926 0.271063 0.491283 0.158171 0.370236 0.278815 0.928741 0.472016 0.683299 0.362881 0.573061 0.249152 0.766245 0.369304 0.363538

Chapter Four

Analysis and Results with Implementation

Random Excursions Variant (6) Random Excursions Variant (7) Random Excursions Variant (8) Random Excursions Variant (9) Random Excursions Variant (10) Random Excursions Variant (11) Random Excursions Variant (12) Random Excursions Variant (13) Random Excursions Variant (14) Random Excursions Variant (15) Random Excursions Variant (16) Random Excursions Variant (17) Random Excursions Variant (18) 16-Lempel Ziv Compression Test

1000000

0.825061 0.683299 0.362881 0.716259 0.122705 0.158371 0.137262 0.749153 0.722561 0.372914 0.481065 0.697105 0.586927 1.000000

Table 4.9 The Statistical Tests for the EUPHRATES Cipher Statistical Tests 1-Block Frequency Test 2-Mono-bit Frequency Test 3-Overlapping Templates Test 4-Non- Overlapping Templates Test 5-Serial Test Serial (1) Serial (2) 6-Aproximate Entropy Test 7-Linear Complexity Test 8-Maurer’s “Universal Statistical Test” 9-Cumulative Sums Test Cumulative Sums (Forward) Cumulative Sums (Reverse) 10-Runs Test 11-Longest Run of Ones Test 12-Binary Matrix Rank Test 13-Spectral DFT Test 14-Random Excursions Test Random Excursions (1) Random Excursions (2) Random Excursions (3) Random Excursions (4) Random Excursions (5) - 96 -

Input Size (n) 2048 128 1000000 1000000 1000000 128 1000000 1000000 128 128 128 38912 1024 1000000

P-Value >=0.01 0.467201 0.285732 0.105854 0.657239 0.572602 0.574923 0.379216 0.670911 0.916923 0.358103 0.387211 0.492764 1.000000 0.729613 0.491872 0.638251 0.671901 0.368419 0.381754 0.810131

Chapter Four

Analysis and Results with Implementation

Random Excursions (6) Random Excursions (7) Random Excursions (8) 15- Random Excursions Variant Test Random Excursions Variant (1) Random Excursions Variant (2) Random Excursions Variant (3) Random Excursions Variant (4) Random Excursions Variant (5) Random Excursions Variant (6) Random Excursions Variant (7) Random Excursions Variant (8) Random Excursions Variant (9) Random Excursions Variant (10) Random Excursions Variant (11) Random Excursions Variant (12) Random Excursions Variant (13) Random Excursions Variant (14) Random Excursions Variant (15) Random Excursions Variant (16) Random Excursions Variant (17) Random Excursions Variant (18) 16-Lempel Ziv Compression Test

1000000

1000000

0.524672 0.692844 0.394877 0.672681 0.431782 0.921145 0.864102 0.665834 0.256751 0.243793 0.583902 0.567549 0.382195 0.184369 0.472861 0.474396 0.671278 0.602463 0.100735 0.305827 0.372865 0.911748

Table 4.10 The Statistical Tests for the FAROQ Cipher Statistical Tests 1-Block Frequency Test 2-Mono-bit Frequency Test 3-Overlapping Templates Test 4-Non- Overlapping Templates Test 5-Serial Test Serial (1) Serial (2) 6-Aproximate Entropy Test 7-Linear Complexity Test 8-Maurer’s “Universal Statistical Test” 9-Cumulative Sums Test Cumulative Sums (Forward) Cumulative Sums (Reverse) 10-Runs Test - 97 -

Input Size (n) 2048 128 1000000 1000000 1000000 128 1000000 1000000 128 128

P-Value >=0.01 0.653911 0.738494 0.247836 0.493088 0.826532 0.819844 0.693773 0.180234 0.710388 0.313993 0.372874 0.517485

Chapter Four

Analysis and Results with Implementation

11-Longest Run of Ones Test 12-Binary Matrix Rank Test 13-Spectral DFT Test 14-Random Excursions Test Random Excursions (1) Random Excursions (2) Random Excursions (3) Random Excursions (4) Random Excursions (5) Random Excursions (6) Random Excursions (7) Random Excursions (8) 15- Random Excursions Variant Test Random Excursions Variant (1) Random Excursions Variant (2) Random Excursions Variant (3) Random Excursions Variant (4) Random Excursions Variant (5) Random Excursions Variant (6) Random Excursions Variant (7) Random Excursions Variant (8) Random Excursions Variant (9) Random Excursions Variant (10) Random Excursions Variant (11) Random Excursions Variant (12) Random Excursions Variant (13) Random Excursions Variant (14) Random Excursions Variant (15) Random Excursions Variant (16) Random Excursions Variant (17) Random Excursions Variant (18) 16-Lempel Ziv Compression Test

4.7

128 38912 1024 1000000

1000000

1000000

0.120334 0.429557 0.302894 0.719466 0.738012 0.518663 0.537885 0.524774 0.378276 0.317484 0.218847 0.329947 0.513477 0.811039 0.716602 0.319927 0.428285 0.124783 0.917784 0.562991 0.528954 0.618734 0.639866 0.286636 0.239578 0.450239 0.728894 0.734478 0.925131 0.737843

Applications of the Proposed Symmetric Models The main motivations for the proposed ciphers are to design

coherent and elegant algorithms ciphers with high margin of security that satisfies the current challenges and future requirements. The proposed symmetric algorithms can be applied efficiently for the following applications: - 98 -

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1. The proposed algorithms ciphers are considered a good choice and so suitable for the secure web browsing and in E-commerce. 2. They can be used in mobile phone applications, cellular phone and PDA. 3. They are sufficient ciphers’ models to be applied in secured

E-mails

system of Pretty Good Privacy (PGP) which can maintain the secrecy and privacy of secure E-mails in the governmental institutes and personal letters. 4. They are provided with high secure financial transfers and trusted business transactions, in addition to submit high privacy to the banks Automata Teller Machine (ATM) services. 5. The

proposed

models

are

very

suitable

for

software/hardware

implementation. So, implementing these models offers a reasonable performance in both hardware and software, since it is based on efficient algorithms that have deep roots in block cipher field. 6. They can be devoted to real times and online applications that need high speed communications like the Fiber Distributed Data Interface (FDDI) 7. All the proposed algorithms have concrete design and accept extra key lengths. The proposed models are implementable on a variety of environments and awkward applications as well as they are suitable for other cryptographic protocols 8. They can be used to protect the E-Voting and online examinations for distant learning and global tests.

4.8

Advantages and Limitations of the Proposed Ciphers There are several pros and cons for any developed system or algorithm

because there is no ideal or perfect work in all aspects of design. So, the proposed ciphers have several advantages and limitations that can be listed as follows: - 99 -

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Analysis and Results with Implementation

4.8.1 The Advantages The proposed symmetric models have several marvellous properties and fascinating characteristics that can be listed as follows: 1. The developed algorithms may be applied to run quickly as there is a coordinating

trade-off

between

the

strong/performances

in

implementation and flexibility. 2. They give a more resistance against all the attacks because they are based on increasing the confusion and diffusion properties. 3. The suggested models can be applied on smart phone’s applications and in restricted devices that require a small size of internal memory. 4.8.2 The Limitations The proposed symmetric ciphers have some limitations and restricted properties which can be listed as follows: 1. Some proposed ciphers on hardware may require larger code and gate count and slower time execution as compared with the original AES, and the other one that has an acceptable size. 2. The reduced key agility and the requirement of longer time for the ciphering key setup than that of the original AES that may be considered a sacrifice to the efficiency for some algorithms on account of the flexibility and implementation factors. 3. In software, the proposed ciphers still use two different structures one for encryption and the other for decryption, although most of the stages are reduced by the reversibility techniques especially in shift rows and mixcolumns. 4. The proposed algorithms have distinctive features that in some aspects, may be less efficient with increased number of rounds that drastically become a - 100 -

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Analysis and Results with Implementation

slower especially with large data entry, as the number of rounds considers an effective factor for the speed.

4.9 Security Analysis for the Proposed Asymmetric Models This is the second part of the dissertation where the magic square construction method is consider the prominent change and memorable innovation that are also based on various techniques that giving more strength to defeat the attacks and increase the probability of resistant against the statistical analysis and the estimation process. The efficiency of the proposed cipher relies on the elapsed time for encryption/decryption and the way through which it produces different cipher-text from a plaintext. With respect to efficiency, as it is well known that most of the public-key cipher suffers from the difficulty of the key generation and the parameters selection for the session establishment and the key agreement. The proposed cipher offers a cost effective and flexible solution for the key exchange (key agility) and encryption/decryption. The adoption of the magic square and magic cube mathematical problem could significantly change the nature of public key cryptography and the manner through which it will be treated, in addition to the behaviour and the style of attacks. A simple comparison has been introduced among three different public key ciphers (RSA, El-Gamal and the proposed model) can be explained in Figure 4.1, which illustrates the consuming time in seconds to achieve the encryption and decryption operation for three messages with different size (1000 char, 2000 char, and 3000 char) respectively.

- 101 -

Chapter Four

Analysis and Results with Implementation

Encryption & Decryption

30 25 20 15 10 5 0

1000 Char

2000 Char

Proposed Cipher

3000 Char

RSA Cipher

ElGAMAL Cipher

Figure 4.1 Comparison of Encryption and Decryption Chart’s Time Figure 4.2 below submits the running time of the signature and verification algorithms for the same message. In this test, there is no need to take different messages lengths, because the execution time will be based on the

Signature & Verfication

message digest of the original message. Comparison of ExcutionTime 10 8 6 4 2 0

Proposed Signature

RSA - Signature

Asymmetric Algorithms

Sign

El-GAMAL Signature Verify

Figure 4.2 Comparison of Signature and Verification Chart’s Chart’s Time

- 102 -

Chapter Four

Analysis and Results with Implementation

4.10 The Analysis of Magic Cube The analysis of magic cube needs a deep understanding in the magic cube’s methods and the types of the magic cubes construction that provide a good insight for the main features of design procedures. Figure 4.3 and Figure

4.4 submit a clear explanation for the characteristics of third order cube with three surfaces that can be compared from several aspects and according to many parameters as stated in Table 4.11.

Figure 4.3 Cube with Three Dimensions

Figure 4.4 Three Layers of the Magic Cube - 103 -

Chapter Four

Analysis and Results with Implementation

Planar

Oblique

Folding

Oblique

Proposed Cube

Planar

Math Math

Total

Pandiagonal Perfect (Nasik)

Traig(3)

Math

Diag(2)

PantriagDiag

Ortho(1)

Math Math Math

Dimensions

Simple Diagonal Pantriagonal

Smallest Possible Order

Types of Magic Cube

Construction Type

Table 4.11 Comparison between the Proposed Cube and other Distinct Cubes

3 5

3-D 3-D

3n2 3n2

-6n

4 4

3n2 +4 3n2+6n+4

-3n

-6

---

---

4

3-D

3n2

--

4n2

7n2

--

--

--

--

8

3-D

3n2

6n

4n2

7n2+6n

3n

0

--

6

7

3-D

3n2

6n2

4

9n2+4

--

6

3n

--

8

3-D

3n2

6n2

4n2

13n2

--

--

3n

6n

6-D

MC1+MC2+…+ MC6

3

Line Summation to S

Regular Magic Square

n ( n 3 + 1) S= 2

ź

ͯ=

/ ( ͯ)

Pandiag onal Magic Square

Only Regular Magic Square (Odd Order, Singly Even or Doubly Even Order)

4.11 Advantages and Limitations of Magic Cube There are several advantages and disadvantages for the construction of the proposed model of magic cube that can be listed as follows:

4.11.1 Advantages of the Proposed Magic Cube The main advantages of the proposed approach can be summarized as follows: 1. It is an easy, elegant and fast method that requires no substantial efforts in its construction, since it depends upon the tradition magic square methods. 2. The proposed method is considered a generalized technique to any order of magic cube. - 104 -

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Analysis and Results with Implementation

3. The proposed method does not require much of thinking or deep understanding in the cube geometry field and does not need a long experience in the cube comprehension design. 4. The vertical track, horizontal track or diagonals track of values give the same magic constant of cube and the same magic sum of cube that keeps the cube magically. 5. The proposed approach works with cube of six dimensions or six faces in opposite to the methods which work with three dimensions. 6. Each face or dimension in the proposed cube represents an independent magic square that keeps the face magically.

4.11.2 Limitations of the Proposed Magic Cube There are some specific limitations for the magic cube’s construction that can be shown as follows: 1. The construction of the proposed magic cube has no independent clue, since it is based mainly on the magic square techniques. 2. The construction of this cube is not based on the specific algorithm of magic square, because it will need different algorithms according to the dimension of square through which it is constructed. 3. In spite of the proposed method being easy and does not need the mathematical background, but it requires some knowledge or experience in the magic square methods. 4. The proposed magic cube is restricted just by the regular magic squares’ techniques and properties.

- 105 -

Chapter Four

Analysis and Results with Implementation

4.12 Applications of the Proposed Magic Cube Magic cube has no direct usage or specific applications. Recently, several published research papers embed the magic square and magic cube properties in many aspects and in several contributions such as: 1. Cryptography and Information Security. 2. Remote Access Control. 3. Applied Mathematics. 4. Number Theory. 5. Determinants and Matrices Field. 6. Coding theory and Error Correctness 7. Artificial Games

4.13 Motivations and Limitations for Proposed Public Key There are several motivations to design the proposed public key model, since most of the public key methods mainly depend upon the DLP and IFP and these methods are highly time consuming. The core idea for the proposed model concentrates on the design, a fast and strong method that is based on the magic square mathematical problem, in order to guarantee the complete security and invulnerability against the malicious attacks. The complexity of this cipher includes that the eavesdroppers should try all possible probabilities of construction the magic squares’ matrices, starting from the unknown random value that acts as a starting value for the construction and unknown dimension value which needs a lot of estimation and guessing. This scheme allows fast encryption and decryption process in addition to the fast signature generation and verification, as well as it enlarges the search space against the brute force attack and consequently increases the complexity. There are several negatives and limitations that are considered one of the main problems and the limited side in the public-key cryptography which involves how to evidence that the - 106 -

Chapter Four

Analysis and Results with Implementation

asymmetric key is authoritative and it has not altered and changed by another key or by an intruder or even an unknown intercepted person. Perhaps the most vulnerable attack on the public key cipher is the man-inthe-middle attack, in which a third malicious party impersonates the personality of the authorized person by the intercepting and modifying the public key. An active adversary in the middle communication manipulates and modifies the messages and the implication deceives the two communicated parties. In order to agree on a key which is exclusively shared between Alice and Bob, these principals must make sure that the messages they receive in a protocol run are indeed from the intended principals. A trusted third party may be used to represent as a certificate authority that is verifying from the identity of persons using the authentication system. The public key in general terms or asymmetric cryptosystem compared with the symmetric cryptosystem take much more time in the established key for encryption and decryption processes since it uses sophisticated mathematical problems in its construction. The random choices for some numbers to construct the magic square whence starting value, difference value and the fixed multiplied value to generate the private key give a more resistant against the attacks. Unacceptable selections for these random numbers represent a basic restriction and may open the door in front of the passive and active attacks. So, one should be careful to ensure that the identity of the parties takes place by using a trusted third party or mediated party. Eventually, the main purpose for the extended magic square to magic cube is to treat with six parties or terminals on network instead of one using the same field and the same module number according to the DH algorithm in order to exchange the secret key. Hence, each face from the cube can be represented as an independent terminal. Thus, the main purpose for the generalized magic cube is to create a simple method based on the folding concept that constructed the magic cube easily. The magic cube gives more potential for the selection of ciphering key through the generation process as each face or dimension in the magic cube can - 107 -

Chapter Four

Analysis and Results with Implementation

be used as a key generation method for the encryption/ decryption process and for the signature algorithm.

4.14 Graphical User Interface (GUI) The present study has been designed with a simple and friendly graphical interface that allows the user to select which one algorithm that can be used in encryption and decryption process for text and files, in addition to the submission of an important option for selecting the size of key according to the sensitivity of information. The main window of the proposed cryptosystems can be shown in Figure 4.5 that consists of three of distinct proposed symmetric algorithms (TIGRIS, EUPHRATES and FAROQ) designed together. The main window of the proposed cryptosystem comprises the choice of a simple example with a random text and random ciphering key of 128-bit encrypted by using the TIGRES cipher. The proposed algorithms of symmetric and asymmetric ciphers designed with Microsoft visual studio 2013 by the C# programming language with two environments that include the graphical interfaces for the first part of the dissertation that represents the symmetric models and the console environment for second part of the asymmetric models. Several assistant programs are designed through the dissertation for various purposes that involve the mathematical operations, statistical testing, timing testing and the magic square construction methods.

- 108 -

Chapter Four

Analysis and Results with Implementation

Figure 4.5 The Main Window for the Symmetric Models

- 109 -

Chapter Five

Conclusions and Recommendations

CHAPTER FIVE CONCLUSIONS AND RECOMMANDATIONS 5.1

Conclusions

This dissertation submits some conclusions and recommendations for the future work that can be listed as follows: 1. There are two types of proposed cryptosystems symmetric and asymmetric models that have been designed specially to be performed efficiently in applications with low covered area resources those that require high level of processing speed and high volume of protection in addition to the high resistance against malicious attacks. 2. The proposed symmetric ciphers include three of distinct block ciphers models that provide an efficient and elegant key schedule (key setup), which is expanded in two directions in rows/columns states that lead to increase the complexity of key generation and eliminate the weak and semi-weak keys in the expansion process. 3. The new designed symmetric models inherited the efficiency features from the original AES cipher, while being upgraded to provide a novel structure with reduced form by the reversible technique in some stages. Therefore; the design of these ciphers are imposed to submit a reasonable implementation for several platforms and in a wider usage of systems, since it is capable of safeguarding against all known attacks and has an elegant construct. 4. During the design process of these ciphers several criteria have been taken into account to produce a conservative design and concrete structure. So the main structure to these proposed ciphers are built according to several

- 110 -

Chapter Five

Conclusions and Recommendations

characteristics represented by efficiency on software and other platforms, ease of analysis and the stability with the absence of weak keys. 5. The whole proposed symmetric algorithms can be implemented completely using the byte operations. The second type of the developed models comprises the improved technique for the public key. So, the proposed model gives a good insight and introduces a smart method in the designing processes that pave the way for the new mathematical comprehension which is related to the probability of dimension for the magic square construction, since the research space and the complexity of magic square increases dramatically with the increasing dimension. The basic idea is focused on the clue of creating a confidential communication channel with a secret sharing between the communicating parties in the presence of malicious adversaries. 6. The magic cube mathematical problem has been exploited, and it plays a vital role in encryption/decryption and signing/verifying operations. It gives a remarkable significant speed and reduces costs as well as improvement in the efficiency and security margin. 7. The proposed method for building the magic cube is considered as the simplest and nearly the fastest method to construct the magic cube, because it is based on the folded procedures and the traditional magic square methods that can be constructed with any order easily, as well as with sequential or non-sequential numbers. There is no existence for any real difficulty in the construction of any cube with this technique, because it depends upon the folded process for the magic square, and does not need a strong mathematical comprehension or experience in the geometrical aspects. The current study represents integrated encryption and decryption protocol that simulates the modern cryptosystems.

- 111 -

Chapter Five

Conclusions and Recommendations

8. Eventually the proposed cryptosystem includes the symmetric and asymmetric models in addition to use a new variant of mathematical problem of magic square and magic cube problems.

5.2

Recommendations

There are several recommendations and suggestions for the future work that can be adopted for the real researches and projects which can be subdivided as follows:

5.2.1 The Symmetric Cipher Recommendations Many suggestions for the symmetric cipher may be taken in future research. They can be listed as follows: 1. Design an invertible block cipher with full reversible stages of SPN structure and with variable block size and variable key length that accepts different lengths of plaintext and cipher key. As a result, the proposed cipher reduces costs and high complexity 2. Developing a hybrid cipher by depending on the AES and Camellia cipher that combines two different structures of (FN) and (SPN). 3. Designing a lightweight cipher for the embedded devices, smart phone, smart card and sensor network. 4. Producing a tiny algorithm for the purpose of education and for the learning of block cipher principles that opens the door in front of the minor researchers and the academic scholars. 5. Exposed the proposed symmetric models to deep analysis and an effective attacks as a future work.

- 112 -

Chapter Five

Conclusions and Recommendations

5.2.2 The Asymmetric Cipher Recommendations Many remarkable recommendations for the asymmetric approach can be stated as follows: 1. Proposing Asymmetric cipher with key exchange based on the pure magic square comprehensions completely. 2. Submitting a new method for developing of signature algorithm based on the magic square and magic cube characteristics. 3. Suggesting a public key cipher with secret sharing based on the magic cubes’ dimensions. 4. Designing an access control algorithm for the authentication scheme by using magic square and magic cube properties. 5. Applied the proposed model with Elliptic curve methods.

5.2.3 Magic Cube and Magic Square Recommendations Some ideas for improving the magic cube and magic square approaches may be devoted in the near time, like the following: 1. Developing a new variant method for the generalized magic square construction with N dimensions. 2. Proposing a new approach for the magic cube construction inside other magic cube. In other words, embedding a magical cube inside another different magic cube of different order. 3. A good suggestion for the generalized magic cube construction of three and six dimensions without relying on the magic square methods.

- 113 -

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