PUBLIC KEY CRYPTOGRAPHY THAT ENHANCES SECURITY OF THE
DIFFIE-HELLMAN ALGORITHIM
PRESENTED BY
IRUNGU JULIUS GIKONYO
ON AUGUST 2018
,ABSTRACT;
Cryptographic algorithms play a vital role in providing the data security against malicious
attacks. Deffie-Hellman algorithm is extensively used in the popular implementations of
Public Key Infrastructures. In asymmetric key cryptography, also called Public Key
cryptography, two different keys (which form a key pair) are used. One key is used for
encryption and only the other corresponding key must be used for decryption. No other key
can decrypt the message – not even the original (i.e. the first) key used for encryption. Our
aim is to implement encryption technique namely; Deffie-Hellman algorithm whereby we
propose a trapdoor function to describe the algorithm for key agreement and public key
encryption using a given commutative ring of integers.
, Contents
ABSTRACT; .............................................................................................................................................................. 2
Contents ................................................................................................................................................................. 3
CHAPTER ONE ......................................................................................................................................................... 5
INTRODUCTION ...................................................................................................................................................... 5
1.1: Introduction and Background of the study ................................................................................................. 5
1.2 Statement of the problem ........................................................................................................................ 6
1.3: Objective of the study ................................................................................................................................. 6
1.4. Significance of the Study ............................................................................................................................. 6
1.5 Methodology ................................................................................................................................................ 6
1.6 LITERATURE REVIEW .................................................................................................................................... 7
CHAPTER TWO ........................................................................................................................................................ 9
2.1. BASIC DEFINITIONS ...................................................................................................................................... 9
2.1.1 MATRICES .............................................................................................................................................. 9
2.1.2 Types of matrices ................................................................................................................................ 10
2.3 MATRIX ARITHMETIC .................................................................................................................................. 11
2.3.1 Matrix addition ................................................................................................................................... 11
2.3.2 Matrix subtraction .............................................................................................................................. 11
2.3.3 Multiplication of matrix by a scalar .................................................................................................... 11
2.3.4 Matrix multiplication .......................................................................................................................... 11
2.4 Functions of a matrix .................................................................................................................................. 12
2.4.1 The transpose of a matrix ................................................................................................................... 12
2.4.2 The determinant ................................................................................................................................. 12
2.4.3 The matrix inverse .............................................................................................................................. 13
2.5 More special matrices ................................................................................................................................ 14
2.5.1Symmetric matrix ................................................................................................................................. 14
2.5.2 Circulant Matrix .................................................................................................................................. 14
2.5.3 The Greatest Common Divisor (GCD) of a circulant matrix. ............................................................... 14
2.5.4 Prime circulant matrix ......................................................................................................................... 14
2.5.5 Doubly circulant coefficient matrix ............................................................................................... 15
Lemma 1 ...................................................................................................................................................... 15
Lemma 2 ...................................................................................................................................................... 16
2.6.0 Eigen values and Eigen vectors ............................................................................................................... 16
CHAPTER THREE .................................................................................................................................................... 17
3.1.1: Definitions .................................................................................................................................................. 17
DIFFIE-HELLMAN ALGORITHIM
PRESENTED BY
IRUNGU JULIUS GIKONYO
ON AUGUST 2018
,ABSTRACT;
Cryptographic algorithms play a vital role in providing the data security against malicious
attacks. Deffie-Hellman algorithm is extensively used in the popular implementations of
Public Key Infrastructures. In asymmetric key cryptography, also called Public Key
cryptography, two different keys (which form a key pair) are used. One key is used for
encryption and only the other corresponding key must be used for decryption. No other key
can decrypt the message – not even the original (i.e. the first) key used for encryption. Our
aim is to implement encryption technique namely; Deffie-Hellman algorithm whereby we
propose a trapdoor function to describe the algorithm for key agreement and public key
encryption using a given commutative ring of integers.
, Contents
ABSTRACT; .............................................................................................................................................................. 2
Contents ................................................................................................................................................................. 3
CHAPTER ONE ......................................................................................................................................................... 5
INTRODUCTION ...................................................................................................................................................... 5
1.1: Introduction and Background of the study ................................................................................................. 5
1.2 Statement of the problem ........................................................................................................................ 6
1.3: Objective of the study ................................................................................................................................. 6
1.4. Significance of the Study ............................................................................................................................. 6
1.5 Methodology ................................................................................................................................................ 6
1.6 LITERATURE REVIEW .................................................................................................................................... 7
CHAPTER TWO ........................................................................................................................................................ 9
2.1. BASIC DEFINITIONS ...................................................................................................................................... 9
2.1.1 MATRICES .............................................................................................................................................. 9
2.1.2 Types of matrices ................................................................................................................................ 10
2.3 MATRIX ARITHMETIC .................................................................................................................................. 11
2.3.1 Matrix addition ................................................................................................................................... 11
2.3.2 Matrix subtraction .............................................................................................................................. 11
2.3.3 Multiplication of matrix by a scalar .................................................................................................... 11
2.3.4 Matrix multiplication .......................................................................................................................... 11
2.4 Functions of a matrix .................................................................................................................................. 12
2.4.1 The transpose of a matrix ................................................................................................................... 12
2.4.2 The determinant ................................................................................................................................. 12
2.4.3 The matrix inverse .............................................................................................................................. 13
2.5 More special matrices ................................................................................................................................ 14
2.5.1Symmetric matrix ................................................................................................................................. 14
2.5.2 Circulant Matrix .................................................................................................................................. 14
2.5.3 The Greatest Common Divisor (GCD) of a circulant matrix. ............................................................... 14
2.5.4 Prime circulant matrix ......................................................................................................................... 14
2.5.5 Doubly circulant coefficient matrix ............................................................................................... 15
Lemma 1 ...................................................................................................................................................... 15
Lemma 2 ...................................................................................................................................................... 16
2.6.0 Eigen values and Eigen vectors ............................................................................................................... 16
CHAPTER THREE .................................................................................................................................................... 17
3.1.1: Definitions .................................................................................................................................................. 17
