Semirings as Building Blocks in Cryptography

133 Pages·2020·5.172 MB·English

by  Mariana Durcheva

Semirings as
Building Blocks in
Cryptography

Semirings as
Building Blocks in
Cryptography
By
Mariana Durcheva
Semirings as Building Blocks in Cryptography

By Mariana Durcheva

This book first published 2020

Cambridge Scholars Publishing

Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK

British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library

Copyright © 2020 by Mariana Durcheva

All rights for this book reserved. No part of this book may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without
the prior permission of the copyright owner.

ISBN (10): 1-5275-4170-3
ISBN (13): 978-1-5275-4170-2
CONTENTS

Acknowledgements .................................................................................. viii

Preface ........................................................................................................ ix

Chapter One ............................................................................................... 1
Introduction

Chapter Two .............................................................................................. 4
The Role of Different Algebraic Structures as Building
Blocks in Public Key Cryptography

2.1 Discrete Logarithms in Finite Fields ................................................ 4
2.1.1 Definitions ............................................................................... 4
2.1.2 Public key cryptosystems based on the
Discrete Logarithm Problem ....................................................... 5
2.1.3 Generic algorithms for solving the
Discrete Logarithm Problem ....................................................... 8
2.1.4 Index Calculus algorithms for finite fields ............................ 10
2.2 Group Based Cryptography ........................................................... 12
2.2.1 The Discrete Logarithm Problem in abelian groups .............. 12
2.2.2 The Discrete Logarithm Problem in nonabelian groups ........ 13
2.2.3 Groups used as platforms for cryptosystems ......................... 17
2.3 The Discrete Logarithm Problem in Rings and Semirings ............. 22
2.3.1 The Diffie-Hellman protocol in terms of formal algebra –
a parallel between groups and rings .......................................... 22
2.3.2 The Discrete Logarithm Problem in noncommutative rings . 24
2.3.3 The Discrete Logarithm Problem and cryptography
using semirings ......................................................................... 26
2.3.4 Semirings and cryptography .................................................. 28

Bibliography to Chapter One and Chapter Two................................... 31
vi Contents

Chapter Three .......................................................................................... 53
The Role of Idempotent Semirings

3.1 Idempotent Algebra ....................................................................... 53
3.1.1 Dioids as lattices .................................................................... 56
3.1.2 Residuated maps .................................................................... 58
3.1.3 Semimodules over idempotent semirings .............................. 59
3.1.4 Residuated lattices ................................................................. 62
3.1.5 Construction of complete dioid ............................................. 63
3.1.6 Matrices defined over idempotent semifields ........................ 65
3.1.7 Residual operations over matrix semimodules ...................... 69
3.1.8 Max-plus algebra ................................................................... 70
3.1.9 Min-plus algebra ................................................................... 73
3.1.10 Max-time algebra ................................................................ 77
3.1.11 Min-time algebra ................................................................. 78
3.2 Linear Equations ............................................................................ 80
3.3 Applications of Idempotent Algebra in Cryptography ................... 84

Bibliography to Chapter Three ............................................................ 87

Chapter Four ........................................................................................... 96
Distributed Multicast Key-Exchange Protocol
Based on Idempotent Semirings

4.1 Definitions and Motivation to Study .............................................. 96
4.2 Distributed Secure Multicast Protocol ........................................... 98

Bibliography to Chapter 4 ....................................................................... 102

Chapter Five .......................................................................................... 104
Endomorphism Semirings and Certain Cryptographic
Protocols Based on Them

5.1 Endomorphism Semirings ............................................................ 104
5.2 Simplices and Simplicial Complexes ........................................... 107
5.3 Cryptography Based on Endomorphism Semirings ..................... 112
5.3.1 Motivation, basic definitions and notations ......................... 112
Semirings as Building Blocks in Cryptography vii

5.3.2 Three key-exchange protocols based
on endomorphism semirings ................................................... 115
5.3.3 Four key exchange protocols using right identities
of the simplex ......................................................................... 117

Bibliography to Chapter 5 .................................................................. 121
ACKNOWLEDGEMENTS

This book is the revised, updated and expanded version of my monograph
“Applications of Some Algebraic structures in cryptography”, published in
2015 in Bulgarian under the editing of Prof. Ivan Trendafilov to whom I
would like to express my sincere gratitude.
PREFACE

Nowadays when the main communication between people takes place
over the Internet, information protection is crucial. With the development
of virtual business and e-commerce, data sharing through unprotected
public channels is expanded. The role of cryptography is to protect users’
data from malicious intrusions. Public-key cryptography underlies the
security features of many issues such as signed and encrypted email,
single sign-on, and Secure Sockets Layer (SSL) communications. Public-
key introduces a concept involving key pairs: one for encrypting, the other
for decrypting. Various approaches are used to create the key exchange
protocol. In this book we show the role of some algebraic structures as
building blocks for these protocols. We first review and analyze some
algebraic structures that have already been used to build cryptographic
protocols – finite fields, groups and rings. Then we discuss the role in
cryptography of the structures that are either not used for such purposes or
used in a limited way. Such structures are idempotent semirings and
semirings of the endomorphisms of a finite chain. Based on these studies,
several key exchange protocols are developed. We hope that “Semirings
as building blocks in cryptography” will be useful not only for
cryptographers and specialists in Applied Algebra, but also for students of
Cryptography or Applied Algebra.