Selda Calkavur - Code Based Secret Sharing Schemes: Applied Combinatorial Coding Theory

CODE BASED SECRET
SHARING SCHEMES

Applied Combinatorial Coding Theory

CODE BASED SECRET
SHARING SCHEMES

Applied Combinatorial Coding Theory

Selda alkavur

Kocaeli University, Turkey

Alexis Bonnecaze

University of Aix-Marseille, France

Romar dela Cruz

University of the Philippines Diliman, Philippines

Patrick Sol

University of Aix-Marseille, France

Published by

World Scientific Publishing Co. Pte. Ltd.

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British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

CODE BASED SECRET SHARING SCHEMES
Applied Combinatorial Coding Theory

Copyright 2022 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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ISBN 978-981-124-832-0 (hardcover)
ISBN 978-981-124-833-7 (ebook for institutions)
ISBN 978-981-124-834-4 (ebook for individuals)

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Foreword

I am very happy that my colleagues from ATI team at Institut de Mathmatiques de Marseille (I2M), Alexis Bonnecaze and Patrick Sol, asked me to write a foreword their book on Code-Based Secret Sharing Schemes (SSS).

Both Alexis and Patrick are experts in Coding Theory, with a lifelong interest in cryptography. To speak about their two coauthors: Selda is a young Turkish cryptographer, and Romar is a young mathematician from Manila.

The book focuses on the interplay between Coding Theory and SSS. While the first and most popular SSS (Shamir scheme) was written in the language of polynomial interpolation over finite fields, it transpired soon that it was in fact related to Reed Solomon codes, the most famous class of codes after the Hamming codes. In subsequent years, research developed in several directions related to combinatorics of codes and designs. The concept of minimal code is an example related to Massey scheme. There are also some application to the so-called real world-like image processing.

The book collects classical applications and gives some examples of research related to the authors practice. I recommend it as a friendly introduction to a domain at the crossroads of mathematics, computer science and engineering.

Preface

A pirate wants to disclose the secret of a treasure location to his crew. To avoid that the greedy sailors compete and fight for the secret, he dissects the map of the location into several pieces, and gives a piece to each crew member. One single piece of the map is not enough to locate the treasure. Several pieces are needed to find it. This small tale is a toy example of a secret-sharing scheme.

Secret-sharing schemes form one of the most important topic in Cryptography. These protocols are used in many areas, applied mathematics, computer science, engineering, etc. A secret-sharing scheme is an encryption method. A secret s is divided into n pieces called shares. Each piece has no information about the secret, but it is retrieved by a specified of combination of these pieces. Specifically, a secret-sharing scheme is called a (k, n)-threshold scheme if the following property holds. Any k out of n pieces can reach the secret s but k 1 or fewer pieces have no information about s. The k elements that are capable of recovering the secret are called minimal access elements. A central entity, called the dealer, distributes the pieces to the participants. This distribution phase constitutes the first step of the protocol. In a second step, the recovery phase, coalitions of participants can pool their respective shares, and, if they are authorized, run an algebraic algorithm to reconstruct the secret.

Secret-sharing schemes were introduced independently by Shamir [].

The aim of this book is twofold. First, we give a self-contained exposition of these classical protocols, and their various generalizations. This is the topic of .

References

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