The Design Of Rijndael: Aes - The Advanced Encryption Standard (information Security And Cryptography)

Joan Daernen Vincent Rijrnen · The Design of Rijndael AES - The Advanced Encryption Standard With 48 Figures and 17 Tables Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris TnL-1Jn Springer Foreword Joan Daemen Proton World International (PWI) Zweefvliegtuigstraat 10 1130 Brussels, Bel gium Vincent Rijmen Cryptomathic NV Lei Sa 3000 Leuven, Belgium Library of Congress Cataloging-in-Publication Data Daemen, Joan, 1965- The design of Rijndael: AES - The Advanced Encryption Standard/Joan Daemen, Vincent R ijmen. p.cm. Includes bibliographical references and index. ISBN 3540425802 (alk. paper) .. . 1. Computer security - Passwords. 2. Data encryption (Computer sCIence) I. RIJmen, Vincent, 1970- II. Title QA76.9.A25 D32 2001 005.8-dc21 2001049851 ACM Subject Classification (1998): E.3, C.2, DA.6, K.6.S ISBN 3-540-42580-2 Springer-Verlag Berlin Heidelberg New York � � This work is subject to copyright. All rights are reserved, whet er the whole o� part o the . material is concerned, specifically the rights of translation, repnntmg, reuse of 11lust�atIOns, recitation, broadcasting, reproduction on microfilm or in any other way, and storage l� ata banks. Duplication of this publication or parts thereof is permitted on y under the P!o:'lSlons of the German Copyright Law of September 9, 1965, in its current verSIOn, and per�lssIOn for use must always be obtained from Springer-Verlag. Violations are liable for prosecutIOn under the German Copyright Law. � ? Springer-Verlag Berlin Heidelberg New York, a member of BertelsmannSpringer Science+ Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, trademarks, etc. in this publication does not imply, even in . the'absence of a specific statement, that such names are exempt from the relevant protectIve laws and regulations and therefore free for general use. Typesetting: Camera-ready by the authors Cover Design: KiinkelLopka, Heidelberg n .. =_L_.l �_ ��=.l ._��_�_�_ cnru 1()O is a commutative ring. For special choices of the reduction polynomial m(x) , the structure becomes a field. Strings of bits are often abbreviated using the hexadecimal notation. corresponds t o the bit string 01010 1 1 1 , or 15 a(x) x b(x) + m(x) x c(x) = gcd (a(x) , m(x) ) . (2 .25) Here gcd (a(x) , m(x)) denotes the greatest common divisor of the polynomials a(x) and m(x) , which is always equal to 1 iff m(x) is irreducible. Applying modular reduction to (2.25) , we get : a(x) x b(x) == 1 ( mod m(x) ) , (2.26) which means that b( x) is the inverse element of a( x) for the definition of the multiplication ' . ' given in (2.24) . Conclusion. Let F be the field GF(p) . With a suitable choice for the reduc­ tion polynomial, the structure < f="" [xl="" l="" n="" ,="" +,="" '=""> is a field with p n elements, usually denoted by GF(p n ) . since: (x 6 + x4 + x 2 + + 1) ffi (x7 + + 1 ) = x7 + x 6 + x4 + x 2 + ( 1 ffi l)x + ( 1 ffi 1