APPLIED CRYPTOGRAPHY SECOND EDITION: Protocols
This is the gap that Bruce Schneier's Applied Cryptography has come to fill. Beginning with the objectives of communication security and elementary.
A Graduate Course in Applied Cryptography
unified framework for doing cryptographic proofs. A reader who masters this framework will be capable of applying it to new constructions that may not be
This is a Chapter from the Handbook of Applied Cryptography by A
ciency of a particular cryptographic scheme based on any one of these algebraic a treatment of several of the most practical exponentiation algorithms.
This is a Chapter from the Handbook of Applied Cryptography by A
Handbook of Applied Cryptography by A. Menezes P. van Oorschot and S. Vanstone. tion to increase the cryptographic security of an encryption process ...
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unified framework for doing cryptographic proofs. A reader who masters this framework will be capable of applying it to new constructions that may not be
A Graduate Course in Applied Cryptography
capable of applying it to new constructions that may not be covered in the book. Part III is about cryptographic protocols such as protocols for user ...
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3205 Applied Cryptography and Network Security. 1. (Preliminary Description). 2. Zvi M. Kedem. 3. Updated December 18 2014.
CRC Press - Handbook of applied Cryptography.pdf
With electronic information the concept of a signature needs to be. Handbook of Applied Cryptography by A. Menezes P. van Oorschot and S. Vanstone.
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This Applied. Cryptography class offers a comprehensive introduction to Modern Cryptography and
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The literature of cryptography has a curious history Secrecy of course has always played a central role but until the First World War important
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This book is about exactly that: constructing practical cryptosystems for which we can argue security under plausible assumptions The book covers many
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The purpose of this book is to give an up-to-date treatise of the principles techniques and algorithms of interest in cryptographic practice Emphasis has
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This article provides a general introduction to the subject of Cryptology Crytography and Crytoanalysis and explains the terminology and the practical
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This is a Chapter from the Handbook of Applied Cryptography by A Menezes practical public-key encryption and signature scheme now referred to as RSA
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6 oct 2015 · Applied Cryptography Second Edition: Protocols Algorthms and Source Code in C 20th Anniversary Edition PDF · Request permissions
Handbook of Applied Cryptography - Free Computer Books
Title: Handbook of Applied Cryptography; Author(s) Alfred J Menezes Hardcover: 780 pages; eBook: PDF and PostScript Files; Language: English
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This free book is about mathematical modeling and proofs to show that a particular cryptosystem satisfies the security properties attributed to it:
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Covering the latest developments in practical cryptographic techniques this new edition shows programmers who design computer applications networks and
How is cryptography applied?
Cryptography uses mathematical functions to transform data and prevent it from being read or tampered with by unauthorized parties. Nearly every computing and communications device uses cryptographic technologies to protect the confidentiality and integrity of information that is communicated and/or stored.What are the five applications of cryptography?
Cryptography in Everyday Life
Authentication/Digital Signatures. Authentication and digital signatures are a very important application of public-key cryptography. Time Stamping. Electronic Money. Secure Network Communications. Anonymous Remailers. Disk Encryption.What are the 3 parts of cryptography?
A basic cryptosystem includes the following components:
Plaintext- This is the data that needs to be protected.Encryption algorithm- This is the mathematical algorithm that takes plaintext as the input and returns ciphertext. Ciphertext- This is the encrypted, or unreadable, version of the plaintext.- Cryptographic principles are the fundamental concepts and techniques that are used in the field of cryptography to secure communication and protect data. These principles include confidentiality, integrity, authentication, non-repudiation, and key management.
Oorschot, and S. Vanstone, CRC Press, 1996.
For further information, seewww.cacr.math.uwaterloo.ca/hac CRC Press has granted the following specic permissions for the electronic version of this book: Permission is granted to retrieve, print and store a single copy of this chapter for personal use. This permission does not extend to binding multiple chapters of the book, photocopying or producing copies for other than personal use of the person creating the copy, or making electronic copies available for retrieval by others without prior permission in writing from CRC Press. Except where over-ridden by the specic permission above, the standard copyright notice from CRC Press applies to this electronic version: Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microlming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specic permission must be obtained in writing from CRC Press for such copying. c ?1997 by CRC Press, Inc.Chapter14
Ef®cient Implementation
Contents in Brief
14.1 Introduction:::::::::::::::::::::::::::::591
14.2 Multiple-precision integer arithmetic::::::::::::::::592
14.3 Multiple-precision modular arithmetic:::::::::::::::599
14.4 Greatest common divisor algorithms::::::::::::::::606
14.5 Chinese remainder theorem for integers::::::::::::::610
14.6 Exponentiation:::::::::::::::::::::::::::613
14.7 Exponent recoding:::::::::::::::::::::::::627
14.8 Notes and further references::::::::::::::::::::630
14.1 Introduction
Many public-key encryption and digital signature schemes, and some hash functions (see x9.4.3), require computations inZ m , the integers modulom(mis a large positive integer whichmayormaynotbeaprime). Forexample,theRSA,Rabin,andElGamalschemesre- quire ef®cient methods for performingmultiplication and exponentiationinZ m . Although Z m is prominent in many aspects of modern applied cryptography, other algebraic struc- turesarealsoimportant. Theseinclude,butarenotlimitedto,polynomialrings,®nite®elds, and ®nite cyclic groups. For example, the group formed by the points on an elliptic curve over a ®nite ®eld has considerable appeal for various cryptographicapplications. The ef®- ciency of a particular cryptographicscheme based on any one of these algebraic structures ing power available, software and/orhardwareoptimization,and mathematicalalgorithms. ing out computationsin the underlyingalgebraic structure. Since many of the most widely implemented techniques rely onZ m , emphasis is placed on ef®cient algorithms for per- forming the basic arithmetic operations in this structure (addition, subtraction, multiplica- tion, division, and exponentiation). Insome cases, severalalgorithmswill bepresentedwhichperformthesame operation. For example, a number of techniquesfor doing modular multiplication and exponentiation are discussed inx14.3 andx14.6, respectively. Ef®ciency can be measured in numerous ways; thus, it is dif®cultto de®nitivelystate which algorithmis the best. An algorithmmay be ef®cientin the time it takesto performa certain algebraicoperation,but quite inef®cient in the amount of storage it requires. One algorithm may require more code space than an- other. Dependingontheenvironmentinwhichcomputationsaretobeperformed,onealgo- rithm may be preferable over another. For example, current chipcard technology provides 591592 Ch.14 Ef®cient Implementation
verylimitedstorageforbothprecomputedvaluesandprogramcode. Forsuchapplications, an algorithm which is less ef®cient in time but very ef®cient in memory requirements may be preferred. The algorithms described in this chapter are those which, for the most part, have re- ceived considerable attention in the literature. Although some attempt is made to point out their relative merits, no detailed comparisons are given.Chapter outline
x14.2 deals with the basic arithmetic operations of addition, subtraction, multiplication, squaring, and division for multiple-precision integers.x14.3 describes the basic arithmetic m . Techniquesdescribedforper- forming modular reduction for an arbitrary modulusmare the classical method (x14.3.1), Montgomery's method (x14.3.2), and Barrett's method (x14.3.3).x14.3.4 describes a re- duction procedure ideally suited to moduli of a special form. Greatest common divisor (gcd) algorithms are the topic ofx14.4, including the binary gcd algorithm (x14.4.1) and Lehmer's gcd algorithm (x14.4.2). Ef®cient algorithms for performing extended gcd com- putations are given inx14.4.3. Modular inverses are also considered inx14.4.3. Garner's algorithmfor implementingthe Chinese remaindertheoremcanbe foundinx14.5.x14.6is a treatment of several of the most practical exponentiation algorithms.x14.6.1 deals with exponentiation in general, without consideration of any special conditions.x14.6.2 looks at exponentiationwhen the base is variable and the exponentis ®xed.x14.6.3considersal- gorithmswhich take advantage of a ®xed-base elementand variable exponent. Techniques ponentmay allow signi®cant performanceenhancements.x14.8 containsfurther notes and references.14.2 Multiple-precision integer arithmetic
This section deals with the basic operations performed on multiple-precision integers: ad- dition, subtraction, multiplication, squaring,and division. The algorithms presentedin this section are commonly referred to as theclassical methods.14.2.1 Radix representation
Positive integers can be represented in variousways, the most commonbeingbase10.For example,a= 123base10meansa=110 2 +2101 +310
0 . For machinecomputations, base2(binary representation) is preferable. Ifa= 1111011base2,thena=2 6 +2 5 2 4 +2 3 +02 2 +2 1 +2 0
14.1 FactIfb2is an integer, then any positive integeracan be expressed uniquely asa=
a n b n +a n-1 b n-1 ++a 1 b+a 0 ,wherea i is an integer with0a i14.3 Note(notation and terminology)
(i) The basebrepresentation of a positive integeragiven in Fact 14.1 is usually written asa=(a n a n-1 a 1 a 0 b . The integersa i ,0in, are calleddigits.a n is called themost signi®cant digitorhigh-order digit;a 0 theleast signi®cant digitor low-order digit.Ifb=10, the standard notation isa=a n a n-1 a 1 a 0 (ii) It is sometimes convenient to pad high-order digits of a basebrepresentation with0's; such a padded number will also be referred to as the basebrepresentation.
(iii) If(a n a n-1 a 1 a 0 b is the basebrepresentationofaanda n 6 =0, thentheprecision orlengthofaisn+1.Ifn=0,thenais calledasingle-precisioninteger; otherwise, ais amultiple-precision integer.a=0is also a single-precision integer. The division algorithm for integers (see De®nition 2.82) provides an ef®cient method for determiningthe basebrepresentationof a non-negativeinteger,for a givenbaseb.This provides the basis for Algorithm 14.4.14.4 AlgorithmRadixbrepresentation
INPUT: integersaandb,a0,b2.
OUTPUT: the basebrepresentationa=(a
n a 1 a 0 b ,wheren0anda n 6 =0ifn1.1.i 0,x a,q b
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