Algorithmic Cryptanalysis

Algorithmic Cryptanalysis

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Features

  • Introduces cryptography from a cryptanalytic perspective
  • Covers linear algebra, sieving, brute force, algorithms based on the birthday paradox, Hadamard–Fourier–Walsh transforms, lattice reduction, and Gröbner bases
  • Presents advanced applications, such as LFSR-based stream ciphers, lattice methods for cryptanalysis, elliptic curves, and index calculus methods
    • Includes exercises, with some hints and solutions offered on http://www.joux.biz/algcrypt/
    • Provides several C codes for download on http://www.joux.biz/algcrypt/

    Summary

    Illustrating the power of algorithms, Algorithmic Cryptanalysis describes algorithmic methods with cryptographically relevant examples. Focusing on both private- and public-key cryptographic algorithms, it presents each algorithm either as a textual description, in pseudo-code, or in a C code program.

    Divided into three parts, the book begins with a short introduction to cryptography and a background chapter on elementary number theory and algebra. It then moves on to algorithms, with each chapter in this section dedicated to a single topic and often illustrated with simple cryptographic applications. The final part addresses more sophisticated cryptographic applications, including LFSR-based stream ciphers and index calculus methods.

    Accounting for the impact of current computer architectures, this book explores the algorithmic and implementation aspects of cryptanalysis methods. It can serve as a handbook of algorithmic methods for cryptographers as well as a textbook for undergraduate and graduate courses on cryptanalysis and cryptography.

    Table of Contents

    BACKGROUND

    A Bird’s-Eye View of Modern Cryptography

    Preliminaries

    Defining security in cryptography

    Elementary Number Theory and Algebra Background

    Integers and rational numbers

    Greatest common divisors in Z

    Modular arithmetic

    Univariate polynomials and rational fractions

    Finite fields

    Vectors spaces and linear maps

    The RSA and Diffie–Hellman cryptosystems

    ALGORITHMS

    Linear Algebra

    Introductory example: multiplication of small matrices over F2

    Dense matrix multiplication

    Gaussian elimination algorithms

    Sparse linear algebra

    Sieve Algorithms

    Introductory example: Eratosthenes’s sieve

    Sieving for smooth composites

    Brute Force Cryptanalysis

    Introductory example: dictionary attacks

    Brute force and the DES algorithm

    Brute force as a security mechanism

    Brute force steps in advanced cryptanalysis

    Brute force and parallel computers

    The Birthday Paradox: Sorting or Not?

    Introductory example: birthday attacks on modes of operation

    Analysis of birthday paradox bounds

    Finding collisions

    Application to discrete logarithms in generic groups

    Birthday-Based Algorithms for Functions

    Algorithmic aspects

    Analysis of random functions

    Number theoretic applications

    A direct cryptographic application in the context of blockwise security

    Collisions in hash functions

    Hellman’s time memory tradeoff

    Birthday Attacks through Quadrisection

    Introductory example: subset sum problems

    General setting for reduced memory birthday attacks

    Extensions of the technique

    Some direct applications

    Fourier and Hadamard–Walsh Transforms

    Introductory example: studying S-boxes

    Algebraic normal forms of boolean functions

    Goldreich–Levin theorem

    Generalization of the Walsh transform to Fp

    Fast Fourier transforms

    Lattice Reduction

    Definitions

    Introductory example: Gauss reduction

    Higher dimensions

    Shortest vectors and improved lattice reduction

    Dual and orthogonal lattices

    Polynomial Systems and Gröbner Bases Computations

    General framework

    Bivariate systems of equations

    Definitions: multivariate ideals, monomial orderings, and Gröbner bases

    Buchberger algorithm

    Macaulay’s matrices

    Faugère’s algorithms

    Algebraic attacks on multivariate cryptography

    On the complexity of Gröbner bases computation

    APPLICATIONS

    Attacks on Stream Ciphers

    LFSR-based keystream generators

    Correlation attacks

    Algebraic attacks

    Extension to some nonlinear shift registers

    The cube attack

    Time memory data tradeoffs

    Lattice-Based Cryptanalysis

    Direct attacks using lattice reduction

    Coppersmith’s small roots attacks

    Elliptic Curves and Pairings

    Introduction to elliptic curves

    The Weil pairing

    The elliptic curve factoring method

    Index Calculus Algorithms

    Introduction to index calculus

    A simple finite field example

    Generalization to finite fields with small enough characteristics

    Introduction to the number field sieve

    Smoothness probabilities

    References

    Author Bio(s)

    Antoine Joux is associate professor at Université de Versailles Saint-Quentin-en-Yvelines.

    Editorial Reviews

    … this book is a must-read/must-have-in-your-shelf for anybody seriously involved in the field of cryptography if only to give an overview of the range of techniques that can be applied to break cryptographic schemes and the cryptographic hurdles that one needs to get over to design secure systems. Also parts of the book can also easily be used as the basis for a cryptography course since every chapter contains exercises (hints and solution to some of them can be found on the author’s website).
    I particularly appreciated the focus on the practicality of the provided material: after an overview in plain English, every algorithm is clearly stated in the form of pseudo-code and many of them are also provided in C code. In that way the reader can easily follow the careful complexity analysis to convince himself that it is the algorithm he wants to use and then write an implementation of that algorithm from the book. The practical aspect of book shines as well through the effort of always presenting the most efficient algorithms for a given task with particular constraints in mind.
    To sum it up, this book is a mine of information on cryptanalysis and goes above and beyond to provide the reader with everything he needs to become a better cryptographer.
    —Alexandre Anzala-Yamajako, SIGACT News, 2012

    With a plethora of algorithms available to them, readers will be ready to appreciate a number of their applications. … The book also provides a number of downloadable codes in C. … Suggestions are made on how to use the book to meet a variety of teaching needs — including accommodating readers with a stronger computer science background. The flexibility in this approach to learning is a welcome feature of the book and makes it an attractive option for those who may need to deliver pertinent material to a variety of audiences. Such audiences will be provided with a clear presentation of key aspects of this vital scientific discipline. The clarity of exposition should also serve well those who use this book as a handbook.
    Contemporary Physics, Volume 52, Issue 3, 2011

    … very nice to see the connection between cryptography and the different algorithms. … A convenient extra of the book are the good references. … Algorithmic Cryptanalysis is a high level book that covers many interesting topics. I would recommend this book for graduate students with a strong mathematical background, a cryptographic background, knowledge in C-programming and an interest in implementing cryptanalytic attacks. As mentioned before, the book covers interesting topics when it comes to implementing an attack which I haven't seen in any other book before in this combination. …
    —IACR Book Reviews, October 2010

    … The aim of the book is to survey work on cryptanalysis (both for symmetric and public key cryptography) and to present background on all major cryptanalytic tools. The author is a leading authority who has made major research contributions in most aspects of the subject. To have such a wide-ranging survey of the area written by someone with such depth of experience will be extremely valuable to students and researchers. … Chapters 3, 4 and 15 give an excellent survey of index calculus algorithms for the discrete logarithm problem in finite fields … the book will certainly be useful to postgraduates and researchers in cryptography and cryptanalysis.
    Mathematical Reviews, Issue 2010h

    The book under review presents a complete panoramic of the different methods and techniques used in modern cryptanalysis … emphasis is in the algorithms, in fact one of the main attractions of the book is the great quantity of algorithms that it presents: some described in pseudocode (listed as algorithms) and others as programs in C language (listed as programs). Additional C implementations can be downloaded from the website www.joux.biz/algcrypt …
    Zentralblatt MATH 1172

    … This book takes an algorithmic approach to the topic and covers a number of algorithms that might be used in the cryptanalysis of different systems. … There is quite a bit of interesting material in the book … The material is very well presented most of the time … This book could be a very good introduction to cryptanalysis for graduate students who have already been introduced to cryptography and have a fair amount of mathematical background. The book could be used in an advanced undergraduate course as well … It would also be quite an interesting read for those studying algorithms, as some of the algorithms presented are quite intriguing …
    —Jeffrey Putnam, Computing Reviews, May 2010

    This is a work suitable for first-year graduate students or advanced undergraduates. … the addition of the online materials makes this book usable by independent readers or industry algorithm implementers in need of a reference work. … Combining practical algorithms and supported by explanation of the relevant theory, this is a good introduction to cryptanalysis that improves on that good recipe by including key details on current computer architecture. This makes this work succeed as both handbook and textbook.
    —Tom Schulte, MAA Reviews, April 2010