Algorithmic Cryptanalysis

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.

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

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