Cryptology ePrint Archive: Report 2016/007

Easing Coppersmith Methods using Analytic Combinatorics: Applications to Public-Key Cryptography with Weak Pseudorandomness

Fabrice Benhamouda and Céline Chevalier and Adrian Thillard and Damien Vergnaud

Abstract: The \emph{Coppersmith methods} is a family of lattice-based techniques to find small integer roots of polynomial equations. They have found numerous applications in cryptanalysis and, in recent developments, we have seen applications where the number of unknowns and the number of equations are non-constant. In these cases, the combinatorial analysis required to settle the complexity and the success condition of the method becomes very intricate.

We provide a toolbox based on \emph{analytic combinatorics} for these studies. It uses the structure of the considered polynomials to derive their generating functions and applies complex analysis techniques to get asymptotics. The toolbox is versatile and can be used for many different applications, including multivariate polynomial systems with arbitrarily many unknowns (of possibly different sizes) and simultaneous modular equations over different moduli. To demonstrate the power of this approach, we apply it to recent cryptanalytic results on number-theoretic pseudorandom generators for which we easily derive precise and formal analysis. We also present new theoretical applications to two problems on RSA key generation and randomness generation used in padding functions for encryption.

Category / Keywords: Coppersmith Methods, Analytic Combinatorics, Cryptanalysis, Pseudorandom Generators, RSA Key Generation, Encryption Padding

Original Publication (with major differences): IACR-PKC-2016

Date: received 4 Jan 2016

Contact author: fabrice ben hamouda at ens fr

Available format(s): PDF | BibTeX Citation

Version: 20160104:224259 (All versions of this report)

Short URL: ia.cr/2016/007