Cryptology ePrint Archive: Report 2013/512

Rounding LLL: Finding Faster Small Roots of Univariate Polynomial Congruences

Jingguo Bi and Phong Q. Nguyen

Abstract: In a seminal work at EUROCRYPT '96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysis and in a few security proofs. However, the running time of the algorithm is a high-degree polynomial, which limits experiments: the bottleneck is an LLL reduction of a high-dimensional matrix with extra-large coefficients. We present in this paper a polynomial speedup over Coppersmith's algorithm. Our improvement is based on a special property of the matrices used by Coppersmith's algorithm, which allows us to speed up the LLL reduction by rounding. The exact speedup depends on the LLL algorithm used: for instance, the speedup is quadratic in the bit-size of the small-root bound if one uses the Nguyen-Stehl\'e L^2 algorithm.

Category / Keywords: public-key cryptography / Coppersmith's Algorithm, Small Roots of Polynomial Equations, LLL, Complexity, Speedup, RSA.

Date: received 17 Aug 2013

Contact author: pnguyen at di ens fr

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Version: 20130817:205128 (All versions of this report)

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