On optimal bounds of small inverse problems and approximate GCD problems with higher degree

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Abstract

We show a relation between optimal bounds of a small inverse problem and an approximate GCD problem. First, we present a lattice based method to solve small inverse problems with higher degree. The problem is a natural extension of small secret exponent attack on RSA cryptosystem introduced by Boneh and Durfee. They reduced this attack to solving a bivariate modular equation: , where A is a given integer and e is a public exponent. They proved that the problem can be solved in polynomial time when d ≤ N 0.292. In this paper, we extend the Boneh-Durfee's result to more general problem. For a monic polynomial h(y) of degree κ ≤ 1), integers C and e, we want to find all small roots of a bivariate modular equation: . We denote by X and Y the upper bound of roots. We present an algorithm for solving the problem and prove that the problem can be solved in polynomial time if and |C| is small enough, where X = e γ and Y = e α . We employ a similar approach as unravelled linearization technique introduced by Herrmann and May in especially evaluating the lattice volume. Interestingly, our algorithm does not rule out the case of C = 0, which implies that our algorithm can solve a univariate unknown modular equation , where p is unknown. Our algorithm achieves the best bound in the literature. Then, we show that our obtained bound is natural under the similar sense of Howgrave-Graham's discussion in CaLC2001 and we prove that our bound, including Boneh-Durfee's bound, is optimal under the reasonable assumption. © 2012 Springer-Verlag.

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APA

Kunihiro, N. (2012). On optimal bounds of small inverse problems and approximate GCD problems with higher degree. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7483 LNCS, pp. 55–69). https://doi.org/10.1007/978-3-642-33383-5_4

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