New results on modular inversion hidden number problem and inversive congruential generator

Abstract

The Modular Inversion Hidden Number Problem (MIHNP), introduced by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001, is briefly described as follows: Let MSB𝛿(𝑧) refer to the δ most significant bits of z. Given many samples(𝑡𝑖,MSB𝛿((𝛼+𝑡𝑖)−1mod𝑝))(ti,MSBδ((α+ti)−1modp)) for random 𝑡𝑖∈ℤ𝑝, the goal is to recover the hidden number 𝛼∈ℤ . MIHNP is an important class of Hidden Number Problem. In this paper, we revisit the Coppersmith technique for solving a class of modular polynomial equations, which is respectively derived from the recovering problem of the hidden number α in MIHNP. For any positive integer constant d, let integer 𝑛=𝑑3+𝑜(1) . Given a sufficiently large modulus p, n+1 samples of MIHNP, we present a heuristic algorithm to recover the hidden number$$\alpha $$ with a probability close to 1 when𝛿/log2𝑝>1𝑑+1+𝑜(1𝑑). The overall time complexity of attack is polynomial in log2𝑝, where the complexity of the LLL algorithm grows as dO(d) and the complexity of the Gröbner basis computation grows as(2d)O(n2). When 𝑑>2, this asymptotic bound outperforms 𝛿/log2𝑝>1/3 which is the asymptotic bound proposed by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001. It is the first time that a better bound for solving MIHNP is given, which implies that the conjecture that MIHNP is hard whenever 𝛿/log2𝑝<1/3 is broken. Moreover, we also get the best result for attacking the Inversive Congruential Generator (ICG) up to now.