New Results on Modular Inversion Hidden Number Problem and Inversive Congruential Generator

The Modular Inversion Hidden Number Problem (MIHNP), introduced by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001, is briefly described as follows: Let$${\mathrm {MSB}}_{\delta }(z)$$ refer to the$$\delta $$ most significant bits of z. Given many samples$$\left(t_{i}, {\mathrm {MSB}}_{\delta }((\alpha + t_{i})^{-1} \bmod {p})\right) $$ for random$$t:i \in \mathbb {Z}_p$$, the goal is to recover the hidden number$$\alpha \in \mathbb {Z}_p$$. 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$$\alpha $$ in MIHNP. For any positive integer constant d, let integer$$n=d^{3+o(1)}$$. 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 when$$\delta /\log _2 p>\frac{1}{d\,+\,1}+o(\frac{1}{d})$$. The overall time complexity of attack is polynomial in$$\log _2 p$$, where the complexity of the LLL algorithm grows as$$d^{\mathcal {O}(d)}$$ and the complexity of the Gröbner basis computation grows as$$(2d)^{\mathcal {O}(n^2)}$$. When$$d> 2$$, this asymptotic bound outperforms$$\delta /\log _2 p>\frac{1}{3}$$ 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$$\delta /\log _2 p<\frac{1}{3}$$ is broken. Moreover, we also get the best result for attacking the Inversive Congruential Generator (ICG) up to now.