The Modular Inversion Hidden Number Problem (MIHNP), which was proposed at Asiacrypt 2001 by Boneh, Halevi, and Howgrave-Graham, is summarized as follows: Assume that the $\delta$ most significant bits of $z$ are denoted by ${\mathrm {MSB}}_{\delta }(z)$ . The goal is to retrieve the hidden number $\alpha \in \mathbb {Z}_{p}$ given many samples $\left ({t_{i}, {\mathrm {MSB}}_{\delta }((\alpha + t_{i})^{-1} \bmod {p})}\right)$ for random $t_{i} \in \mathbb {Z}_{p}$ . MIHNP is a significant subset of Hidden Number Problems. Eichenauer and Lehn introduced the Inversive Congruential Generator (ICG) in 1986. It is basically characterized as follows: For iterated relations $v_{i+1}=(av^{-1}_{i}+b)\bmod {p}$ with a secret seed $v_{0} \in \mathbb {Z}_{p}$ , each iteration produces $\mathrm {MSB}_{\delta }(v_{i+1})$ where $i \geq 0$ . The ICG family of pseudorandom number generators is a significant subclass of number-theoretic pseudorandom number generators. Sakai-Kasahara scheme is an identity-based encryption (IBE) system proposed by Sakai and Kasahara. It is one of the few commercially implemented identity-based encryption schemes. We explore the Coppersmith approach for solving a class of modular polynomial equations, which is derived from the recovery issue for the hidden number $\alpha$ in MIHNP and the secret seed $v_{0}$ in ICG, respectively. Take a positive integer $n=d^{3+o(1)}$ for some positive integer constant $d$ . We propose a heuristic technique for recovering the hidden number $\alpha$ or secret seed $v_{0}$ with a probability close to 1 when $\delta /\log _{2} p>\frac {1}{d+1}+o\left({\frac {1}{d}}\right)$ . The attack’s total time complexity is polynomial in the order of $\log _{2} p$ , with the complexity of the LLL algorithm increasing as $d^{\mathcal {O}(d)}$ and the complexity of the Gröbner basis computation increasing as $d^{\mathcal {O}(n)}$ . When $d> 2$ , this asymptotic bound surpasses the asymptotic bound $...