Paper 2017/1131 Original: eprint.iacr.org/2017/1131 ↗

A Certain Family of Subgroups of $\mathbb Z_n^\star$ Is Weakly Pseudo-Free under the General Integer Factoring Intractability Assumption

Abstract

Let $\mathbb G_n$ be the subgroup of elements of odd order in the group $\mathbb Z_n^\star$ and let $\mathcal U(\mathbb G_n)$ be the uniform probability distribution on $\mathbb G_n$. In this paper, we establish a probabilistic polynomial-time reduction from finding a nontrivial divisor of a composite number $n$ to finding a nontrivial relation between $l$ elements chosen independently and uniformly at random from $\mathbb G_n$, where $l\ge1$ is given in unary as a part of the input. Assume that finding a nontrivial divisor of a random number in some set $N$ of composite numbers (for a given security parameter) is a computationally hard problem. Then, using the above-mentioned reduction, we prove that the family $\{(\mathbb G_n,\mathcal U(\mathbb G_n))\}_{n\in N}$ of computational abelian groups is weakly pseudo-free. The disadvantage of this result is that the probability ensemble $\{\mathcal U(\mathbb G_n)\}_{n\in N}$ is not polynomial-time samplable. To overcome this disadvantage, we construct a polynomial-time computable function $\nu\colon D\to N$ (where $D\subseteq\{0,1\}^*$) and a polynomial-time samplable probability ensemble $\{\mathcal G_d\}_{d\in D}$ (where $\mathcal G_d$ is a distribution on $\mathbb G_{\nu(d)}$ for each $d\in D$) such that the family $\{(\mathbb G_{\nu(d)},\mathcal G_d)\}_{d\in D}$ of computational abelian groups is weakly pseudo-free.