## Abstract

Let ${\mathbb{G}}_{n}$ be the subgroup of elements of odd order in the group ${\mathbb{Z}}_{n}^{\star}$, and let $\mathcal{\mathcal{U}}({\mathbb{G}}_{n})$ 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 $l\ge 1$ 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 $(({\mathbb{G}}_{n},\mathcal{\mathcal{U}}({\mathbb{G}}_{n}))\mid n\in N)$ of computational abelian groups is weakly pseudo-free.
The disadvantage of this result is that the probability ensemble $(\mathcal{\mathcal{U}}({\mathbb{G}}_{n})\mid n\in N)$ is not polynomial-time samplable.
To overcome this disadvantage, we construct a polynomial-time computable function $\nu :D\to N$ (where $D\subseteq {\{0,1\}}^{*}$) and a polynomial-time samplable probability ensemble $({\mathcal{\mathcal{G}}}_{d}\mid d\in D)$ (where ${\mathcal{\mathcal{G}}}_{d}$ is a distribution on ${\mathbb{G}}_{\nu (d)}$ for each $d\in D$) such that the family $(({\mathbb{G}}_{\nu (d)},{\mathcal{\mathcal{G}}}_{d})\mid d\in D)$ of computational abelian groups is weakly pseudo-free.

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