## Cryptology ePrint Archive: Report 2017/1131

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

Mikhail Anokhin

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.

Category / Keywords: foundations / families of computational groups, weak pseudo-freeness, abelian groups, factoring

Original Publication (with major differences): Groups, Complexity, Cryptology, 10(2):99-110, November 2018
DOI:
10.1515/gcc-2018-0007

Date: received 22 Nov 2017, last revised 6 Nov 2018

Contact author: anokhin at mccme ru

Available format(s): PDF | BibTeX Citation

Note: We have added a reference to the journal version of the paper, changed equation numbering style, and made some other minor changes.

Short URL: ia.cr/2017/1131

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