A certain family of subgroups of ℤ𝑛⋆ is weakly pseudo-free under the general integer factoring intractability assumption

Mikhail Anokhin

doi:10.1515/gcc-2018-0007

Published by De Gruyter October 17, 2018

A certain family of subgroups of ℤ_𝑛^⋆ is weakly pseudo-free under the general integer factoring intractability assumption

Mikhail Anokhin

From the journal Groups Complexity Cryptology

https://doi.org/10.1515/gcc-2018-0007

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Abstract

Let 𝔾n be the subgroup of elements of odd order in the group ℤn⋆, and let 𝒰⁢(𝔾n) be the uniform probability distribution on 𝔾n. 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 𝔾n, where l≥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 ((𝔾n,𝒰(𝔾n))∣n∈N) of computational abelian groups is weakly pseudo-free. The disadvantage of this result is that the probability ensemble (𝒰(𝔾n)∣n∈N) is not polynomial-time samplable. To overcome this disadvantage, we construct a polynomial-time computable function ν:D→N (where D⊆{0,1}*) and a polynomial-time samplable probability ensemble (𝒢d∣d∈D) (where 𝒢d is a distribution on 𝔾ν⁢(d) for each d∈D) such that the family ((𝔾ν⁢(d),𝒢d)∣d∈D) of computational abelian groups is weakly pseudo-free.

Keywords: Family of computational groups; weakly pseudo-free family of computational groups; abelian group; general integer factoring intractability assumption

MSC 2010: 68Q17; 94A60; 11Y05; 20K99

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Received: 2017-11-28

Published Online: 2018-10-17

Published in Print: 2018-11-01

A certain family of subgroups of ℤ_𝑛^⋆ is weakly pseudo-free under the general integer factoring intractability assumption

Abstract

References

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