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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus


CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2018: 0.38

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1869-6104
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A certain family of subgroups of ℤ𝑛 is weakly pseudo-free under the general integer factoring intractability assumption

Mikhail AnokhinORCID iD: http://orcid.org/0000-0002-3960-3867
Published Online: 2018-10-17 | DOI: https://doi.org/10.1515/gcc-2018-0007

Abstract

Let 𝔾n be the subgroup of elements of odd order in the group n, 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 l1 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))nN) of computational abelian groups is weakly pseudo-free. The disadvantage of this result is that the probability ensemble (𝒰(𝔾n)nN) is not polynomial-time samplable. To overcome this disadvantage, we construct a polynomial-time computable function ν:DN (where D{0,1}*) and a polynomial-time samplable probability ensemble (𝒢ddD) (where 𝒢d is a distribution on 𝔾ν(d) for each dD) such that the family ((𝔾ν(d),𝒢d)dD) 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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About the article

Received: 2017-11-28

Published Online: 2018-10-17

Published in Print: 2018-11-01


Citation Information: Groups Complexity Cryptology, Volume 10, Issue 2, Pages 99–110, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2018-0007.

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