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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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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


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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