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- W2897601298 abstract "Abstract Let {mathbb{G}_{n}} be the subgroup of elements of odd order in the group {mathbb{Z}^{star}_{n}} , 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 {lgeq 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{U}(mathbb{G}_{n}))mid nin N)} of computational abelian groups is weakly pseudo-free. The disadvantage of this result is that the probability ensemble {(mathcal{U}(mathbb{G}_{n})mid nin N)} is not polynomial-time samplable. To overcome this disadvantage, we construct a polynomial-time computable function {nucolon Dto N} (where {Dsubseteq{0,1}^{*}} ) and a polynomial-time samplable probability ensemble {(mathcal{G}_{d}mid din D)} (where {mathcal{G}_{d}} is a distribution on {mathbb{G}_{nu(d)}} for each {din D} ) such that the family {((mathbb{G}_{nu(d)},mathcal{G}_{d})mid din D)} of computational abelian groups is weakly pseudo-free." @default.
- W2897601298 created "2018-10-26" @default.
- W2897601298 creator A5029443980 @default.
- W2897601298 date "2018-11-01" @default.
- W2897601298 modified "2023-09-26" @default.
- W2897601298 title "A certain family of subgroups of ℤ𝑛⋆ is weakly pseudo-free under the general integer factoring intractability assumption" @default.
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- W2897601298 doi "https://doi.org/10.1515/gcc-2018-0007" @default.
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