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I. A. Kruglov

Shifted products of independent random variables with values in finite groups

We consider sequences of random variables

x ^(N) = ζ₁ · ζ₂ · . . . · ζ_N , ω^(N) = ξ₁ · ζ₁ · ξ₂ · ζ₂ · . . . · ξ_N · ζ _N ,

where (ξ_N , ζ_N), N ≥ 1, is a sequence of independent identically distributed random variables with values in the Cartesian product G × G of a finite group (G; ·). We investigate the degree of dependence of the random variables x^(N) and ω^(N). Such problems arise in the study of a class of information security algorithms. In connection to this problem, we study the random variable with values in G whose distribution coincides with the conditional distribution of the random variable ω^(N) under condition that x^(N) = a, where a is such that P(x^(N) = a) > 0. We give conditions of convergence and limit distributions of as N → ∞, where s_N is a sequence of integers tending to infinity in such a way that P(x^(sN) = a) > 0.

Discrete Mathematics and Applications, Walter de Gruyter

Print ISSN: 0924-9266
Volume: 17, 04/2007
Pages: 37 - 46

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