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On Deterministic Reduction of Factoring Integers to Computing the Exponents of Elements in Modular Group

Pomykała J.

Fundamenta Informaticae

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2017

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Vol. 152, nr 3

289--295

EN

In the paper we prove that all but at most x/A(x) positive integers n ≤ x can be completely factored in deterministic polynomial time C(x), querying the prime decomposition exponent oracle at most D(x) times. The functions A(x), C(x) and D(x) have the polynomial growth (of log x) at infinity.

2

Large Sieve, Miller-Rabin Compositeness Witnesses and Integer Factoring Problem

Durnoga K., Pomykała J.

Fundamenta Informaticae

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2017

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Vol. 156, nr 2

179--185

EN

G. Miller in his seminal paper from the mid 1970s has proven that the problem of factoring integers reduces to computing Euler’s totient function Φ under the Extended Riemann Hypothesis. We show, unconditionally, that such a deterministic polynomial reduction exists for a large class of integers.

3

Small Generating Sets and DLPC Problem

Pomykała J.

Fundamenta Informaticae

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2016

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Vol. 145, nr 2

143--150

EN

In the paper we investigate the set of odd, squarefree positive integers n that can be factored completely in polynomial timeO(log6+ε n), given the prime decomposition of orders ordnb for b ≤ logη n, (η > 2), which is closely related to DLPC problem. We prove that the number of n ≤ x that may not be factored in deterministic time O(log6+εn), is at most (η - 2)-1x(log x)-c(η-2), for some c > 0 and arbitrary ε > 0.

4

Dedekind Sums, Mean Square Value of L-Functions at s = 1 and Upper Bounds on Relative Class Numbers

Louboutin S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2016

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Vol. 64, no. 2/3

165--174

EN

Explicit formulas for the quadratic mean value at s = 1 of the Dirichlet L-functions associated with the set Xf¯ of the ϕ(f)/2 odd Dirichlet characters mod f are known. They have been used to obtain explicit upper bounds for relative class numbers of cyclotomic number fields. Here we present a generalization of these results: we show that explicit formulas for quadratic mean values at s = 1 of Dirichlet L-functions associated with subsets of Xf¯ can be obtained. As an application we use them to obtain explicit upper bounds for relative class numbers of imaginary subfields of cyclotomic number fields.

5

Explicit Formulas for the Fourth Moment of Exponential Sums and Dirichlet Character Sums

Luo A., Chen H., Chen J., Cai G.

Przegląd Elektrotechniczny

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2013

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R. 89, nr 3b

122--124

EN

Exponential sums and Dirichlet character sums have important applications in electronic science and technology, such as ecommerce, e-government and so on. A new proof is proposed and some interesting computational formula and transformation formula are obtained.

PL

W artykule przedstawiono zagadnienie rozwiązywania wyrażenia zawierającego sumy wykładnicze oraz sumy operatorów Dirichleta. Zaproponowano nowy dowód, formułę obliczeń oraz formułę transformacji.