Wyniki wyszukiwania - BazTech

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Optymalizacja procesu poszukiwania nierozkładalnych wielomianów osadowych

Augustynowicz P., Paszkiewicz A.

Przegląd Telekomunikacyjny + Wiadomości Telekomunikacyjne

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2018

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nr 8-9

786--789, CD

PL

W artykule omówiono optymalizację procesu poszukiwania nierozkładalnego wielomianu osadowego nad GF(2) zadanego, wysokiego stopnia poprzez wybór odpowiedniego punktu startowego i sposobu kolejkowania badanych wielomianów. Postawione tezy sprawdzono empirycznie oraz zilustrowano odpowiednimi przykładami.

EN

This article addresses the problem of sedimentary irreducible polynomials over GF(2) search process optimization. Different queueing strategies and starting points were tested for optimality. Obtained results were empirically verified by computational experiments.

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Empiryczna weryfikacja hipotezy o stopniu wewnętrznym osadowych wielomianów nieprzywiedlnych nad GF(2)

Augustynowicz P., Paszkiewicz A.

Przegląd Telekomunikacyjny + Wiadomości Telekomunikacyjne

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2017

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nr 8-9

799--802, CD

PL

Artykuł stanowi sprawozdanie z rozległego eksperymentu obliczeniowego dotyczącego weryfikacji hipotezy o osadowych wielomianach nierozkładalnych nad GF(2). Prezentujemy pełną listę wszystkich nierozkładalnych wielomianów osadowych, przy czym najmłodszych leksykograficznie o stopniu wewnętrznym nie większym, niż 21 nie spełniających znanej hipotezy.

EN

This paper is a report of a vast computational experiment concerning verification of a hypothesis on irreducible sedimentary polynomials over GF(2). We present a complete list of all irreducible sedimentary polynomials but lexicographically minimal up to inner degree not exceeding 21 which contradict the famous hypothesis.

3

New Trinomials Xⁿ + X + 1 and Xⁿ + X ² + 1 Irreducible over GF(2)

Bartosik P., Paszkiewicz A.

Electronics and Telecommunications Quarterly

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2009

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Vol. 55, No 2

355-361

EN

We extend the limit of investigations for trinomials irreducible over GF(2), having the form Xⁿ + g(X), where deg (g(X)) = 1 or deg (g(X)) = 2 and complete the existing list of irreducible trinomials with that form by a dozen of new elements. We checked all degrees n below 500000 while searching for that polynomials. A large part of computations were performed by a new programming package developed especially for computations in finite fields with characteristic two. This package is a bit more than twice faster than Shoup's NTL package for trinomials and about six times faster than NTL in the case of pentanomials. We also complete the list of Mersenne irreducible polynomials for which a trinomial does not exist by pentanomials and irreducible polynomials which are lexicographicaly youngest.

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Irreducible Pentanomials and their Applications to Effective Implementations of Arithmetic in Binary Fields

Paszkiewicz A.

Electronics and Telecommunications Quarterly

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2009

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Vol. 55, No 2

363-375

EN

There are only few main classes of irreducible polynomials which are used for designing arithmetic in Galois Fields with characteristic two. These are: irreducible trinomials, pentanomials, all-one polynomials (AOP) and equally spaced irreducible polynomials (ESP). The most critical and time consuming arithmetical operations in Galois Fields are multiplication and modular reduction. A special structure of the modular polynomial defining the arithmetic allows significant speedup of these operations. The best class of binary irreducible polynomials are trinomials, but for about one half of degrees below 30000 an irreducible trinomial does not exist. By exhaustive computation we established that for all degrees n between 4 and 30000 an irreducible pentanomial always exists. Therefore using irreducible pentanomials for defining the arithmetic of Galois Fields have practical interest. In the paper we investigate a function describing the number of binary irreducible pentanomials of a given degree n greater than 3 and study its properties. We also analyze the complexity of a circuit (the number of XOR and AND gates) implementing multiplication in the finite field represented by general irreducible pentanomials.