InforLorV4, PascalFrancis, Corpus, bibRecord, 000769

A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377

Identifieur interne : 000769 ( PascalFrancis/Corpus ); précédent : 000768; suivant : 000770

A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377

Auteurs : Richard P. Brent ; Samuli Larvala ; Paul Zimmermann

Source :

Mathematics of computation [ 0025-5718 ] ; 2003.

RBID : Pascal:03-0388460

Descripteurs français

Pascal (Inist)
- Calcul numérique, Théorie nombre, Polynôme, Représentation irréductible, Génération nombre aléatoire, Carroyage, Méthode réduction, Algorithme rapide, Trinôme irréductible, Exposant Mersenne, Nombre Mersenne, Trinôme primitif.

English descriptors

KwdEn :
- Fast algorithm, Irreducible representation, Irreducible trinomials, Mersenne exponent, Mersenne number, Number theory, Numerical computation, Polynomial, Primitive trinomials, Random number generation, Reduction method, Squaring.

Abstract

The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r /2+O(1) bits of memory and significantly fewer memory references and bit-operations than the standard algorithm. If 2^r - 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al. The two results for r = 3021377 are primitive trinomials of the highest known degree.

Notice en format standard (ISO 2709)

Pour connaître la documentation sur le format Inist Standard.

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A14	`01`			`@1 Oxford University Computing Laboratory, Wolfson Building, Parks Road @2 Oxford, OX1 3QD @3 GBR @Z 1 aut.`
A14	`02`			`@1 Helsinki University of Technology @2 Espoo @3 FIN @Z 2 aut.`
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C01	`01`		`ENG`	@0 The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r /2+O(1) bits of memory and significantly fewer memory references and bit-operations than the standard algorithm. If 2^r - 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al. The two results for r = 3021377 are primitive trinomials of the highest known degree.
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C03	`04`	`X`	`SPA`	`@0 Representación irreductible @5 04`
C03	`05`	`X`	`FRE`	`@0 Génération nombre aléatoire @5 05`
C03	`05`	`X`	`ENG`	`@0 Random number generation @5 05`
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C03	`06`	`X`	`SPA`	`@0 Cuadriculación @5 07`
C03	`07`	`X`	`FRE`	`@0 Méthode réduction @5 08`
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C03	`07`	`X`	`SPA`	`@0 Método reducción @5 08`
C03	`08`	`X`	`FRE`	`@0 Algorithme rapide @5 09`
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C03	`08`	`X`	`SPA`	`@0 Algoritmo rápido @5 09`
C03	`09`	`X`	`FRE`	`@0 Trinôme irréductible @4 CD @5 96`
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C03	`11`	`X`	`FRE`	`@0 Nombre Mersenne @4 CD @5 98`
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N21				`@1 272`

Format Inist (serveur)

NO :	PASCAL 03-0388460 INIST
ET :	A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377
AU :	BRENT (Richard P.); LARVALA (Samuli); ZIMMERMANN (Paul)
AF :	Oxford University Computing Laboratory, Wolfson Building, Parks Road/Oxford, OX1 3QD/Royaume-Uni (1 aut.); Helsinki University of Technology/Espoo/Finlande (2 aut.); LORIA/INRIA Lorraine, 615 rue du Jardin Botanique, BP 101/54602 Villers-lès-Nancy/France (3 aut.)
DT :	Publication en série; Niveau analytique
SO :	Mathematics of computation; ISSN 0025-5718; Coden MCMPAF; Etats-Unis; Da. 2003; Vol. 72; No. 243; Pp. 1443-1452; Bibl. 27 ref.
LA :	Anglais
EA :	The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r /2+O(1) bits of memory and significantly fewer memory references and bit-operations than the standard algorithm. If 2^r - 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al. The two results for r = 3021377 are primitive trinomials of the highest known degree.
CC :	001A02C02; 001A02C03; 001A02I02; 001D02A05
FD :	Calcul numérique; Théorie nombre; Polynôme; Représentation irréductible; Génération nombre aléatoire; Carroyage; Méthode réduction; Algorithme rapide; Trinôme irréductible; Exposant Mersenne; Nombre Mersenne; Trinôme primitif
ED :	Numerical computation; Number theory; Polynomial; Irreducible representation; Random number generation; Squaring; Reduction method; Fast algorithm; Irreducible trinomials; Mersenne exponent; Mersenne number; Primitive trinomials
SD :	Cálculo numérico; Teoría números; Polinomio; Representación irreductible; Generación número aleatorio; Cuadriculación; Método reducción; Algoritmo rápido; Número real
LO :	INIST-5227.354000118344630190
ID :	03-0388460

Links to Exploration step

Pascal:03-0388460

Le document en format XML

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<server><NO>PASCAL 03-0388460 INIST</NO>
<ET>A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377</ET>
<AU>BRENT (Richard P.); LARVALA (Samuli); ZIMMERMANN (Paul)</AU>
<AF>Oxford University Computing Laboratory, Wolfson Building, Parks Road/Oxford, OX1 3QD/Royaume-Uni (1 aut.); Helsinki University of Technology/Espoo/Finlande (2 aut.); LORIA/INRIA Lorraine, 615 rue du Jardin Botanique, BP 101/54602 Villers-lès-Nancy/France (3 aut.)</AF>
<DT>Publication en série; Niveau analytique</DT>
<SO>Mathematics of computation; ISSN 0025-5718; Coden MCMPAF; Etats-Unis; Da. 2003; Vol. 72; No. 243; Pp. 1443-1452; Bibl. 27 ref.</SO>
<LA>Anglais</LA>
<EA>The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r /2+O(1) bits of memory and significantly fewer memory references and bit-operations than the standard algorithm. If 2<sup>r</sup>
 - 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al. The two results for r = 3021377 are primitive trinomials of the highest known degree.</EA>
<CC>001A02C02; 001A02C03; 001A02I02; 001D02A05</CC>
<FD>Calcul numérique; Théorie nombre; Polynôme; Représentation irréductible; Génération nombre aléatoire; Carroyage; Méthode réduction; Algorithme rapide; Trinôme irréductible; Exposant Mersenne; Nombre Mersenne; Trinôme primitif</FD>
<ED>Numerical computation; Number theory; Polynomial; Irreducible representation; Random number generation; Squaring; Reduction method; Fast algorithm; Irreducible trinomials; Mersenne exponent; Mersenne number; Primitive trinomials</ED>
<SD>Cálculo numérico; Teoría números; Polinomio; Representación irreductible; Generación número aleatorio; Cuadriculación; Método reducción; Algoritmo rápido; Número real</SD>
<LO>INIST-5227.354000118344630190</LO>
<ID>03-0388460</ID>
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A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377

A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377

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