A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377
Identifieur interne : 007B78 ( Main/Exploration ); précédent : 007B77; suivant : 007B79A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377
Auteurs : Richard P. Brent [Royaume-Uni] ; Samuli Larvala [Finlande] ; Paul Zimmermann [France]Source :
- Mathematics of computation [ 0025-5718 ] ; 2003.
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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 2r - 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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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.</div></front></TEI><affiliations><list><country><li>Finlande</li><li>France</li><li>Royaume-Uni</li></country><region><li>Grand Est</li><li>Lorraine (région)</li></region><settlement><li>Villers-lès-Nancy</li></settlement></list><tree><country name="Royaume-Uni"><noRegion><name sortKey="Brent, Richard P" sort="Brent, Richard P" uniqKey="Brent R" first="Richard P." last="Brent">Richard P. 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