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Conference Papers Year : 2003

Algorithms for finding almost irreducible and almost primitive trinomials

Abstract

Consider polynomials over $\GF(2)$. We describe efficient algorithms for finding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree~$r$ for all Mersenne exponents $r = \pm 3 \mmod 8$ in the range $5 < r <10^7$, although there is no irreducible trinomial of degree~$r$. We also give trinomials with a primitive factor of degree $r = 2^k$ for $3 \le k \le 12$. These trinomials enable efficient representations of the finite field $\GF(2^r)$. We show how trinomials with large primitive factors can be used efficiently in applications where primitive trinomials would normally be used.

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Other [cs.OH]

Dates and versions

inria-00099724 , version 1 (26-09-2006)

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Richard P. Brent, Paul Zimmermann. Algorithms for finding almost irreducible and almost primitive trinomials. Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams, 2003, Banff, Canada, France. ⟨inria-00099724⟩
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