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Algorithms for finding almost irreducible and almost primitive trinomials

Richard P. Brent Paul Zimmermann 1 
1 SPACES - Solving problems through algebraic computation and efficient software
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
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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Submitted on : Tuesday, September 26, 2006 - 9:40:41 AM
Last modification on : Friday, February 4, 2022 - 3:09:51 AM

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  • HAL Id : inria-00099724, version 1
  • ARXIV : 2105.06013



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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