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arXiv · 2105.06013

Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials

Richard P. Brent·Paul Zimmermann
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Abstract

Consider polynomials over ${\rm GF}(2)$. 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 $r = \pm 3 \bmod 8$ 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 ${\rm GF}(2^r)$. 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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Richard P. Brent, Paul Zimmermann. 2021-05-13. Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials. https://doi.org/10.1090/fic/041

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