Algorithms for finding almost irreducible and almost primitive trinomials
Résumé
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.