Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials

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 = ±3 mod 8 in the range 5 < r < 10, although there is no irreducible trinomial of degree r. We also give trinomials with a primitive factor of degree r = 2 for 3 ≤ k ≤ 12. These trinomials enable efficient representations of the finite field GF(2). We show how trinomials with large primitive factors can be used efficiently in applications where primitive trinomials would normally be used.