Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials

TL;DR

This paper presents efficient algorithms for finding trinomials over GF(2) with large irreducible or primitive factors, providing new examples and applications for finite field representations.

Contribution

It introduces algorithms for discovering trinomials with large primitive factors and supplies explicit examples for various degrees, enhancing finite field implementations.

Findings

Identified trinomials with primitive factors for all Mersenne exponents in a specified range.

Constructed trinomials with primitive factors of degrees 2^k for 3 ≤ k ≤ 12.

Demonstrated applications of these trinomials in efficient finite field representations.

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.