A Short Note on Irreducible Trinomials in Binary Fields

In this paper, we analyse the irreducibility of trinomials defined over F2[X]. For elliptic curve cryptography, prime extensions fields F2p are recommended to avoid current attacks like the Weil descent attack [3]. One often represents F2p as a quotient F2[X]/(f(X)), where f is an irreducible polynomial over F2 of degree p. Performance reasons impose that irreducible polynomials have the shortest number of non zero terms. More precisely, the reduction polynomial plays a fundamental role in the basic field operations and particularly in modular reductions. This in turn is related to the number of carries in each modular reduction and this is where the structure of the polynomial plays a crucial role. Recommended binary fields for elliptic curve cryptosystems, as in norms IEEEP1363, ANSI X9.62 or SEC1, are produced together with irreducible trinomials or pentanomials when no irreducible trinomials exist. However criteria for existence or non-existence of irreducible trinomials or pentanomials over given extensions are not clearly stated in the cryptographic literature (but cf. [4, § 4.5.2]). The purpose of this short note is a first attempt to clarify some parts of this problem. We propose, following the work of [1, 6], to give a proof of the following theorem.