Montgomery multiplier and squarer for a class of finite fields

Montgomery multiplication in GF(2^m) is defined by a(x)b(x)r^-1(x) mod f(x), where the field is generated by a root of the irreducible polynomial f(x), a(x) and b(x) are two field elements in GF(2^m), and r(x) is a fixed field element in GF(2 ^m). In this paper, first, a slightly generalized Montgomery multiplication algorithm in GF(2^m) is presented. Then, by choosing r(x) according to f (x), we show that efficient architectures of bit-parallel Montgomery multiplier and squarer can be obtained for the fields generated with an irreducible trinomial. Complexities of the Montgomery multiplier and squarer in terms of gate counts and time delay of the circuits are investigated and found to be as good as or better than that of previous proposals for the same class of fields