Multiplication in Cyclotomic Rings and its Application to Finite Fields

A representation of finite fields that has proved useful when implementing finite field arithmetic in hardware is based on an isomorphism between subrings and fields. In this paper, we present an unified formulation for multiplication in cyclotomic rings and cyclotomic fields in that most arithmetic operations are done on vectors. From this formulation we can generate optimized algorithms for multiplication. For example, one of the proposed algorithms requires approximately half the number of coordinate-level multiplications at the expense of extra coordinate-level additions. Our method is then applied to the finite fields GF(q m) to further reduce the number of operations. We then present optimized algorithms for multiplication in finite fields with type-I and type-II optimal normal bases.