Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms

This contribution introduces a class of Galois eld used to achieve fast nite eld arithmetic which we call an Optimal Extension Field OEF This approach is well suited for implementation of public key cryptosystems based on elliptic and hyperelliptic curves Whereas previous reported optimizations focus on nite elds of the form GF p and GF m an OEF is the class of elds GF p for p a prime of special form and m a positive integer Modern RISC workstation proces sors are optimized to perform integer arithmetic on integers of size up to the word size of the processor Our construction employs well known techniques for fast nite eld arithmetic which fully exploit the fast in teger arithmetic found on these processors In this paper we describe our methods to perform the arithmetic in an OEF and the methods to construct OEFs We provide a list of OEFs tailored for processors with and bit word sizes We report on our application of this ap proach to construction of elliptic curve cryptosystems and demonstrate a substantial performance improvement over all previous reported software implementations of Galois eld arithmetic for elliptic curves