On the optimal computation of finite field exponentiation

It has been shown that the optimal computation of finite field exponentiation is closely related to the problem of finding a suitable addition chain with the shortest possible length. However, obtaining the shortest addition chain for a given arbitrary exponent is a NP-hard problem. Hence in general, we are forced to use some kind of heuristic in order to compute field exponentiation with a semi-optimal number of underlying arithmetic operations. In this paper we present a novel heuristic for that problem which is based on an immune artificial system strategy. The results obtained by our scheme yield the shortest reported lengths for the exponents typically used when computing field multiplicative inverses for error-correcting and elliptic curve cryptographic applications.