Efficient algorithms for elliptic curve cryptosystems

Abstract

This contribution describes three algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm is an entirely new approach which accelerates the multiplications of points which is the core operation in elliptic curve public-key systems. The algorithm works in conjunction with the fc-ary or sliding window method. The algorithm explores computational advantages by computing repeated point doublings directly through closed formulae rather than from individual point doublings. This approach reduces the number of inversions in the underlying finite field at the cost of extra multiplications. For many practical implementations, where field inversion is at least four times as costly as field multiplication, the new approach proofs to be faster than traditional point multiplication methods. The second algorithm deals with efficient inversion in composite Galois fields of the form GF((2n)m). Based on an idea of Itoh and Tsujii, we optimize the algorithm for software implementation of elliptic curves. The algorithm reduced inversion in the composite field to inversion in the subfield GF(2n). The third algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in GF((2n)m). We provide a detailed complexity analysis of the algorithm for the case that subfield arithmetic is performed through table look-up. We apply all three algorithms to an implementation of an elliptic curve system over (GF((216)11). We provide absolute performance measures for the field operations and for an entire point multiplication.