An Efficient Multiplication Algorithm using Binomial Residue Representation

In this paper, we propose an extension of the algorithm proposed by Bajard, Imbert and Negre in (Bajar et al., 2006), refered as BIN algorithm. We use binomial residue representation of field elements instead of the Lagrange representation of (Bajar et al., 2006). Specifically, every elements in Fpk is represented by a set of residue modulo fixed binomials. We propose two versions of our algorithm, one in general form with a sub-quadratic complexity equal to O(k1.5) operations in Fp. The second one is optimized with the use of FFT. In this case the cost is O(k log(k)) operations in Fp. For fields GF(pk) suitable for elliptic curve cryptography our algorithm roughly improves the time delay of (Bajar et al., 2006) by 45%.