Efficient multiplication using type 2 optimal normal bases

Abstract

In this paper we propose a new structure for multiplication using optimal normal bases of type 2. The multiplier uses an efficient linear transformation to convert the normal basis representations of elements of double-struck F signqn to suitable polynomials of degree at most n over double-struck F signq. These polynomials are multiplied using any method which is suitable for the implementation platform, then the product is converted back to the normal basis using the inverse of the above transformation. The efficiency of the transformation arises from a special factorization of its matrix into sparse matrices. This factorization -which resembles the FFT factorization of the DFT matrix -allows to compute the transformation and its inverse using O(n log n) operations in double-struck F signq, rather than O(n 2) operations needed for a general change of basis. Using this technique we can reduce the asymptotic cost of multiplication in optimal normal bases of type 2 from 2M(n) + O(n) reported by Gao et al. (2000) to M(n) + O(n log n) operations in double-struck F signq, where M(n) is the number of double-struck F signq-operations to multiply two polynomials of degree n -1 over double-struck F signq. We show that this cost is also smaller than other proposed multipliers for n > 160, values which are used in elliptic curve cryptography. © Springer-Verlag Berlin Heidelberg 2007.