Normal Basis Multiplication Algorithms for GF(2n) (Full Version)

In this paper, we propose a new normal basis multiplication algorithm for GF(2n). This algorithm can be used to design not only fast software algorithms but also low complexity bit-parallel multipliers in some GF(2n)s. Especially, for some values of n that no Gaussian normal basis exists in GF(2n), i.e., 8|n, this algorithm provides an alternative way to construct low complexity normal basis multipliers. Two improvements on a recently proposed software normal basis multiplication algorithm are also presented. Time and memory complexities of these normal basis multiplication algorithms are compared with respect to software performance. It is shown that they have some specific behavior in different applications. For example, GF(2571) is one of the five binary fields recommended by NIST for ECDSA (Elliptic Curve Digital Signature Algorithm) applications. In this field, our experiments show that the new algorithm is even faster than the polynomial basis Montgomery multiplication algorithm: 525 us v. 819 us