We present a family of hyperelliptic curves whose Jacobians are suitable for cryptographic use, and whose parameters can be specified in a highly efficient way. This is done via complex multiplication and identity-based parameters. We also present some novel computational shortcuts for these families.
1. INTRODUCTION. The groups of invertible matrices over finite fields are among the first groups we meet in a beginning course in modern algebra. Eventually, we find out about simple groups and that the unique simple group of order 168 has two representations as a group of matrices. And this is where we learn that the group of 2 × 2 unimodular matrices over… (More)
Gauss's Cyclotomic Formula [3, pp.425-428, p.467] is a neglected mathematical wonder. Theorem 1.1. (Gauss) Let p be an odd prime and set p ′ = (−1) (p−1)/2 p. Then there exist integer polynomials R(x, y) and S(x, y) such that 4(x p + y p) x + y = R(x, y) 2 − p ′ S(x, y) 2 .
Let A be a k × k matrix over a ring R; let GM (A, R) be the digraph with vertex set R k , and an arc from v to w if and only if w = Av. In this paper, we determine the numbers and lengths of the cycles of GM (A, R) for k = 2 in the following two cases. (a) R = F F q , the q–element finite field, and (b) R = Z Z/nZ Z and GCD(n, det(A)) = 1. This extends… (More)