Fast Scalar Multiplications on Hyperelliptic Curve Cryptosystems

Scalar multiplication is the key operation in hyperelliptic curve cryptosystem. By making use of Euclidean lengths of algebraic integral numbers in a related algebraic integer ring, we discuss the Frobenius expansions of algebraic numbers, theoretically and experimentally show that the multiplier in a scalar multiplication can be reduced and converted into a Frobenius expansion of length approximate to the field extension degree, and then propose an efficient scalar multiplication algorithm. Our method is an extension of the results given by Müller, Smart and Günther et al. If some (optimal) normal basis is employed, then, for some hyperelliptic curves over finite fields, our method will make the computations of scalar multiplications be lessened about fifty-five percent compared with the signed binary method.