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Counting rational points on Jacobian varieties of hyperellip- tic curves over finite fields is very important for constructing hyperellip- tic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running times. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y2 = x5 + ax over given large prime fields p, e.g. 80-bit fields. For these curves, we also determine the necessary condition to be suitable for HCC, that is, to satisfy that the order of the Jacobian group is of the form l ¢ c where l is a prime number greater than about 2160 and c is a very small integer. We show some examples of suitable curves for HCC obtained by using our algorithm. We also treat curves of type y2 = x5 + a where a is not square in p.

Counting Points for Hyperelliptic Curves of Type y 2=x 5+ax over Finite Prime Fields