Baby Step Giant Step Algorithms in Point Counting of Hyperelliptic Curves

SUMMARY Counting the number of points of Jacobian varieties of hyperelliptic curves over finite fields is necessary for construction of hyperelliptic curve cryptosystems. Recently Gaudry and Harley proposed a practical scheme for point counting of hyperelliptic curves. Their scheme consists of two parts: firstly to compute the residue modulo a positive integer m of the order of a given Jacobian variety, and then search for the order by a square-root algorithm. In particular, the parallelized Pollard’s lambda-method was used as the square-root algorithm, which took 50CPU days to compute an order of 127 bits. This paper shows a new variation of the baby step giant step algorithm to improve the square-root algorithm part in the Gaudry-Harley scheme. With knowledge of the residue modulo m of the characteristic polynomial of the Frobenius endomorphism of a Jacobian variety, the proposed algorithm provides a speed up by a factor m, instead of √ m in square-root algorithms. Moreover, implementation results of the proposed algorithm is presented including a 135-bit prime order computed about 15 hours on Alpha 21264/667 MHz and a 160-bit order.