Abstract
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 algorithm for point counting of hyperelliptic curves. Their algorithm consists of two parts: firstly to compute the residue modulo an 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 algorithm. 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 in 16 hours on Alpha 21264/667MHz.
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Matsuo, K., Chao, J., Tsujii, S. (2002). An Improved Baby Step Giant Step Algorithm for Point Counting of Hyperelliptic Curves over Finite Fields. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_36
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DOI: https://doi.org/10.1007/3-540-45455-1_36
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