Baby Step Giant Step Algorithms in Point Counting of Hyperelliptic Curves

Kazuto MATSUO; Jinhui CHAO; Shigeo TSUJII

Baby Step Giant Step Algorithms in Point Counting of Hyperelliptic Curves

Kazuto MATSUO, Jinhui CHAO, Shigeo TSUJII

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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 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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E86-A No.5 pp.1127-1134

Publication Date: 2003/05/01

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Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)

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Kazuto MATSUO, Jinhui CHAO, Shigeo TSUJII, "Baby Step Giant Step Algorithms in Point Counting of Hyperelliptic Curves" in IEICE TRANSACTIONS on Fundamentals, vol. E86-A, no. 5, pp. 1127-1134, May 2003, doi: .
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 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 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.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/e86-a_5_1127/_p

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@ARTICLE{e86-a_5_1127,
author={Kazuto MATSUO, Jinhui CHAO, Shigeo TSUJII, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Baby Step Giant Step Algorithms in Point Counting of Hyperelliptic Curves},
year={2003},
volume={E86-A},
number={5},
pages={1127-1134},
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 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 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.},
keywords={},
doi={},
ISSN={},
month={May},}

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TY - JOUR
TI - Baby Step Giant Step Algorithms in Point Counting of Hyperelliptic Curves
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1127
EP - 1134
AU - Kazuto MATSUO
AU - Jinhui CHAO
AU - Shigeo TSUJII
PY - 2003
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E86-A
IS - 5
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - May 2003
AB - 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 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.
ER -