An Improved Baby Step Giant Step Algorithm for Point Counting of Hyperelliptic Curves over Finite Fields

Matsuo, Kazuto; Chao, Jinhui; Tsujii, Shigeo

doi:10.1007/3-540-45455-1_36

Cited by 27 publications

(32 citation statements)

References 13 publications

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“…(Note that one could reduce the number of "baby steps" by a constant factor, but at the expense of increasing the number of "giant steps" by the same factor.) Very recently, Gaudry and Schost [GS04] designed a space-efficient variant of a baby step-giant step algorithm by Matsuo, Chao and Tsujii [MCT02] that likely can be modified for our application. This should enable us to compute even larger examples, and is future work.…”

Section: Resultsmentioning

confidence: 99%

Point counting on Picard curves in large characteristic

2005

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Section: Resultsmentioning

confidence: 99%

Point counting on Picard curves in large characteristic

2005

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“…Therefore, for medium characteristic p (say 10 9 , see [5]), point counting is easier than for very large p. We mentioned that in the non-small characteristic case, once the group order has been computed modulo some large integer, the computation is finished using a Pollard lambda method. Matsuo, Chao and Tsujii [21] proposed a Babystep/Giant-step algorithm that speeds up this last phase. With this device and using the Cartier-Manin trick, they performed a point counting computation of cryptographical size for a medium characteristic field.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we improve on the methods of [11], so that, combined with the algorithm of [21], we can reach cryptographical size over prime fields. Our improvements are concerned with the construction and the manipulation of torsion elements in the Schoof-like algorithm of [11].…”

Section: Introductionmentioning

confidence: 99%

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Construction of Secure Random Curves of Genus 2 over Prime Fields

Gaudry

Schost

2004

Advances in Cryptology - EUROCRYPT 2004

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Abstract. For counting points of Jacobians of genus 2 curves defined over large prime fields, the best known method is a variant of Schoof's algorithm. We present several improvements on the algorithms described by Gaudry and Harley in 2000. In particular we rebuild the symmetry that had been broken by the use of Cantor's division polynomials and design a faster division by 2 and a division by 3. Combined with the algorithm by Matsuo, Chao and Tsujii, our implementation can count the points on a Jacobian of size 164 bits within about one week on a PC.

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“…Applying optimized baby-step giant-step methods given constraints on |β| is not new; this technique is often used in conjunction with -adic methods, as in [41]. The novelty here is that the constraints are obtained generically.…”

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confidence: 99%

A generic approach to searching for Jacobians

Sutherland

2009

Math. Comp.

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Abstract. We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N 1/12 ) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over low-degree extension fields, where in genus 2 we find Jacobians over F p 2 and trace zero varieties over F p 3 with near-prime orders up to 372 bits in size. For p = 2 61 − 1, the average time to find a group with 244-bit near-prime order is under an hour on a PC.

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An Improved Baby Step Giant Step Algorithm for Point Counting of Hyperelliptic Curves over Finite Fields

Cited by 27 publications

References 13 publications

Point counting on Picard curves in large characteristic

Point counting on Picard curves in large characteristic

Construction of Secure Random Curves of Genus 2 over Prime Fields

A generic approach to searching for Jacobians

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