A double large prime variation for small genus hyperelliptic index calculus

Gaudry, Pierrick; Thomé, Emmanuel; Thériault, Nicolas; Diem, Claus

doi:10.1090/s0025-5718-06-01900-4

Cited by 89 publications

(108 citation statements)

References 26 publications

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“…This smaller value of N max only lowers the time for the construction of the tree by a constant factor but decreases the storage requirements substantially. This is analogous to the situation in [GTTD07].…”

Section: C U Forms a Generating System Conditionally To Any Outcmentioning

confidence: 52%

“…A corresponding result for hyperelliptic curves in imaginary quadratic representation and with cyclic degree 0 class groups was obtained in [GTTD07], and on a heuristic basis an obvious generalization of the algorithm for the theorem in [GTTD07] already gives rise to the result in Theorem 1. A related work is [Nag07], in which a similar algorithm is given.…”

Section: Introductionmentioning

confidence: 98%

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On the discrete logarithm problem in class groups of curves

Diem

2010

Math. Comp.

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Abstract. We study the discrete logarithm problem in degree 0 class groups of curves over finite fields, with particular emphasis on curves of small genus. We prove that for every fixed g ≥ 2, the discrete logarithm problem in degree 0 class groups of curves of genus g can be solved in an expected time ofÕ(q 2− 2 g ), where F q is the ground field. This result generalizes a corresponding result for hyperelliptic curves given in imaginary quadratic representation with cyclic degree 0 class group, and just as this previous result, it is obtained via an index calculus algorithm with double large prime variation.Generalizing this result, we prove that for fixed g 0 ≥ 2 the discrete logarithm problem in class groups of all curves C/F q of genus g ≥ g 0 can be solved in an expected time ofÕ((q g ) 2 g 0 (1− 1 g 0 ) ) and in an expected time ofAs a complementary result we prove that for any fixed n ∈ N with n ≥ 2 the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields F q n , q a prime power, can be solved in an expected time of O(q 2− 2 n ). Furthermore, we give an algorithm for the efficient construction of a uniformly randomly distributed effective divisor of a specific degree, given the curve and its L-polynomial.

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Section: C U Forms a Generating System Conditionally To Any Outcmentioning

confidence: 52%

Section: Introductionmentioning

confidence: 98%

On the discrete logarithm problem in class groups of curves

Diem

2010

Math. Comp.

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“…They can be used for factoring large integers [14] and for computing discrete logarithms in finite fields [2,10,1] and in some elliptic or hyperelliptic curve groups [6,8,5,9,7].…”

Section: Introductionmentioning

confidence: 99%

Faster Index Calculus for the Medium Prime Case Application to 1175-bit and 1425-bit Finite Fields

Joux

2013

Advances in Cryptology – EUROCRYPT 2013

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Abstract. Many index calculus algorithms generate multiplicative relations between smoothness basis elements by using a process called Sieving. This process allows to filter potential candidate relations very quickly, without spending too much time to consider bad candidates. However, from an asymptotic point of view, there is not much difference between sieving and straightforward testing of candidates. The reason is that even when sieving, some small amount time is spend for each bad candidates. Thus, asymptotically, the total number of candidates contributes to the complexity. In this paper, we introduce a new technique: Pinpointing, which allows us to construct multiplicate relations much faster, thus reducing the asymptotic complexity of relations' construction. Unfortunately, we only know how to implement this technique for finite fields which contain a medium-sized subfield. When applicable, this method improves the asymptotic complexity of the index calculus algorithm in the cases where the sieving phase dominates. In practice, it gives a very interesting boost to the performance of state-of-the-art algorithms. We illustrate the feasability of the method with a discrete logarithm record in medium prime finite fields of sizes 1175 bits and 1425 bits.

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“…The existence of various index calculus algorithms has centered attention on hyperelliptic curves of genus g ≤ 3 [15, 18,24]. We similarly focus on the hyperelliptic case, although our results may be applied to any family of low genus curves.…”

Section: Introductionmentioning

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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A double large prime variation for small genus hyperelliptic index calculus

Cited by 89 publications

References 26 publications

On the discrete logarithm problem in class groups of curves

On the discrete logarithm problem in class groups of curves

Faster Index Calculus for the Medium Prime Case Application to 1175-bit and 1425-bit Finite Fields

A generic approach to searching for Jacobians

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