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An L(1/3 + ε) algorithm for the discrete logarithm problem in low degree curves
 Draft, 2006, http://www.lix.polytechnique.fr/Labo/Andreas.Enge/vorabdrucke/l13.pdf. References in notes
"... Abstract. The discrete logarithm problem in Jacobians of curves of high genus g over finite fields Fq is known to be computable with subexponential complexity Lqg(1/2, O(1)). We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and su ..."
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Cited by 11 (3 self)
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Abstract. The discrete logarithm problem in Jacobians of curves of high genus g over finite fields Fq is known to be computable with subexponential complexity Lqg(1/2, O(1)). We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of Lqg(1/3, O(1)), and a discrete logarithm computation takes subexponential time of Lqg(1/3 + ε,o(1)) for any positive ε. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms. 1
Smooth ideals in hyperelliptic function fields
 Math.Comp., posted on October 4, 2001, PII
"... Abstract. Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of highgenus hyperelliptic curves over finite fields. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently de ..."
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Cited by 9 (7 self)
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Abstract. Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of highgenus hyperelliptic curves over finite fields. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense. We explicitly show how these density results can be derived. All proofs are purely combinatorial and do not exploit analytic properties of generating functions. 1.
An L(1/3) Discrete Logarithm Algorithm for Low Degree Curves, 2009, http://hal.inria.fr/inria00383941/en/, Accepted for publication in Journal of Cryptology
"... We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in X and Y are low with respect to their genera. The finite base fields Fq are arbitrary, but their sizes should not grow too fast compared to the genus. For such families, the g ..."
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Cited by 4 (0 self)
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We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in X and Y are low with respect to their genera. The finite base fields Fq are arbitrary, but their sizes should not grow too fast compared to the genus. For such families, the group structure and discrete logarithms can be computed in subexponential time of Lqg(1/3, O(1)). The runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve. 1
Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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Cited by 3 (1 self)
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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
On the Largest Degree of an Irreducible Factor of a Polynomial in F_q X]
, 1997
"... Introduction. Let F q [X] be the semigroup of monic polynomials f over a finite field F q having q elements. There exists a fairly extensive bibliography of papers dealing with the value distribution problems of various maps F q [X] ! R when the polynomials f are taken "at random". Usually, ..."
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Cited by 1 (0 self)
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Introduction. Let F q [X] be the semigroup of monic polynomials f over a finite field F q having q elements. There exists a fairly extensive bibliography of papers dealing with the value distribution problems of various maps F q [X] ! R when the polynomials f are taken "at random". Usually, the probability measure n (: : : ) := q \Gamman #ff : ffif = n; : : : g; where ffif := deg f , is applied. We mention here the investigations [1], [5], [713], [1720], [25]. On the other hand, there exists a parallel theory investigating the value distribution of the maps Sn ! R, where Sn denotes the symmetric group of order n, when a permutation oe 2 Sn is taken with the equal probability 1=n! (see, for instance, [3], [6], [10], [12], [14], [21], [23], [26]). Observe that despite the fact that the same analytic or probabilistic methods can be applied, the problems arising in these two theories have been considered separately. To demonstrate a new point of view, we quote a corollary
Discrete logarithms: The past and the future
 Designs, Codes, and Cryptography
, 2000
"... The first practical public key cryptosystem to be published, the DiffieHellman key exchange algorithm, was based on the assumption that discrete logarithms are hard to compute. This intractability hypothesis is also the foundation for the presumed security of a variety of other public key schemes. ..."
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The first practical public key cryptosystem to be published, the DiffieHellman key exchange algorithm, was based on the assumption that discrete logarithms are hard to compute. This intractability hypothesis is also the foundation for the presumed security of a variety of other public key schemes. While there have been substantial advances in discrete log algorithms in the last two decades, in general the discrete log still appears to be hard, especially for some groups, such as those from elliptic curves. Unfortunately no proofs of hardness are available in this area, so it is necessary to rely on experience and intuition in judging what parameters to use for cryptosystems. This paper presents a brief survey of the current state of the art in discrete logs. 1. Introduction Many of the popular public key cryptosystems are based on discrete exponentiation. If G is a group, such as the multiplicative group of a finite field or the group of points on an elliptic curve, and g is an elem...
On some densities in the set of permutations
"... The asymptotic density of random permutations with given properties of the kth shortest cycle length is examined. The approach is based upon the saddle point method applied for appropriate sums of independent random variables. 1 ..."
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The asymptotic density of random permutations with given properties of the kth shortest cycle length is examined. The approach is based upon the saddle point method applied for appropriate sums of independent random variables. 1