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54
Short signatures from the Weil pairing
, 2001
"... Abstract. We introduce a short signature scheme based on the Computational DiffieHellman assumption on certain elliptic and hyperelliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signa ..."
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Cited by 712 (28 self)
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Abstract. We introduce a short signature scheme based on the Computational DiffieHellman assumption on certain elliptic and hyperelliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signatures are typed in by a human or signatures are sent over a lowbandwidth channel. 1
Computing Hilbert Class Polynomials
"... Abstract. We present and analyze two algorithms for computing the Hilbert class polynomial HD. The first is a padic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CMcurves over f ..."
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Cited by 17 (9 self)
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Abstract. We present and analyze two algorithms for computing the Hilbert class polynomial HD. The first is a padic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CMcurves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for computing HD, and we show that all methods have comparable run times. 1
The maximum or minimum number of rational points on genus three curves over finite fields
, 2001
"... We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the SerreWeil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq. ..."
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Cited by 16 (0 self)
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We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the SerreWeil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq.
Real polynomials with all roots on the unit circle and abelian varieties over finite fields
 J. Number Theory
, 1998
"... Version 19980315 Abstract. In this paper we prove several theorems about abelian varieties over finite fields by studying the set of monic real polynomials of degree 2n all of whose roots lie on the unit circle. In particular, we consider a set Vn of vectors in R n that give the coefficients of such ..."
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Cited by 16 (2 self)
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Version 19980315 Abstract. In this paper we prove several theorems about abelian varieties over finite fields by studying the set of monic real polynomials of degree 2n all of whose roots lie on the unit circle. In particular, we consider a set Vn of vectors in R n that give the coefficients of such polynomials. We calculate the volume of Vn and we find a large easilydescribed subset of Vn. Using these results, we find an asymptotic formula — with explicit error terms — for the number of isogeny classes of ndimensional abelian varieties over Fq. We also show that if n> 1, the set of group orders of ndimensional abelian varieties over Fq contains every integer in an interval of length roughly q n − 1 2 centered at q n + 1. Our calculation of the volume of Vn involves the evaluation of the integral over the simplex { (x1,..., xn) ∣ 0 ≤ x1 ≤ · · · ≤ xn ≤ 1} of the determinant of the n × n matrix [ x e] i−1 j, where the ei are positive real numbers. 1.
CONGRUENCES BETWEEN MODULAR FORMS GIVEN BY THE DIVIDED β FAMILY IN HOMOTOPY THEORY
"... Abstract. We characterize the 2line of the plocal AdamsNovikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p ≥ 5. We give a similar characterization of the 1line, reinterpreting a computation of A. Baker. These results are then used t ..."
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Cited by 16 (3 self)
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Abstract. We characterize the 2line of the plocal AdamsNovikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p ≥ 5. We give a similar characterization of the 1line, reinterpreting a computation of A. Baker. These results are then used to deduce that, for ℓ a prime which generates Z × p, the spectrum Q(ℓ) detects the α and β families in the stable stems. Contents
Jacobians in isogeny classes of abelian surfaces over finite fields
 Ann. Inst. Fourier (Grenoble
"... Abstract. We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus2 curves over finite fields. 1. ..."
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Cited by 15 (2 self)
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Abstract. We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus2 curves over finite fields. 1.
ANALYTIC PROBLEMS FOR ELLIPTIC CURVES
, 2005
"... Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the dist ..."
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Cited by 13 (0 self)
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Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups. (This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv). Contents
Geometric Methods for Improving the Upper Bounds on the Number of Rational Points on Algebraic Curves over Finite Fields
 Journal of Algebraic Geometry
, 2001
"... Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g dened over a nite eld Fq come either from Serre's renement of the Weil bound if the genus is small compared to q, or from Oesterle's optimization o ..."
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Cited by 11 (1 self)
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Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g dened over a nite eld Fq come either from Serre's renement of the Weil bound if the genus is small compared to q, or from Oesterle's optimization of the explicit formulae method if the genus is large. This paper presents three methods for improving these bounds. The arguments used are the indecomposability of the theta divisor of a curve, Galois descent, and HondaTate theory. Examples of improvements on the bounds include lowering them for a wide range of small genus when q = 2 3 ; 2 5 ; 2 13 ; 3 3 ; 3 5 ; 5 3 ; 5 7 , and when q = 2 2s , s > 1. For large genera, isolated improvements are obtained for q = 3; 8; 9. 1
Families of curves and weight distributions of codes
 Bull. AMS
, 1995
"... Abstract. In this expository paper we show how one can, in a uniform way, calculate the weight distributions of some wellknown binary cyclic codes. The codes are related to certain families of curves, and the weight distributions are related to the distribution of the number of rational points on t ..."
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Abstract. In this expository paper we show how one can, in a uniform way, calculate the weight distributions of some wellknown binary cyclic codes. The codes are related to certain families of curves, and the weight distributions are related to the distribution of the number of rational points on the curves. 1.