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39
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 571 (29 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
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 13 (1 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.
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 11 (6 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 9 (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.
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 9 (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
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 9 (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.
Point counting on Picard curves in large characteristic
 Math. Comp
, 2005
"... Abstract. We present an algorithm for computing the cardinality of the Jacobian of a random Picard curve over a finite field. If the underlying field is a prime field Fp, the algorithm has complexity O ( √ p). 1. ..."
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Cited by 7 (0 self)
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Abstract. We present an algorithm for computing the cardinality of the Jacobian of a random Picard curve over a finite field. If the underlying field is a prime field Fp, the algorithm has complexity O ( √ p). 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 6 (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
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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Cited by 5 (0 self)
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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.
Values of zeta functions at s = 1/2
, 2005
"... We study the behaviour near s = 1 of zeta functions of varieties over finite fields Fq 2 with q a square. The main result is an Eulercharacteristic formula for the square of the special value at s = 1. The Eulercharacteristic is constructed from the Weilétale cohomology of a certain 2 supersingul ..."
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Cited by 4 (0 self)
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We study the behaviour near s = 1 of zeta functions of varieties over finite fields Fq 2 with q a square. The main result is an Eulercharacteristic formula for the square of the special value at s = 1. The Eulercharacteristic is constructed from the Weilétale cohomology of a certain 2 supersingular elliptic curve.