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122
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
Faltings, Degeneration of abelian varieties
, 1990
"... An abelian variety A defined over a finite field Fq admits sufficiently many complex multiplications, as Tate showed in [27]. For some details about complex multiplication, see §1.1. Is A the reduction of an abelian variety with sufficiently many complex multiplications in characteristic zero? We fo ..."
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Cited by 142 (8 self)
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An abelian variety A defined over a finite field Fq admits sufficiently many complex multiplications, as Tate showed in [27]. For some details about complex multiplication, see §1.1. Is A the reduction of an abelian variety with sufficiently many complex multiplications in characteristic zero? We formulate several versions of this “CMlifting problem ” in §1.2. Honda
Supersingular curves in cryptography
, 2001
"... Frey and Rück gave a method to map the discrete logarithm problem in the divisor class group of a curve over ¢¡ into a finite field discrete logarithm problem in some extension. The discrete logarithm problem in the divisor class group can therefore be solved as long ¥ as is small. In the elliptic ..."
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Cited by 98 (9 self)
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Frey and Rück gave a method to map the discrete logarithm problem in the divisor class group of a curve over ¢¡ into a finite field discrete logarithm problem in some extension. The discrete logarithm problem in the divisor class group can therefore be solved as long ¥ as is small. In the elliptic curve case it is known that for supersingular curves one ¥§¦© ¨ has. In this paper curves of higher genus are studied. Bounds on the possible values ¥ for in the case of supersingular curves are given. Ways to ensure that a curve is not supersingular are also given. 1.
Cycles of quadratic polynomials and rational points on a genus 2 curve
, 1996
"... It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), who ..."
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Cited by 44 (13 self)
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It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), whose rational points had been previously computed. We prove there are none with N = 5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal(Q/Q)stable 5cycles, and show that there exist Gal(Q/Q)stable Ncycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N = 6.
Moduli of abelian varieties and pdivisible groups: density of Hecke orbits and a conjecture by Grothendieck
 CONFERENCE ON ARITHMETIC GEOMETRY,GÖTTINGEN
, 2006
"... In the week 7 – 11 August 2006 we gave a course, and here are notes for that course. Our main topic is: geometry and arithmetic of Ag ⊗ Fp, the moduli space of polarized abelian varieties of dimension g in positive characteristic. We illustrate properties, and some of the available techniques by tre ..."
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Cited by 18 (11 self)
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In the week 7 – 11 August 2006 we gave a course, and here are notes for that course. Our main topic is: geometry and arithmetic of Ag ⊗ Fp, the moduli space of polarized abelian varieties of dimension g in positive characteristic. We illustrate properties, and some of the available techniques by treating two topics: an
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
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