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58
Constructing Elliptic Curve Cryptosystems in Characteristic 2
, 1998
"... Since the group of an elliptic curve defined over a finite field F_q... The purpose of this paper is to describe how one can search for suitable elliptic curves with random coefficients using Schoof's algorithm. We treat the important special case of characteristic 2, where one has certain simp ..."
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Cited by 18 (1 self)
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Since the group of an elliptic curve defined over a finite field F_q... The purpose of this paper is to describe how one can search for suitable elliptic curves with random coefficients using Schoof's algorithm. We treat the important special case of characteristic 2, where one has certain simplifications in some of the algorithms.
A padic algorithm to compute the Hilbert class polynomial
 in ASIACRYPT ’98 Springer LNCS 1514
, 2007
"... Abstract. Classicaly, the Hilbert class polynomial P ∆ ∈ Z[X] of an imaginary quadratic discriminant ∆ is computed using complex analytic techniques. In 2002, Couveignes and Henocq [5] suggested a padic algorithm to compute P∆. Unlike the complex analytic method, it does not suffer from problems c ..."
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Cited by 14 (4 self)
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Abstract. Classicaly, the Hilbert class polynomial P ∆ ∈ Z[X] of an imaginary quadratic discriminant ∆ is computed using complex analytic techniques. In 2002, Couveignes and Henocq [5] suggested a padic algorithm to compute P∆. Unlike the complex analytic method, it does not suffer from problems caused by rounding errors. In this paper we complete the outline given in [5] and we prove that, if the Generalized Riemann Hypothesis holds true, the expected runtime of the padic algorithm is eO(∆). We illustrate the algorithm by computing the polynomial P−639 using a 643adic algorithm. 1.
Average Frobenius distribution of elliptic curves
, 2005
"... The SatoTate conjecture asserts that given an elliptic curve without complex multiplication, the primes whose Frobenius elements have their trace in a given interval (2α √ p, 2β √ p) 1 − t2 dt. We prove that this conjecture is true on average in a have density given by 2 π more general setting. ..."
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Cited by 14 (5 self)
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The SatoTate conjecture asserts that given an elliptic curve without complex multiplication, the primes whose Frobenius elements have their trace in a given interval (2α √ p, 2β √ p) 1 − t2 dt. We prove that this conjecture is true on average in a have density given by 2 π more general setting.
Gaussian hypergeometric functions and traces
 of Hecke operators, International Mathematical Research Notices (2004
"... Abstract. We establish a simple inductive formula for the trace Trnewk (Γ0(8), p) of the pth Hecke operator on the space Snewk (Γ0(8)) of newforms of level 8 and weight k in terms of the values of 3F2hypergeometric functions over the finite field Fp. Using this formula when k = 6, we prove a conje ..."
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Cited by 9 (0 self)
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Abstract. We establish a simple inductive formula for the trace Trnewk (Γ0(8), p) of the pth Hecke operator on the space Snewk (Γ0(8)) of newforms of level 8 and weight k in terms of the values of 3F2hypergeometric functions over the finite field Fp. Using this formula when k = 6, we prove a conjecture of Koike relating Trnew 6 (Γ0(8), p) to the values 6F5(1)p and 4F3(1)p. Furthermore, we find new congruences between Tr new k (Γ0(8), p) and generalized Apéry numbers.
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 8 (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.
Modularity of a certain CalabiYau threefold
 Monatsh. Math
"... Abstract. The Langlands program predicts that certain CalabiYau threefolds are modular in the sense that their Lseries correspond to the Mellin transforms of weight 4 newforms. Here we prove that the Lfunction of the threefold given by P4 i=1(xi + x −1 i) = 0 is η ..."
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Cited by 8 (1 self)
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Abstract. The Langlands program predicts that certain CalabiYau threefolds are modular in the sense that their Lseries correspond to the Mellin transforms of weight 4 newforms. Here we prove that the Lfunction of the threefold given by P4 i=1(xi + x −1 i) = 0 is η
Computing modular polynomials
 London Math. Soc., Journal of Computational Mathematics
, 2005
"... The ℓ th modular polynomial, φℓ(x,y), parameterizes pairs of elliptic curves with an isogeny of degree ℓ between them. Modular polynomials provide the defining equations for modular curves, and are useful in many different aspects of computational number theory and cryptography. For example, computa ..."
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Cited by 7 (3 self)
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The ℓ th modular polynomial, φℓ(x,y), parameterizes pairs of elliptic curves with an isogeny of degree ℓ between them. Modular polynomials provide the defining equations for modular curves, and are useful in many different aspects of computational number theory and cryptography. For example, computations with modular polynomials have been used to speed elliptic curve pointcounting
Group structures of elementary supersingular abelian varieties over finite fields
 J. Number Theory
, 2000
"... Let A be a supersingular abelian variety over a finite field k which is kisogenous to a power of a simple abelian variety over k. Write the characteristic polynomial of the Frobenius endomorphism of A relative to k as f = g e for a monic irreducible polynomial g and a positive integer e. We show th ..."
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Cited by 7 (1 self)
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Let A be a supersingular abelian variety over a finite field k which is kisogenous to a power of a simple abelian variety over k. Write the characteristic polynomial of the Frobenius endomorphism of A relative to k as f = g e for a monic irreducible polynomial g and a positive integer e. We show that the group of krational points A(k) onAis isomorphic to (Z g(1) Z) e unless A's simple component is of dimension 1 or 2, in which case we prove that A(k) is isomorphic to (Z g(1) Z) a _ (Z (g(1) 2) Z_Z 2Z) b for some nonnegative integers a, b with a+b=e. In particular, if the characteristic of k is 2 or A is simple of dimension greater than 2, then A(k)$(Z g(1) Z) e.
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