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37
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 simplific ..."
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Cited by 17 (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 13 (4 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.
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.
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
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.
Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm
 Acta Arith
"... Abstract. Let Fq (q = pr) be a finite field. In this note the number of irreducible polynomials of degree m in Fq[x] with prescribed trace and norm coefficients is calculated in certain special cases and general bounds for that number are obtained. As a corollary, sharp bounds are obtained for the n ..."
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Cited by 4 (2 self)
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Abstract. Let Fq (q = pr) be a finite field. In this note the number of irreducible polynomials of degree m in Fq[x] with prescribed trace and norm coefficients is calculated in certain special cases and general bounds for that number are obtained. As a corollary, sharp bounds are obtained for the number of elements in Fq3 with prescribed trace and norm over Fq improving the estimates by Katz. Next, simple necessary and sufficient conditions are given when a Kloosterman sum over F2r is divisible by three. This result generalizes the earlier result by Charpin, Helleseth, and Zinoviev obtained only in the case r odd. Finally, a new elementary proof for the value distribution of a Kloosterman sum over the field F3r, obtained by Katz and Livne, is given. 1.