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ON HYPERELLIPTIC CURVES
"... Abstract. We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even w ..."
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Abstract. We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even
An elementary introduction to hyperelliptic curves
"... This paper presents an elementary introduction to some of the theory of hyperelliptic curves over finite fields of arbitrary characteristic that has cryptographic relevance. Cantor’s algorithm for adding in the jacobian of a hyperelliptic curve and a proof of correctness of the algorithm are presen ..."
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Cited by 72 (4 self)
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This paper presents an elementary introduction to some of the theory of hyperelliptic curves over finite fields of arbitrary characteristic that has cryptographic relevance. Cantor’s algorithm for adding in the jacobian of a hyperelliptic curve and a proof of correctness of the algorithm
THE HARMONIC VOLUMES OF HYPERELLIPTIC CURVES
, 2005
"... Abstract. We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka [10]. 1. ..."
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Abstract. We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka [10]. 1.
GENERATORS OF JACOBIANS OF HYPERELLIPTIC CURVES
"... Abstract. This paper provides a probabilistic algorithm to determine generators of the mtorsion subgroup of the Jacobian of a hyperelliptic curve of genus two. 1. ..."
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Abstract. This paper provides a probabilistic algorithm to determine generators of the mtorsion subgroup of the Jacobian of a hyperelliptic curve of genus two. 1.
Fields of moduli of hyperelliptic curves
, 2004
"... Let F be an algebraically closed field with char(F) ̸ = 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let ι be the hyperelliptic involution of X. We show that X can be defined over its field of moduli relative to the extension F/K if Aut(X)/〈ι 〉 is not cyclic. ..."
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Cited by 9 (0 self)
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Let F be an algebraically closed field with char(F) ̸ = 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let ι be the hyperelliptic involution of X. We show that X can be defined over its field of moduli relative to the extension F/K if Aut(X)/〈ι 〉 is not cyclic
PAIRINGS ON HYPERELLIPTIC CURVES
, 2009
"... We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairingbased cryptography. We also showcase the hyperelliptic pairings proposed to date, and develop a unifying framework. We discuss the techni ..."
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Cited by 1 (0 self)
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We assemble and reorganize the recent work in the area of hyperelliptic pairings: We survey the research on constructing hyperelliptic curves suitable for pairingbased cryptography. We also showcase the hyperelliptic pairings proposed to date, and develop a unifying framework. We discuss
Counting hyperelliptic curves
 Adv. Math
, 2009
"... We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k = Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q − 1 and q + 1. As a byproduct we obtain ..."
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Cited by 5 (1 self)
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We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k = Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q − 1 and q + 1. As a byproduct we
COMPACTIFYING MODULI OF HYPERELLIPTIC CURVES
, 2007
"... Abstract. We construct a new compactification of the moduli space Hg of smooth hyperelliptic curves of genus g. We compare our compactification with other wellknown remarkable compactifications of Hg. 1. ..."
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Abstract. We construct a new compactification of the moduli space Hg of smooth hyperelliptic curves of genus g. We compare our compactification with other wellknown remarkable compactifications of Hg. 1.
MODULAR EQUATIONS FOR HYPERELLIPTIC CURVES
, 2004
"... We define modular equations describing the ℓtorsion subgroups of the Jacobian of a hyperelliptic curve. Over a finite base field, we prove factorization properties that extend the wellknown results used in Atkin’s improvement of Schoof’s genus 1 point counting algorithm. ..."
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Cited by 11 (3 self)
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We define modular equations describing the ℓtorsion subgroups of the Jacobian of a hyperelliptic curve. Over a finite base field, we prove factorization properties that extend the wellknown results used in Atkin’s improvement of Schoof’s genus 1 point counting algorithm.
Ate Pairing on Hyperelliptic Curves
 ADVANCES IN CRYPTOLOGY  EUROCRYPT 2007, SPRINGERVERLAG LNCS 4515
, 2007
"... In this paper we show that the Ate pairing, originally defined for elliptic curves, generalises to hyperelliptic curves and in fact to arbitrary algebraic curves. It has the following surprising properties: The loop length in Miller’s algorithm can be up to g times shorter than for the Tate pairin ..."
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Cited by 16 (3 self)
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In this paper we show that the Ate pairing, originally defined for elliptic curves, generalises to hyperelliptic curves and in fact to arbitrary algebraic curves. It has the following surprising properties: The loop length in Miller’s algorithm can be up to g times shorter than for the Tate
Results 1  10
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7,212