Results 1  10
of
21
A CRT algorithm for constructing genus 2 curves over finite fields
, 2007
"... Abstract. — We present a new method for constructing genus 2 curves over a finite field Fn with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discretelog based cryptosystems. Our algorithm prov ..."
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Cited by 19 (7 self)
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Abstract. — We present a new method for constructing genus 2 curves over a finite field Fn with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discretelog based cryptosystems. Our algorithm provides an alternative to the traditional CM method for constructing genus 2 curves. For a quartic CM field K with primitive CM type, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem (CRT) and a bound on the denominators to construct the class polynomials. We also provide an algorithm for determining endomorphism rings of ordinary Jacobians of genus 2 curves over finite fields, generalizing the work of Kohel for elliptic curves. Résumé (Un algorithme fondé sur le théorème chinois pour construire des courbes de genre 2 sur des corps finis) Nous présentons une nouvelle méthode pour construire des courbes de genre 2 sur un corps fini Fn avec un nombre donné de points sur sa jacobienne. Cette méthode a des applications importantes en cryptographie, où des groupes d’ordre premier sont employés pour former des cryptosystèmes fondés sur le logarithme discret. Notre algorithme fournit une alternative à la méthode traditionnelle de multiplication complexe pour construire des courbes de genre 2. Pour un corps quartique K à multiplication complexe de type primitif, nous calculons les polynômes de classe d’Igusa modulo p pour certain petit premiers p et employons le théorème chinois et une borne sur les dénominateurs pour construire les polynômes de classe. Nous fournissons également un algorithme pour déterminer les anneaux d’endomorphismes des jacobiennes de courbes ordinaires de genre 2 sur des corps finis, généralisant le travail de Kohel pour les courbes elliptiques.
Zhu: On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field
 J. Number Theory
"... Abstract. We prove that for every field k and every positive integer n, there exists an absolutely simple ndimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k, n) denote the fraction of the isogeny cla ..."
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Cited by 17 (2 self)
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Abstract. We prove that for every field k and every positive integer n, there exists an absolutely simple ndimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k, n) denote the fraction of the isogeny classes of ndimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every n we have S(Fq, n) → 1 as q → ∞ over the prime powers. 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 10 (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.
Constructing pairingfriendly genus 2 curves over prime fields with ordinary Jacobians
 IN: PROCEEDINGS OF PAIRING 2007, LNCS 4575
, 2007
"... We provide the first explicit construction of genus 2 curves over finite fields whose Jacobians are ordinary, have large primeorder subgroups, and have small embedding degree. Our algorithm is modeled on the CocksPinch method for constructing pairingfriendly elliptic curves [5], and works for a ..."
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Cited by 10 (2 self)
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We provide the first explicit construction of genus 2 curves over finite fields whose Jacobians are ordinary, have large primeorder subgroups, and have small embedding degree. Our algorithm is modeled on the CocksPinch method for constructing pairingfriendly elliptic curves [5], and works for arbitrary embedding degrees k and prime subgroup orders r. The resulting abelian surfaces are defined over prime fields Fq with q ≈ r 4. We also provide an algorithm for constructing genus 2 curves over prime fields Fq with ordinary Jacobians J having the property that J[r] ⊂ J(Fq) or J[r] ⊂ J(F q k) for any even k.
Isogeny classes of abelian varieties with no principal polarizations, to appear
 in Moduli of Abelian Varieties (Texel Island
, 1999
"... Abstract. We provide a simple method of constructing isogeny classes of abelian varieties over certain fields k such that no variety in the isogeny class has a principal polarization. In particular, given a field k, a Galois extension ℓ of k of odd prime degree p, and an elliptic curve E over k that ..."
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Cited by 9 (3 self)
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Abstract. We provide a simple method of constructing isogeny classes of abelian varieties over certain fields k such that no variety in the isogeny class has a principal polarization. In particular, given a field k, a Galois extension ℓ of k of odd prime degree p, and an elliptic curve E over k that has no complex multiplication over k and that has no kdefined pisogenies to another elliptic curve, we construct a simple (p − 1)dimensional abelian variety X over k such that every polarization of every abelian variety isogenous to X has degree divisible by p 2. We note that for every odd prime p and every number field k, there exist ℓ and E as above. We also provide a general framework for determining which finite group schemes occur as kernels of polarizations of abelian varieties in a given isogeny class. Our construction was inspired by a similar construction of Silverberg and Zarhin; their construction requires that the base field k have positive characteristic and that there be a Galois extension of k with a certain nonabelian Galois group. 1.
Computing endomorphism rings of jacobians of genus 2 curves
 In Symposium on Algebraic Geometry and its Applications, Tahiti
, 2006
"... Abstract. We present probabilistic algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present algorithms for computing the field of definit ..."
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Cited by 9 (5 self)
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Abstract. We present probabilistic algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups J[ℓ d] for prime powers ℓ d. We use these algorithms to create the first implementation of Eisenträger and Lauter’s algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem [EL], and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute p 3 curves for many small primes p. 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.
Families of genus 2 curves with small embedding degree.” Cryptology ePrint Archive, Report 2007/001
, 2007
"... Abstract. In cryptographic applications, hyperelliptic curves of small genus have the advantage of providing a group of comparable size to that of elliptic curves, while working over a field of smaller size. Pairingfriendly hyperelliptic curves are those for which the order of the Jacobian is divis ..."
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Cited by 7 (1 self)
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Abstract. In cryptographic applications, hyperelliptic curves of small genus have the advantage of providing a group of comparable size to that of elliptic curves, while working over a field of smaller size. Pairingfriendly hyperelliptic curves are those for which the order of the Jacobian is divisible by a large prime, whose embedding degree is small enough for pairing computations to be feasible, and whose minimal embedding field is large enough for the discrete logarithm problem in it to be difficult. We give a sequence of Fqisogeny classes for a family of Jacobians of genus 2 curves over Fq, for q = 2 m, and the corresponding small embedding degrees. We give examples of the parameters for such curves with embedding degree k < (log q) 2, such as k =
Improved upper bounds for the number of points on curves over finite fields
 Ann. Inst. Fourier
, 2006
"... Abstract. We give new arguments that improve the known upper bounds on the maximal number Nq(g) of rational points of a curve of genus g over a finite field Fq, for a number of pairs (q, g). Given a pair (q, g) and an integer N, we determine the possible zeta functions of genusg curves over Fq with ..."
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Cited by 6 (3 self)
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Abstract. We give new arguments that improve the known upper bounds on the maximal number Nq(g) of rational points of a curve of genus g over a finite field Fq, for a number of pairs (q, g). Given a pair (q, g) and an integer N, we determine the possible zeta functions of genusg curves over Fq with N points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genusg curve over Fq with N points must have a lowdegree map to another curve over Fq, and often this is enough to give us a contradiction. In particular, we are able to provide eight previously unknown
Principally polarizable isogeny classes of abelian surfaces over finite fields
, 2008
"... Let A be an isogeny class of abelian surfaces over Fq with Weil polynomial x 4 +ax 3 +bx 2 +aqx+q 2. We show that A does not contain a surface that has a principal polarization if and only if a 2 − b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3. ..."
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Cited by 3 (2 self)
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Let A be an isogeny class of abelian surfaces over Fq with Weil polynomial x 4 +ax 3 +bx 2 +aqx+q 2. We show that A does not contain a surface that has a principal polarization if and only if a 2 − b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3.