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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.
Constructing pairingfriendly genus 2 curves over prime fields with ordinary Jacobians
 In: proceedings of Pairing 2007, LNCS 4575
, 2007
"... Abstract. 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 wor ..."
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Cited by 11 (2 self)
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Abstract. 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. 1
EXPLICIT CMTHEORY FOR LEVEL 2STRUCTURES ON ABELIAN SURFACES
"... Abstract. For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CMfield K, the Igusa invariants j1(A), j2(A), j3(A) generate an unramified abelian extension of the reflex field of K. In this paper we give an explicit geometric description of the Galois ..."
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Cited by 5 (0 self)
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Abstract. For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CMfield K, the Igusa invariants j1(A), j2(A), j3(A) generate an unramified abelian extension of the reflex field of K. In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on j1(A), j2(A), j3(A). Our description can be expressed by maps between various Siegel modular varieties, and we can explicitly compute the action for ideals of small norm. We use the Galois action to modify the CRT method for computing Igusa class polynomials, and our run time analysis shows that this yields a significant improvement. Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby implying that the ‘isogeny volcano ’ algorithm to compute endomorphism rings of ordinary elliptic curves over finite fields does not have a straightforward generalization to computing endomorphism rings of abelian surfaces over finite fields. 1.
Improved CRT algorithm for class polynomials in genus 2.” In: Algorithmic Number Theory — ANTSX. Edited by Everett Howe and Kiran Kedlaya
 Mathematical Science Publishers
"... Abstract. We present a generalization to genus 2 of the probabilistic algorithm in Sutherland [28] for computing Hilbert class polynomials. The improvement over the algorithm presented in [5] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring whi ..."
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Cited by 4 (0 self)
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Abstract. We present a generalization to genus 2 of the probabilistic algorithm in Sutherland [28] for computing Hilbert class polynomials. The improvement over the algorithm presented in [5] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring which is the maximal order: rather we present a probabilistic algorithm for “going up ” to a maximal curve (a curve with maximal endomorphism ring), once we find any curve in the right isogeny class. Then we use the structure of the Shimura class group and the computation of (ℓ, ℓ)isogenies to compute all isogenous maximal curves from an initial one. This article is an extended version of the version published at ANTS X. 1.
Computing endomorphism rings of abelian varieties
, 2012
"... Generalizing a method of Sutherland and the author for elliptic curves [5, 1], we design a subexponential algorithm for computing the endomorphism ring structure of ordinary abelian varieties of dimension two over finite fields. Although its correctness and complexity bound rely on several assumptio ..."
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Generalizing a method of Sutherland and the author for elliptic curves [5, 1], we design a subexponential algorithm for computing the endomorphism ring structure of ordinary abelian varieties of dimension two over finite fields. Although its correctness and complexity bound rely on several assumptions, we report on practical computations showing that it performs very well and can easily handle previously intractable cases. Note. Certain results of this paper previously appeared in the author’s thesis [2]. 1
EXPLICIT CMTHEORY IN DIMENSION 2
"... Abstract. For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CMfield K, the Igusa invariants j1(A), j2(A), j3(A) generate an abelian extension of the reflex field of K. In this paper we give an explicit description of the Galois action of the class g ..."
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Cited by 1 (0 self)
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Abstract. For a complex abelian surface A with endomorphism ring isomorphic to the maximal order in a quartic CMfield K, the Igusa invariants j1(A), j2(A), j3(A) generate an abelian extension of the reflex field of K. In this paper we give an explicit description of the Galois action of the class group of this reflex field on j1(A), j2(A), j3(A). We give a geometric description which can be expressed by maps between various Siegel modular varieties. We can explicitly compute this action for ideals of small norm, and this allows us to improve the CRT method for computing Igusa class polynomials. Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby implying that the ‘isogeny volcano ’ algorithm to compute endomorphism rings of ordinary elliptic curves over finite fields does not have a straightforward generalization to computing endomorphism rings of abelian surfaces over finite fields. 1.