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18
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
 Journal of Number Theory
"... Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the se ..."
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Cited by 15 (7 self)
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Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log DE, where DE is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed. 1.
MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES
, 2010
"... We present a new algorithm to compute the classical modular polynomial Φl in the rings Z[X, Y] and (Z/mZ)[X, Y], for a prime l and any positive integer m. Our approach uses the graph of lisogenies to efficiently compute Φl mod p for many primes p of a suitable form, and then applies the Chinese R ..."
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Cited by 9 (2 self)
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We present a new algorithm to compute the classical modular polynomial Φl in the rings Z[X, Y] and (Z/mZ)[X, Y], for a prime l and any positive integer m. Our approach uses the graph of lisogenies to efficiently compute Φl mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(l3 (log l) 3 log log l), and compute Φl mod m using O(l2 (log l) 2 + l2 log m) space. We have used the new algorithm to compute Φl with l over 5000, and Φl mod m with l over 20000. We also consider several modular functions g for which Φ g l is smaller than Φl, allowing us to handle l over 60000.
Finding composite order ordinary elliptic curves using the cockspinch method. Cryptology ePrint Archive, Report 2009/533
, 2009
"... Abstract. We apply the CocksPinch method to obtain pairingfriendly composite order groups with prescribed embedding degree associated to ordinary elliptic curves, and we show that new security issues arise in the composite order setting. 1. ..."
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Cited by 8 (0 self)
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Abstract. We apply the CocksPinch method to obtain pairingfriendly composite order groups with prescribed embedding degree associated to ordinary elliptic curves, and we show that new security issues arise in the composite order setting. 1.
CONSTRUCTING PAIRINGFRIENDLY HYPERELLIPTIC CURVES USING WEIL RESTRICTION
"... Abstract. A pairingfriendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large primeorder subgroup. In this paper we construct pairingfriendly genus 2 curves over finite fields Fq whose Jacobians are ordinary and simple, but not absolutely simpl ..."
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Cited by 6 (0 self)
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Abstract. A pairingfriendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large primeorder subgroup. In this paper we construct pairingfriendly genus 2 curves over finite fields Fq whose Jacobians are ordinary and simple, but not absolutely simple. We show that constructing such curves is equivalent to constructing elliptic curves over Fq that become pairingfriendly over a finite extension of Fq. Our main proof technique is Weil restriction of elliptic curves. We describe adaptations of the CocksPinch and BrezingWeng methods that produce genus 2 curves with the desired properties. Our examples include a parametric family of genus 2 curves whose Jacobians have the smallest recorded ρvalue for simple, nonsupersingular abelian surfaces. 1.
CLASS INVARIANTS BY THE CRT METHOD
, 1001
"... Abstract. We adapt the CRTapproach to computing Hilbertclass polynomials to handle a wide range of class invariants. Forsuitable discriminantsD, this improves its performance by a large constant factor, more than 200 in the most favourable circumstances. This has enabled recordbreaking construction ..."
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Cited by 4 (1 self)
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Abstract. We adapt the CRTapproach to computing Hilbertclass polynomials to handle a wide range of class invariants. Forsuitable discriminantsD, this improves its performance by a large constant factor, more than 200 in the most favourable circumstances. This has enabled recordbreaking constructions of elliptic curves via the CM method, including examples with D > 10 15. 1.
Computing endomorphism rings of elliptic curves under the GRH
 Journal of Mathematical Cryptology
"... We design a probabilistic algorithm for computing endomorphism rings of ordinary elliptic curves defined over finite fields that we prove has a subexponential runtime in the size of the base field, assuming solely the generalized Riemann hypothesis. Additionally, we improve the asymptotic complexity ..."
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Cited by 4 (3 self)
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We design a probabilistic algorithm for computing endomorphism rings of ordinary elliptic curves defined over finite fields that we prove has a subexponential runtime in the size of the base field, assuming solely the generalized Riemann hypothesis. Additionally, we improve the asymptotic complexity of previously known, heuristic, subexponential methods by describing a faster isogenycomputing routine. 1
Computing (ℓ,ℓ)isogenies in polynomial time on Jacobians of genus 2 curves. 2011. IACR ePrint
"... Abstract. In this paper, we compute ℓisogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in ℓ, when ℓ is an odd prime which is coprime to the characteristic. We use level n symmetric theta structure where n = 2 or n = 4. In a second part of this pa ..."
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Cited by 4 (2 self)
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Abstract. In this paper, we compute ℓisogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in ℓ, when ℓ is an odd prime which is coprime to the characteristic. We use level n symmetric theta structure where n = 2 or n = 4. In a second part of this paper we explain how to convert between Mumford coordinates of Jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4. Combined with the preceding algorithm, this gives a method to compute (ℓ, ℓ)isogenies in polynomial time on Jacobians of genus 2 curves. 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.
MODULAR POLYNOMIALS FOR GENUS 2
, 2009
"... Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to ..."
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Cited by 3 (2 self)
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Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them.
Optimized equations for X1(N) via simulated annealing
"... Some models of an algebraic curve C are better than others. In particular, lowdegree models are useful: • practically, to locate points on C quickly; • theoretically, to prove upper bounds on the gonality of C. Given a defining equation F (r, s) = 0 for C, we seek a bilinear transformation that re ..."
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Some models of an algebraic curve C are better than others. In particular, lowdegree models are useful: • practically, to locate points on C quickly; • theoretically, to prove upper bounds on the gonality of C. Given a defining equation F (r, s) = 0 for C, we seek a bilinear transformation that reduces the degree of F in one of its variables. Our motivating example is the modular curve X1(N), which parameterizes elliptic curves with a point of order N. A general method to obtain a defining equation F (r, s) = 0 for X1(N) (more precisely, the affine curve Y1(N)) is given in [4]. Roots of F may then be used to construct elliptic curves with a point of order N (over a finite field, say), an idea exploited in [6]. The efficiency of this construction depends critically on F, which is typically larger and of higher degree than necessary. We wish to construct an optimized equation f(x, y) = 0, together with an explicit birational map φ that relates the roots of f and F.