Computing endomorphism rings of elliptic curves under the GRH

Bisson, Gaetan

doi:10.1515/jmc.2011.008

Cited by 17 publications

(38 citation statements)

References 17 publications

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“…We begin by recalling a well-known fact about class groups. This map allows us to define the concept of a relation analogously to Bisson [3,4]. Note that we do not follow Bisson and Sutherland's definition of relation in [6] because we do not have the benefit of an easily computable modular polynomial.…”

Section: Class Group Relationsmentioning

confidence: 99%

Computing the endomorphism ring of an ordinary abelian surface over a finite field

Springer

2019

Journal of Number Theory

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We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this algorithm only requires the heuristic assumptions required by the algorithm of Biasse and Fieker [2] which computes the class group of an order in a number field in subexponential time. Thus we avoid the multiple heuristic assumptions on isogeny graphs and polarized class groups which were previously required. The output of the algorithm is an ideal in the maximal totally real subfield of the endomorphism algebra, generalizing the elliptic curve case.

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Section: Class Group Relationsmentioning

confidence: 99%

Computing the endomorphism ring of an ordinary abelian surface over a finite field

Springer

2019

Journal of Number Theory

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“…To treat the ordinary case, we rely on the algorithm in [8], which gives a GRHbased Las Vegas algorithm to compute the index u p of End(E p ) in the maximal order of the imaginary quadratic field End(E p ) ⊗ Q with expected running time…”

Section: Computing Frobenius Conjugacy Classesmentioning

confidence: 99%

“…The first is Schoof's algorithm [55,56], which computes the trace t ∈ Z of the Frobenius endomorphism in time polynomial in log q. The second is a Las Vegas algorithm to compute the endomorphism ring End(E p ) when E p is ordinary, due to Bisson and the author [8,9]; under the GRH its expected running time is subexponential in log q. It follows from a theorem of Duke and Tóth [25] that the pair (t, End(E p )) determines an integer matrix A p whose reduction modulo m lies in the conjugacy class ρ E,m (Frob p ) for every positive integer m. The third is Miller's algorithm to compute the Weil pairing [49], which we use to compute the rank of the -torsion subgroup of E p (F p ) in quasicubic time.…”

Section: Introductionmentioning

confidence: 99%

Computing Images of Galois Representations Attached to Elliptic Curves

Sutherland

2016

Forum of Mathematics, Sigma

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Let E be an elliptic curve without complex multiplication (CM) over a number field K , and let G E ( ) be the image of the Galois representation induced by the action of the absolute Galois group of K on the -torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G E ( ) up to local conjugacy for all primes by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G E ( ) up to one of at most two isomorphic conjugacy classes of subgroups of GL 2 (Z/ Z) that have the same semisimplification, each of which occurs for an elliptic curve isogenous to E. Under the GRH, their running times are polynomial in the bit-size n of an integral Weierstrass equation for E, and for our Monte Carlo algorithm, quasilinear in n. We have applied our algorithms to the non-CM elliptic curves in Cremona's tables and the Stein-Watkins database, some 140 million curves of conductor up to 10 10 , thereby obtaining a conjecturally complete list of 63 exceptional Galois images G E ( ) that arise for E/Q without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images G E ( ) that arise for non-CM elliptic curves over quadratic fields with rational j-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the j-invariant is irrational.

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“…, b k of the lattice of relations between elements of B such that each b i ∈ Z k has entries with absolute value bounded by e (1+o(1)) log 1/3 |D| . We follow the approach of Bisson [2], which consists of rewriting relations between ideals of B and generators of Cl(O) with respect to short decompositions obtained from Corollary 2. For each invertible fractional ideal a, we define σ(a) ∈ Z k to be one of the vectors v with entries bounded by e -σ(g …”

Section: Existence Of Short Relationsmentioning

confidence: 99%

Fast heuristic algorithms for computing relations in the class group of a quadratic order, with applications to isogeny evaluation

Biasse¹,

Fieker²,

Jacobson³

2016

LMS J. Comput. Math.

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In this paper, we present novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded. We show how our methods can be used to improve the computation of large-degree isogenies and endomorphism rings of elliptic curves defined over finite fields. For these problems, we obtain improved heuristic complexity results in almost all cases and significantly improved performance in practice. The speed-up is especially high in situations where the ideal class group can be computed in advance.

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Computing endomorphism rings of elliptic curves under the GRH

Cited by 17 publications

References 17 publications

Computing the endomorphism ring of an ordinary abelian surface over a finite field

Computing the endomorphism ring of an ordinary abelian surface over a finite field

Computing Images of Galois Representations Attached to Elliptic Curves

Fast heuristic algorithms for computing relations in the class group of a quadratic order, with applications to isogeny evaluation

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