On the evaluation of modular polynomials

Sutherland, Andrew V.

doi:10.2140/obs.2013.1.531

Cited by 24 publications

(26 citation statements)

References 37 publications

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“…The order-subgroups of E [ ] are precisely the kernels of -isogenies from E to other elliptic curves, and the set of all such -isogenies (up to isomorphism) corresponds to the set of roots of Φ ( j(E ), x) in F q . The classical modular polynomial Φ (X,Y ), of degree + 1 (in X and Y ) over Z, is defined by the property that Φ ( j(E 1 ), j(E 2 )) = 0 precisely when there exists an -isogeny [36]. Alternatively, we can use precomputed databases of modular polynomials over Z, reducing them modulo p and specializing them at j(E ).…”

Section: Modular Polynomials and Isogeniesmentioning

confidence: 99%

Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

Ballentine

Guillevic

García

et al. 2017

Association for Women in Mathematics Series

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Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof's algorithm. While we are currently missing the tools we need to generalize Elkies' methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this article, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves. IntroductionEfficiently computing the number of points on the Jacobian of a genus 2 curve over a finite field is an important problem in experimental number theory and numbertheoretic cryptography. When the characteristic of the finite field is small, Kedlaya's algorithm and its descendants provide an efficient solution (see [18], [13], and [12]), while in extremely small characteristic we have extremely fast AGM-style algorithms (see for example [25], [26], and [3]). However, the running times of these algorithms are exponential in the size of the field characteristic; the hardest case, therefore (and also the most important case for contemporary cryptographic applications) is where the characteristic is large, or even where the field is a prime field.So let q be a power of a large prime p, and let C be a genus-2 curve over F q . Our fundamental problem is to compute the number of F q -rational points on the Jacobian J C of C .

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Section: Modular Polynomials and Isogeniesmentioning

confidence: 99%

Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

Ballentine

Guillevic

García

et al. 2017

Association for Women in Mathematics Series

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“…Other modular polynomials can be used on restricted classes of curves, for example the Weber polynomials. Also, the methods of [18] allow us to compute Φ (X, j(E)), given j(E) and , significantly faster than the computation of Φ . However, our numerical experiments highlighted the high cost of computing the action of p given Φ , where = N (p).…”

Section: Numerical Experimentsmentioning

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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“…We can determine the class of a prime by finding out how many roots Φ (j(E), x) has in F q . We define a subroutine EvaluatedModularPolynomial( , E), which computes Φ (j(E), x) in O( 3 (log ) 3 log log ) bit operations (under the Generalized Riemann Hypothesis) using the method of [26], assuming that log q = Θ( ). (Note that, in practice, one generally uses precomputed modular polynomials over Z.…”

Section: Atkin Elkies and Volcanic Primesmentioning

confidence: 99%

Computing cardinalities of -curve reductions over finite fields

Morain¹,

Scribot

Smith³

2016

LMS J. Comput. Math.

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We present a specialized point-counting algorithm for a class of elliptic curves over F p 2 that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F p 2 with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof-Elkies-Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.

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On the evaluation of modular polynomials

Cited by 24 publications

References 37 publications

Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

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

Computing cardinalities of -curve reductions over finite fields

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