Computing Modular Polynomials
نویسندگان
چکیده
منابع مشابه
Computing Modular Polynomials
The l modular polynomial, φl(x, y), parameterizes pairs of elliptic curves with an isogeny of degree l between them. Modular polynomials provide the defining equations for modular curves, and are useful in many different aspects of computational number theory and cryptography. For example, computations with modular polynomials have been used to speed elliptic curve point-counting algorithms ([B...
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The `th modular polynomial, φ`(x, y), parameterizes pairs of elliptic curves with an isogeny of degree ` between them. Modular polynomials provide the defining equations for modular curves, and are useful in many different aspects of computational number theory and cryptography. For example, computations with modular polynomials have been used to speed elliptic curve point-counting algorithms (...
متن کاملComputing modular polynomials in quasi-linear time
We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in the literature, has a complexity that is essentially (up to logarithmic factors) linear in the size of the computed polynomials. In particular, it obtains th...
متن کاملFactoring Modular Polynomials
This paper gives an algorithm to factor a polynomial f (in one variable) over rings like Z=rZ for r 2 Z or F q y]=rF q y] for r 2 F q y]. The Chinese Remainder Theorem reduces our problem to the case where r is a prime power. Then factorization is not unique, but if r does not divide the discriminant of f , our (probabilistic) algorithm produces a description of all (possibly exponentially many...
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Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we generalize this concept to the genus 2 case. We give the theoretical framework describing the genus 2 modular polynomials and discuss how to explicitly compute them.
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ژورنال
عنوان ژورنال: LMS Journal of Computation and Mathematics
سال: 2005
ISSN: 1461-1570
DOI: 10.1112/s1461157000000954