Action of Modular Correspondences around CM Points

Couveignes, Jean-Marc; Hénocq, Thiérry

doi:10.1007/3-540-45455-1_19

Cited by 22 publications

(46 citation statements)

References 9 publications

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“…As the degree of P ∆ and the size of its coefficients both grow like |∆| 1/2 for ∆ → −∞, the run time can be no better than O(|∆|). This is the actual run time [9] for the classical analytic approach using the modular function j : H → C. The same is true for the more recent non-archimedean approach [8], [5], [4] to the evaluation of P ∆ , which approximates the roots of P ∆ by a Newton iteration process over Q for a suitable small prime . For both methods, it is possible to reduce the run time by sizable constant factors if one replaces the j-function by 'smaller' modular functions [12], [27], [4], [23].…”

Section: Complex Multiplication Constructionsmentioning

confidence: 88%

“…Rough as this heuristic analysis may be, it 'explains' why in the example N = 10 30 given in [5,Section 6] to illustrate the non-archimedean approach to computing class polynomials, examining the primes p at distance < 10 6 from the end points of H N leads to a fundamental discriminant D ≈ −10 8 . As examining the primes in an interval of length N α to achieve |D| < N β gives rise to a run time O(N max{α,β} ), we can achieve a heuristic run time O(N The extreme case (α, β) = (ε, 1/2) corresponds to taking p as close as possible to the end points of H N , a case we already discussed.…”

Section: Complex Multiplication Constructionsmentioning

confidence: 99%

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Efficient CM-constructions of elliptic curves over finite fields

Bröker

Stevenhagen

2007

Math. Comp.

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Abstract. We present an algorithm that, on input of an integer N ≥ 1 together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) has order N . Although it is unproved that this can be done for all N , a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2 ω(N ) log N , where ω(N ) is the number of distinct prime factors of N . In the cryptographically relevant case where N is prime, an expected run time O((log N ) 4+ε ) can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order N = 10 2004 and N = nextprime(10 2004 ) = 10 2004 +4863.

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Section: Complex Multiplication Constructionsmentioning

confidence: 88%

Section: Complex Multiplication Constructionsmentioning

confidence: 99%

Efficient CM-constructions of elliptic curves over finite fields

Bröker

Stevenhagen

2007

Math. Comp.

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“…Alternatively, we could use the method of [12,8] for computing the class polynomial and getting a proven running time ofÕ(h 2 ), but assuming a suitable Riemann hypothesis.…”

Section: It Can Be Shown That S(d) = O((log H)mentioning

confidence: 99%

Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

Morain

2007

Math. Comp.

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Abstract. The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time currently cannot be proven rigorously, but heuristic arguments show that it should run in timeÕ((log N ) 5 ) to prove the primality of N . An asymptotically fast version of it, attributed to J. O. Shallit, is expected to run in timeÕ((log N ) 4 ). We describe this version in more detail, leading to actual implementations able to handle numbers with several thousands of decimal digits.

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“…There are currently three known algorithms to compute P j ∆ ∈ Z[X]: a complex analytic [8], a p-adic [4,7] and a 'multi prime' approach [1,2,22]. If the Generalized Riemann Hypothesis (GRH) holds true, the run time of all three algorithms is O(|∆|); see [2].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 5 we give the geometric interpretation of Φ f l and prove reduction properties of the curve Φ f l = 0. Our algorithm is an extension of the p-adic algorithm for the j-function [4,7], which we briefly recall in Section 2. In Section 3 we recall properties of the modular function field and give a 'weak version' of Shimura reciprocity linking modular functions and ring class fields.…”

Section: Introductionmentioning

confidence: 99%