Some mathematical remarks on the polynomial selection in NFS

Barbulescu, Razvan; Lachand, Armand

doi:10.1090/mcom/3112

Cited by 6 publications

(5 citation statements)

References 22 publications

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“…Putting the asymptotic development in the form of the statement. Based on Theorem 9, one can reproduce mutatis mutandis the arguments in the proof of Theorem 1.1 in [BL17] and obtain…”

Section: Main Theoremmentioning

confidence: 99%

ECM and the Elliott–Halberstam conjecture for quadratic fields

Barbulescu,

Jouve

2024

Acta Arith.

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“…Putting the asymptotic development in the form of the statement. Based on Theorem 9, one can reproduce mutatis mutandis the arguments in the proof of Theorem 1.1 in [BL17] and obtain…”

Section: Main Theoremmentioning

confidence: 99%

ECM and the Elliott–Halberstam conjecture for quadratic fields

Barbulescu,

Jouve

2024

Acta Arith.

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“…where val is the average value of the valuation in of the set of integers that we study, the average being defined rigorously in the sequel of this section. Indeed, Barbulescu and Lachand in Theorem 1.1 of [BL17] proved that β(F) = α(F) for any quadratic polynomial F of primitive fundamental negative discriminant.…”

Section: Open Question 51 Let E Be An Elliptic Curve Without Complex ...mentioning

confidence: 99%

A classification of ECM-friendly families of elliptic curves using modular curves

Barbulescu

Shinde

2021

Math. Comp.

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In this work, we establish a link between the classification of ECM-friendly elliptic curves and Mazur’s program B, which consists in parameterizing all the families of elliptic curves with exceptional Galois image. Motivated by Barbulescu et al. [ANTS X–proceedings of the tenth algorithmic number theory symposium, Berkeley, CA, 2013], we say an elliptic curve is ECM-friendly if it does not have complex multiplication and if its Galois image is exceptional for some level. Building upon two recent works which treated the case of congruence subgroups of prime-power level which occur for infinitely many j j -invariants, we prove that there are exactly 1525 families of rational elliptic curves with distinct Galois images which are cartesian products of subgroups of prime-power level. This makes a complete list of rational families of ECM-friendly elliptic curves with cartesian Galois images, out of which less than 23 were known in the literature. We furthermore refine a heuristic of Montgomery to compare these families and conclude that the best 4 families which can be put in a = − 1 a=-1 twisted Edwards’ form are new.

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“…Another potential issue with this definition is the convergence of the series defining α(f ). We leave it as a conjecture, since adapting the proof of [5] goes beyond the scope of this article.…”

Section: Quality Criteriamentioning

confidence: 99%

Collecting relations for the number field sieve in

Gaudry¹,

Grémy²,

Videau³

2016

LMS J. Comput. Math.

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In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in F p 6 with the number field sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-q strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various polynomial selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-q and various enumeration strategies.

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Some mathematical remarks on the polynomial selection in NFS

Cited by 6 publications

References 22 publications

ECM and the Elliott–Halberstam conjecture for quadratic fields

ECM and the Elliott–Halberstam conjecture for quadratic fields

A classification of ECM-friendly families of elliptic curves using modular curves

Collecting relations for the number field sieve in

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