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Download Algorithmic Number Theory: 9th International Symposium, by Henri Darmon (auth.), Guillaume Hanrot, François Morain, PDF

By Henri Darmon (auth.), Guillaume Hanrot, François Morain, Emmanuel Thomé (eds.)

This booklet constitutes the refereed lawsuits of the ninth foreign Algorithmic quantity conception Symposium, ANTS 2010, held in Nancy, France, in July 2010. The 25 revised complete papers offered including five invited papers have been conscientiously reviewed and chosen for inclusion within the e-book. The papers are dedicated to algorithmic facets of quantity thought, together with user-friendly quantity thought, algebraic quantity conception, analytic quantity concept, geometry of numbers, algebraic geometry, finite fields, and cryptography.

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Extra resources for Algorithmic Number Theory: 9th International Symposium, ANTS-IX, Nancy, France, July 19-23, 2010. Proceedings

Example text

These bounds, combined with an improvement of the methods of [6], allows us to prove all the above mentioned heuristics of on the factorization of integers from the NICE cryptosystems. 2 Preliminaries and Notation In this section we recall some definitions and properties concerning binary quadratic forms. For a more detailed account of the theory see [5,4,9]. Then, we summarize some results on the norm of a matrix. Quadratic Forms. A binary quadratic form f is a homogeneous polynomial of degree two in two variables f Ôx, y Õ ax2 bxy cy 2 with Ôa, b, cÕ È Z3 which we abbreviate as f Ôa, b, cÕ.

Thus M 2 Mm 2 Ô1 hm Õ2 4 a0 ß am ¤ 4h2m 16 a0 ß ar . 2 2 One still has Δδr γr Ô a0 ar δr2 cr γr2 Õ2 Ô21 a0 Õ2 , because cr am by Lemma 2. This concludes the proof of items 1Õ and 2Õ of Theorem 2. It remains the complexity issue, proved in the following paragraph. Complexity. We now prove the number of iterations performed by RedGL2. Two steps before the end, at iteration r ¡ 2 of RedGL2, we know that the Ôar ¡2 , br ¡2 , cr ¡2 Õ satisfies Δ ar¡2 , because the distance form fr¡2 between the roots of fm 1 is smaller than 1.

Thus M 2 Mm 2 Ô1 hm Õ2 4 a0 ß am ¤ 4h2m 16 a0 ß ar . 2 2 One still has Δδr γr Ô a0 ar δr2 cr γr2 Õ2 Ô21 a0 Õ2 , because cr am by Lemma 2. This concludes the proof of items 1Õ and 2Õ of Theorem 2. It remains the complexity issue, proved in the following paragraph. Complexity. We now prove the number of iterations performed by RedGL2. Two steps before the end, at iteration r ¡ 2 of RedGL2, we know that the Ôar ¡2 , br ¡2 , cr ¡2 Õ satisfies Δ ar¡2 , because the distance form fr¡2 between the roots of fm 1 is smaller than 1.

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