By Ernie Croot, Andrew Granville, Robin Pemantle, Prasad Tetali (auth.), Alfred J. van der Poorten, Andreas Stein (eds.)
ISBN-10: 3540794557
ISBN-13: 9783540794554
ISBN-10: 3540794565
ISBN-13: 9783540794561
This e-book constitutes the refereed lawsuits of the eighth foreign Algorithmic quantity idea Symposium, ANTS 2008, held in Banff, Canada, in may well 2008.
The 28 revised complete papers offered including 2 invited papers have been conscientiously reviewed and chosen for inclusion within the ebook. The papers are prepared in topical sections on elliptic curves cryptology and generalizations, mathematics of elliptic curves, integer factorization, K3 surfaces, quantity fields, aspect counting, mathematics of functionality fields, modular types, cryptography, and quantity theory.
Read or Download Algorithmic Number Theory: 8th International Symposium, ANTS-VIII Banff, Canada, May 17-22, 2008 Proceedings PDF
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Additional info for Algorithmic Number Theory: 8th International Symposium, ANTS-VIII Banff, Canada, May 17-22, 2008 Proceedings
Example text
In this case, we cannot rigourously prove that a is not principal because this is dependent on the truth of the GRH. 1) has no solutions, this could be a substantial problem. For the remainder of this paper we will concentrate our efforts on how to deal with problem 2 above. We will develop a Las Vegas algorithm for solving this problem which executes in time O(hΔ ). Our inputs to this process, besides D, N and the candidate ideal a are RΔ and h, where RΔ ∈ Q, |RΔ −log2 Δ | < 1. We can produce values for RΔ and h by using the index calculus techniques √described by Jacobson [9] and Maurer [19] in time bounded above by LΔ [1/2, 2 + o(1)], but we don’t have a proof that they are correct because of the assumption of the GRH.
Now Burgess’s theorem tells us that N (x) − M (x)/2ω(n) x1− 1/4+δ if x ≥ n , the prime number theorem that ω(n) ≤ log n/ log y = o(log x), and (7) that Ψ (x, y) ≥ x1− /2 as y > L . Hence N (x, y) ∼ Ψ (n, y)/2ω(n) . The number of integers a ≤ x which are coprime to n and a square mod n is ∼ (φ(n)/n)(x/2ω(n) ), and φ(n) = n(1 + O(1/y))ω(n) ∼ n, so the result follows. 4 Making the Numbers Smaller In Pomerance’s quadratic sieve the factoring stage of the algorithm is sped up by having the ai be the reduced values of a polynomial, so that every p-th ai is divisible by p, if any aj is.
1, pp. 411–422 (1995) 36 E. Croot et al. 17. : Multiplicative independence for random integers. J. ) Analytic Number Theory: Proc. Conf. in Honor of Heini Halberstam, Birh¨ auser, vol. 2, pp. 703–711 (1996) 18. : Smooth numbers and the quadratic sieve. , Stevenhagen, P. ) Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, Mathematical Sciences Research Institute Publications 44 (to appear, 2007) 19. : The multiple polynomial quadratic sieve. Math. Comp. 48, 329–339 (1987) 20.
Algorithmic Number Theory: 8th International Symposium, ANTS-VIII Banff, Canada, May 17-22, 2008 Proceedings by Ernie Croot, Andrew Granville, Robin Pemantle, Prasad Tetali (auth.), Alfred J. van der Poorten, Andreas Stein (eds.)
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