New PDF release: Algorithmic Number Theory: 4th International Symposium,

By Jin-Yi Cai (auth.), Wieb Bosma (eds.)

ISBN-10: 3540449949

ISBN-13: 9783540449942

ISBN-10: 3540676953

ISBN-13: 9783540676959

This publication constitutes the refereed lawsuits of the 4th overseas Algorithmic quantity conception Symposium, ANTS-IV, held in Leiden, The Netherlands, in July 2000.
The publication provides 36 contributed papers that have passed through a radical around of reviewing, choice and revision. additionally incorporated are four invited survey papers. one of the subject matters addressed are gcd algorithms, primality, factoring, sieve equipment, cryptography, linear algebra, lattices, algebraic quantity fields, classification teams and fields, elliptic curves, polynomials, functionality fields, and gear sums.

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Additional info for Algorithmic Number Theory: 4th International Symposium, ANTS-IV Leiden, The Netherlands, July 2-7, 2000. Proceedings

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C. Lagarias. The computational complexity of simultaneous diophantine approximation problems. SIAM Journal of Computing, 14:196–209, 1985. 48. J. C. Lagarias, H. W. Lenstra, and C. P. Schnorr. Korkin-Zolotarev Bases and Successive Minima of a Lattice and its Reciprocal Lattice. Combinatorica, 10(4):333348, 1990. 49. J. C. Lagarias and A. M. Odlyzko. Solving low-density subset sum problems. In Proc. 24th IEEE Symposium on Foundations of Computer Science, 1983, 1 – 10. 50. A. K. Lenstra, H. W. Lenstra, and L.

Bn } is the Gram-Schmidt orthogonalization of B. Then it is shown in [48] that λ(B) ≤ λ1 (L) ≤ nλ(B). In particular this gives a way to provide an “NP proof” of a lower bound for λ1 (L) up to a factor of n by guessing an appropriate basis B ∗ and then calculating B. 2 Thus if NP = coNP, then approximating λ1 (L) up to a factor n is not NP-hard in the sense of Karp reductions. More precisely, if NP = coNP, then there is no deterministic polynomial time reduction σ from SAT, σ(ϕ) = (L, λ), such that if ϕ ∈ SAT, then λ1 (L) ≤ λ, and if ϕ ∈ SAT, then λ1 (L) ≥ nλ.

Lattice problems for a special kind of lattice defined by certain graphs have been studied in [18]. 5 Non-NP-Hardness Results To what extent can we expect to improve further the approximation factor√for SVP and remain NP-hard? The current proof appears not feasible beyond 2. On the other hand, the best polynomial time approximation algorithms of Lov´ asz and Schnorr are exponential in the approximation factor. For polynomially bounded factors, transference theorems provide evidence that beyond a factor of Θ(n), the approximate SVP is not NP-hard.

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Algorithmic Number Theory: 4th International Symposium, ANTS-IV Leiden, The Netherlands, July 2-7, 2000. Proceedings by Jin-Yi Cai (auth.), Wieb Bosma (eds.)


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