Refine
Document Type
- Report (5) (remove)
Language
- English (5)
Has Fulltext
- yes (5)
Is part of the Bibliography
- no (5)
Institute
- Informatik (5)
- Mathematik (5)
Given a real vector alpha =(alpha1 ; : : : ; alpha d ) and a real number E > 0 a good Diophantine approximation to alpha is a number Q such that IIQ alpha mod Zk1 ", where k \Delta k1 denotes the 1-norm kxk1 := max 1id jx i j for x = (x1 ; : : : ; xd ). Lagarias [12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector ff 2 Q d , a rational number " ? 0 and a number N 2 N+ , decide whether there exists a number Q with 1 Q N and kQff mod Zk1 ". We prove that, unless ...
We study the approximability of the following NP-complete (in their feasibility recognition forms) number theoretic optimization problems: 1. Given n numbers a1 ; : : : ; an 2 Z, find a minimum gcd set for a1 ; : : : ; an , i.e., a subset S fa1 ; : : : ; ang with minimum cardinality satisfying gcd(S) = gcd(a1 ; : : : ; an ). 2. Given n numbers a1 ; : : : ; an 2 Z, find a 1-minimum gcd multiplier for a1 ; : : : ; an , i.e., a vector x 2 Z n with minimum max 1in jx i j satisfying P n...
We address to the problem to factor a large composite number by lattice reduction algorithms. Schnorr has shown that under a reasonable number theoretic assumptions this problem can be reduced to a simultaneous diophantine approximation problem. The latter in turn can be solved by finding sufficiently many l_1--short vectors in a suitably defined lattice. Using lattice basis reduction algorithms Schnorr and Euchner applied Schnorrs reduction technique to 40--bit long integers. Their implementation needed several hours to compute a 5% fraction of the solution, i.e., 6 out of 125 congruences which are necessary to factorize the composite. In this report we describe a more efficient implementation using stronger lattice basis reduction techniques incorporating ideas of Schnorr, Hoerner and Ritter. For 60--bit long integers our algorithm yields a complete factorization in less than 3 hours.
We study the following problem: given x element Rn either find a short integer relation m element Zn, so that =0 holds for the inner product <.,.>, or prove that no short integer relation exists for x. Hastad, Just Lagarias and Schnorr (1989) give a polynomial time algorithm for the problem. We present a stable variation of the HJLS--algorithm that preserves lower bounds on lambda(x) for infinitesimal changes of x. Given x \in {\RR}^n and \alpha \in \NN this algorithm finds a nearby point x' and a short integer relation m for x'. The nearby point x' is 'good' in the sense that no very short relation exists for points \bar{x} within half the x'--distance from x. On the other hand if x'=x then m is, up to a factor 2^{n/2}, a shortest integer relation for \mbox{x.} Our algorithm uses, for arbitrary real input x, at most \mbox{O(n^4(n+\log \alpha))} many arithmetical operations on real numbers. If x is rational the algorithm operates on integers having at most \mbox{O(n^5+n^3 (\log \alpha)^2 + \log (\|q x\|^2))} many bits where q is the common denominator for x.