Improved methods for calculating vectors of short length in a lattice, including a complexity analysis

Authors:
U. Fincke and M. Pohst

Journal:
Math. Comp. **44** (1985), 463-471

MSC:
Primary 11H50; Secondary 11H55

MathSciNet review:
777278

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of in a suitable box. However, it suffices to consider those which lie in a suitable ellipsoid having a much smaller volume than the box. We show in this paper that searching through that ellipsoid is in many cases much more efficient. If combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions. Several randomly constructed numerical examples illustrate the superiority of our new method over the known ones.

**[1]**U. Dieter,*How to calculate shortest vectors in a lattice*, Math. Comp.**29**(1975), 827–833. MR**0379386**, 10.1090/S0025-5718-1975-0379386-6**[2]**U. Fincke & M. Pohst, "Some applications of a Cholesky-type method in algebraic number theory." (To appear.)**[3]**Ulrich Fincke and Michael Pohst,*On reduction algorithms in nonlinear integer mathematical programming*, Operations research proceedings 1983 (Mannheim, 1983) Springer, Berlin, 1984, pp. 289–295. MR**856594****[4]**Donald E. Knuth,*The art of computer programming*, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR**0378456****[5]**A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász,*Factoring polynomials with rational coefficients*, Math. Ann.**261**(1982), no. 4, 515–534. MR**682664**, 10.1007/BF01457454**[6]**A. Odlyzko, "Cryptoanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir's fast signature system." Preprint, 1983.**[7]**M. Pohst, "On the computation of lattice vectors of minimal length, successive minima and reduced bases with applications,"*ACM SIGSAM Bull.*, v. 15, 1981, pp. 37-44.

Retrieve articles in *Mathematics of Computation*
with MSC:
11H50,
11H55

Retrieve articles in all journals with MSC: 11H50, 11H55

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1985-0777278-8

Article copyright:
© Copyright 1985
American Mathematical Society