论文信息 - Finding a Shortest Vector in a Two-Dimensional Lattice Modulo m

Finding a Shortest Vector in a Two-Dimensional Lattice Modulo m

We find the shortest non-zero vector in the lattice of all integer multiples of the vector (a, b) modulo m, for given integers 0 < a, b < m. We reduce the problem to the computation of a Minkowski-reduced basis for a planar lattice and thereby show that the problem can be solved in O(log m(log log m)2) bit operations.

Günter Rote

[1] Bettina Helfrich,et al. Algorithms to Construct Minkowski Reduced an Hermite Reduced Lattice Bases , 1985, Theor. Comput. Sci..

[2] Alfred V. Aho,et al. The Design and Analysis of Computer Algorithms , 1974 .

[3] Loo Keng Hua,et al. Introduction to number theory , 1982 .

[4] U. Fincke,et al. Improved methods for calculating vectors of short length in a lattice , 1985 .

[5] Brigitte Vallée,et al. La Réduction Des Réseaux, Autour De L'Algorithme De Lenstra, Lenstra, Lovász , 1989, RAIRO Theor. Informatics Appl..

[6] Arnold Schönhage. Storage Modification Machines , 1980, SIAM J. Comput..

[7] Mody Lempel,et al. An Algorithm for Finding a Shortest Vector in a Two-Dimensional Modular Lattice , 1994, Theor. Comput. Sci..

[8] Ravi Kannan,et al. Minkowski's Convex Body Theorem and Integer Programming , 1987, Math. Oper. Res..