A 3-Dimensional Lattice Reduction Algorithm Igor A. Semaev

Source: A 3-Dimensional Lattice Reduction Algorithm | Semantic Scholar

A 3-Dimensional Lattice Reduction Algorithm Igor A. Semaev

Source: A 3-Dimensional Lattice Reduction Algorithm | Semantic Scholar

Lattice reduction From Wikipedia, the free encyclopedia Jump to navigation Jump to search Lattice reduction in two dimensions: the black vectors are the given basis for the lattice (represented by blue dots), the red vectors are the reduced basis In mathematics, the goal of lattice basis reduction is given an integer lattice basis as input, to find a basis with short, nearly orthogonal vectors. This is realized using different algorithms, whose running time is usually at least exponential in the dimension of the lattice.

Source: Lattice reduction – Wikipedia

DistanceFunction

Forn≥3, the LLL algorithm [13] due to Lenstra, Lenstra andLov ́asz, computes a reduced basis of ann-dimensional lattice in polynomial time. However, the notion ofreduction is weaker than in the casen=2, and the exact complexity of the algorithm (even in the worst-case, and for small dimensions) is not precisely known. The LLL algorithm uses as a main procedure theGauss Algorithm.

Source: Lattice reduction in two dimensions: analyses under realistic probabilistic models – document

The aim of this paper is a reduction algorithm for a basis b1,b2, b3 of a 3-dimensional lattice in ℝn for fixed n ≥ 3. We give a definition of the reduced basis which is equivalent to that of the Minkowski reduced basis of a 3-dimensional lattice. We prove that for b 1 , b2, b3 ε ℤn, n ≥ 3 and |b1|, |b2|, |b3 | ≤ M, our algorithm takes O(log2 M) binary operations, without using fast integer arithmetic, to reduce this basis and so to find the shortest vector in the lattice.

Source: A 3-Dimensional Lattice Reduction Algorithm | SpringerLink

In two dimensionsFor a basis consisting of just two vectors, there is a simple and efficient method of reduction closely analogous to the Euclidean algorithm for the greatest common divisor of two integers. As with the Euclidean algorithm, the method is iterative; at each step the larger of the two vectors is reduced by adding or subtracting an integer multiple of the smaller vector.

Source: Lattice reduction – Wikipedia

Extended Euclidean Algorithm Example