# A 3-Dimensional Lattice Reduction Algorithm | Semantic Scholar

A 3-Dimensional Lattice Reduction Algorithm Igor A. Semaev

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

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.

# A 3-Dimensional Lattice Reduction Algorithm | SpringerLink

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.

# Lattice reduction – Wikipedia

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.