A 3-Dimensional Lattice Reduction Algorithm Igor A. Semaev
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
Forn≥3, the LLL algorithm  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.
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.
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
Bézout’s identity — Let a and b be integers with greatest common divisor d. Then, there exist integers x and y such that ax + by = d. More generally, the integers of the form ax + by are exactly the multiples of d.
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Lattice generated by vectors orthogonal to an integer vector Ask Question
Tel Aviv University, Fall 2004Lattices in Computer ScienceLecture 8Dual LatticesLecturer: Oded RegevScribe: Gillat Kol
CSE 206A: Lattice Algorithms and Applications Spring 2014
The dual lattice
Instructor: Daniele Micciancio UCSD CSE
torus is a surface having genus one
is there a possibility to “transform” a matrix (inheriting the input-output “weights”) into a graph object inheriting the matrix relations?
you can convert it to an edge-weighted graph using
WeightedAdjacencyGraph. This will give you a complete graph (a
Graph expression) in which each vertex is also connected to itself. I am not sure how much sense it makes treat this matrix as a graph, given the full connectivity.
Since you have a fully connected graph, it does not matter in your case, but be aware that
WeightedAdjacencyGraph represents missing edges with
Infinity, not with zero. This is somewhat annoying because it is inconsistent with
WeightedAdjacencyMatrix, which uses
Infinity does make sense in some (not all) applications, it is much more common to see data that uses 0.
If V=R2 and W=R2, then T:R2R2 is a linear transformation if and only if there exists a 22 matrix A such that T(v)=Av for all vR2. Matrix A is called the standard matrix for T. The columns of A are T01 and T10 , respectively. Since each linear transformation of the plane has a unique standard matrix, we will identify linear transformations of the plane by their standard matrices. It can be shown that if A is invertible, then the linear transformation defined by A maps parollelograms to parallelograms. We will often illustrate the action of a linear transformation T:R2R2 by looking at the image of a unit square under T
the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects scalar multiples of vectors based at A. [If A, E, F are collinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′).] Geometrically T transforms the grid based on ABCD to that based in A′B′C′D′.Affine transformations do not respect lengths or angles; they multiply area by a constant factor
In some cases, however, you may prefer to convert the system of linear equations into a matrix equation, and then apply matrix manipulation operations to solve it. This approach is often useful when the system of equations arises as part of a general algorithm, and you do not know in advance how many variables will be involved.
The vector and tensor case is more complicated because of the need to account for the change of basis vectors.
American Mathematical Society
Source: MSC2010 database
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