# A 3-Dimensional Lattice Reduction Algorithm | Semantic Scholar

A 3-Dimensional Lattice Reduction Algorithm Igor A. Semaev

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

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.

# 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.

# Bézout’s identity – Wikipedia

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.

# linear algebra – Lattice generated by vectors orthogonal to an integer vector – Mathematics Stack Exchange

Lattice generated by vectors orthogonal to an integer vector Ask Question

# torus is a surface having genus one

torus is a surface having genus one

# mathematica graphing a 2 by 2 matrix as an object

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 `0`. While `Infinity` does make sense in some (not all) applications, it is much more common to see data that uses 0.

# Geometry of Linear Transformations of the Plane – HMC Calculus Tutorial

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

# Affine transformation – Wikipedia

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

# Solving Linear Systems—Wolfram Language Documentation

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.

# Changing Coordinate Systems—Wolfram Language Documentation

The vector and tensor case is more complicated because of the need to account for the change of basis vectors.

# 2010 Mathematics Subject Classification

American Mathematical Society

Source: MSC2010 database

MathSciNet