Finding Reduced Basis for Lattices
Let n be a positive integer. A subset L of the n-dimensional real vector space is
called a lattice if there exists a basis b1,b2,…,bn of such that
The bi’s span L.
n is the rank of L.
We will consider only
Determinant of L:
The bi’s are written as column
vectors. Apparently, this positive
real number doesn’t depend on the
choice of the basis.
Let be linearly independent. Suppose it is a basis for
We perform the Gram-Schmidt process:
Forms an orthogonal basis of L
Dividing by shortens our vectors.
A basis b1, ,bn of a lattice is called reduced if :
* ¾ can be replaced by any ¼<y<1
* | | is Euclidean length.
Factoring polynomials with rational coeffecients
An irreducible polynomial over a field is
non-constant and cannot be
represented as the product of at-least 2
Reducible (over ):
How to find, for a given non-zero
polynomial in its decomposition into
Factor primitive polynomials
(gcd of all coeffecients of f is 1)
Into irreducible factors in
Given , and
Find such that: