Generating a Lattice of a given Genus

Abstract

A full-rank lattice L in R is a discrete subgroup of R which is the set of all integer linear combinations of n-linearly independent vectors, say b1, · · · ,bn i.e., L = { ∑n i=1 zibi | z1, · · · , zn ∈ Z}. The matrix B = [b1, · · · ,bn] is called the basis of the lattice and the matrix Q = B′B is called a Gram matrix of the lattice. A lattice is integral if its Gram matrix has only integer entries. Integral lattices have been studied by mathematicians as positive definite quadratic forms, defined by the equation x′Qx, where x = (x1, · · · , xn) and Q ∈ Zn×n. One of the classical problems in this area is the classification of quadratic forms. Two quadratic forms are equivalent over Z if one can be obtained by the other using a unimodular transformation. If two quadratic forms are equivalent over Z then they are equivalent over the ring Z/pZ, for all primes p and positive integers k. The converse is not true. This leads to the classification of integral quadratic forms into equivalence classes, called the genus. A genus is a set of quadratic forms which are equivalent over Z/pZ for all primes p and positive integers k. The main result of this thesis is to generate a quadratic form of a given genus in randomized polynomial time. Of independent interest is a polynomial time algorithm to generate a uniform random solution of the equation x′Qx ≡ t mod p.