3 research outputs found
Fast Arithmetics Using Chinese Remaindering
In this paper, some issues concerning the Chinese remaindering representation
are discussed. Some new converting methods, including an efficient
probabilistic algorithm based on a recent result of von zur Gathen and
Shparlinski \cite{Gathen-Shparlinski}, are described. An efficient refinement
of the NC division algorithm of Chiu, Davida and Litow
\cite{Chiu-Davida-Litow} is given, where the number of moduli is reduced by a
factor of
Lattice Compression of Polynomial Matrices
This thesis investigates lattice compression of polynomial matrices
over finite fields. For an m x n matrix, the goal of lattice
compression is to find an m x (m+k) matrix, for some relatively
small k, such that the lattice span of two matrices are
equivalent. For any m x n polynomial matrix with degree bound
d, it can be compressed by multiplying by a random n x (m+k)
matrix B with degree bound s. In this thesis, we prove that
there is a positive probability that
L(A)=L(AB) with k(s+1)=\Theta(\log(md)). This
is shown to hold even when s=0 (i.e., where B is a matrix of
constants). We also design a competitive probabilistic lattice
compression algorithm of the Las Vegas type that has a positive
probability of success on any input and requires
O~(nm^{\theta-1}B(d)) field operations