Arithmetic Circuits and the Hadamard Product of Polynomials

Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity testing. Our main results are the following. 1. We show that noncommutative polynomial identity testing for algebraic branching programs over rationals is complete for the logspace counting class \ceql, and over fields of characteristic $p$ the problem is in \ModpL/\Poly. 2.We show an exponential lower bound for expressing the Raz-Yehudayoff polynomial as the Hadamard product of two monotone multilinear polynomials. In contrast the Permanent can be expressed as the Hadamard product of two monotone multilinear formulas of quadratic size.Comment: 20 page

On Efficient Noncommutative Polynomial Factorization via Higman Linearization

In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F of polynomials in noncommuting variables x_1,x_2,..., x_n over the field F. We obtain the following result: Given a noncommutative arithmetic formula of size s computing a noncommutative polynomial f in F as input, where F=F_q is a finite field, we give a randomized algorithm that runs in time polynomial in s, n and log q that computes a factorization of f as a product f=f_1f_2\cdots f_r, where each f_i is an irreducible polynomial that is output as a noncommutative algebraic branching program. The algorithm works by first transforming f into a linear matrix L using Higman's linearization of polynomials. We then factorize the linear matrix L and recover the factorization of f. We use basic elements from Cohn's theory of free ideals rings combined with Ronyai's randomized polynomial-time algorithm for computing invariant subspaces of a collection of matrices over finite fields

Some Sieving Algorithms for Lattice Problems

ABSTRACT. Westudythealgorithmiccomplexityoflatticeproblemsbasedonthesievingtechnique due to Ajtai, Kumar, and Sivakumar [AKS01]. Given a k-dimensional subspace M ⊆ R n and a full rankintegerlattice L ⊆ Q n,thesubspaceavoidingproblemSAP,definedbyBlömerandNaewe[BN07], is to find a shortest vector in L \ M. We first give a 2 O(n+klogk) time algorithm to solve the subspace avoiding problem. Applying this algorithm weobtain thefollowing results. 1. We give a 2 O(n) time algorithm to compute i th successive minima of a full rank lattice L ⊂ Q n if i isO ( n logn). 2. Wegivea2 O(n) timealgorithmtosolvearestrictedclosestvectorproblemCVPwheretheinputs fulfilapromise about thedistance of theinputvector fromthelattice. 3. We also show that unrestricted CVP has a 2 O(n) exact algorithm if there is a 2 O(n) time exact algorithm for solving CVP with additional input v i ∈ L,1 ≤ i ≤ n, where ‖v i‖p is the i th successive minima of L foreach i. We also give a new approximation algorithm for SAP and the Convex Body Avoiding problem which is a generalization of SAP. Several of our algorithms work for gauge functions as metric, where the gauge function has anatural restriction and isaccessed byan oracle.