Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\F_q$ of characteristic $p$ (equivalently, constructing the bigger field $\F_{q^{r^e}}$). Both these problems have famous randomized ... more >>>

In a recent work of Bei, Chen and Zhang (STOC 2013), a trial and error model of computing was introduced, and applied to some constraint satisfaction problems. In this model the input is hidden by an oracle which, for a candidate assignment, reveals some information about a violated constraint if ... more >>>

We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix $M$ whose entries are homogeneous linear polynomials over the integers. Given a linear subspace $\mathcal{B}$ of the $n \times n$ matrices over some field $\mathbb{F}$, we consider ... more >>>

The problem of finding a nontrivial factor of a polynomial $f(x)$ over a finite field $\mathbb{F}_q$ has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the ... more >>>

We present new deterministic algorithms for several cases of the maximum rank matrix completion
problem (for short matrix completion), i.e. the problem of assigning values to the variables in
a given symbolic matrix as to maximize the resulting matrix rank. Matrix completion belongs to
the fundamental problems in computational complexity ... more >>>

In this paper we develop techniques that eliminate the need of the Generalized
Riemann Hypothesis (GRH) from various (almost all) known results about deterministic
polynomial factoring over finite fields. Our main result shows that given a
polynomial f(x) of degree n over a finite field k, we ... more >>>

In this work we relate the deterministic
complexity of factoring polynomials (over
finite
fields) to certain combinatorial objects we
call
m-schemes. We extend the known conditional
deterministic subexponential time polynomial
factoring algorithm for finite fields to get an
underlying m-scheme. We demonstrate ... more >>>