We show that over the field of rational numbers, there is a deterministic algorithm for factorization of constant depth circuits that makes calls to a (whitebox or blackbox) PIT algorithm for such circuits. The time complexity of the algorithm (and hence the number of calls to the PIT oracle) is ... more >>>

We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb{F}_2^n \to \mathbb{F}_2^m$.
Concretely, we want an $F$ and an $A = A(m,n)>0$ as small as possible, so that for every affine map $L: \mathbb{F}_2^n \to \mathbb{F}_2^m$ (of the form $L(x) = Mx + b$) we have:
$$
agree(F,L) ... more >>>

We study factoring algorithms for general sparse polynomials and sparse polynomials of bounded individual degree and prove the following results.
1. We give a deterministic polynomial-time algorithm which takes as input an $n$-variate $s$-sparse polynomial $f$ of bounded individual degree $d$ and outputs a list of circuits which contains ... more >>>

Modular composition is the problem of computing the coefficient vector of the polynomial $f(g(x)) \bmod h(x)$, given as input the coefficient vectors of univariate polynomials $f$, $g$, and $h$ over an underlying field $\mathbb{F}$. While this problem is known to be solvable in nearly-linear time over finite fields due to ... more >>>

We show that the GCD of two univariate polynomials can be computed by (piece-wise) algebraic circuits of constant depth and polynomial size over any sufficiently large field, regardless of the characteristic. This extends a recent result of Andrews \& Wigderson who showed such an upper bound over fields of zero ... more >>>

We show that algebraic formulas and constant-depth circuits are \emph{closed} under taking factors. In other words, we show that if a multivariate polynomial over a field of characteristic zero has a small constant-depth circuit or formula, then all its factors can be computed by small constant-depth circuits or formulas ... more >>>

We present a polynomial-time pseudo-deterministic algorithm for constructing irreducible polynomial of degree $d$ over finite field $\mathbb{F}_q$. A pseudo-deterministic algorithm is allowed to use randomness, but with high probability it must output a canonical irreducible polynomial. Our construction runs in time $\tilde{O}(d^4 \log^4{q})$.

Our construction extends Shoup's deterministic algorithm ... more >>>