Deterministically Factoring Sparse Polynomials into Multilinear Factors and Sums of Univariate Polynomials

We present the first efficient deterministic algorithm for factoring sparse polynomials that split into multilinear factors and sums of univariate polynomials. Our result makes partial progress towards the resolution of the classical question posed by von zur Gathen and Kaltofen in [von zur Gathen/Kaltofen, J. Comp. Sys. Sci., 1985] to devise an efficient deterministic algorithm for factoring (general) sparse polynomials. We achieve our goal by introducing essential factorization schemes which can be thought of as a relaxation of the regular factorization notion

On Some Computations on Sparse Polynomials

In arithmetic circuit complexity the standard operations are +,x. Yet, in some scenarios exponentiation gates are considered as well. In this paper we study the question of efficiently evaluating a polynomial given an oracle access to its power. Among applications, we show that: * A reconstruction algorithm for a circuit class c can be extended to handle f^e for f in C. * There exists an efficient deterministic algorithm for factoring sparse multiquadratic polynomials. * There is a deterministic algorithm for testing a factorization of sparse polynomials, with constant individual degrees, into sparse irreducible factors. That is, testing if f = g_1 x ... x g_m when f has constant individual degrees and g_i-s are irreducible. * There is a deterministic reconstruction algorithm for multilinear depth-4 circuits with two multiplication gates. * There exists an efficient deterministic algorithm for testing whether two powers of sparse polynomials are equal. That is, f^d = g^e when f and g are sparse

Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree

In this paper we study the problem of deterministic factorization of sparse polynomials. We show that if $f \in \mathbb{F}[x_{1},x_{2},\ldots ,x_{n}]$ is a polynomial with $s$ monomials, with individual degrees of its variables bounded by $d$, then $f$ can be deterministically factored in time $s^{\mathrm{poly}(d) \log n}$. Prior to our work, the only efficient factoring algorithms known for this class of polynomials were randomized, and other than for the cases of $d=1$ and $d=2$, only exponential time deterministic factoring algorithms were known. A crucial ingredient in our proof is a quasi-polynomial sparsity bound for factors of sparse polynomials of bounded individual degree. In particular we show if $f$ is an $s$-sparse polynomial in $n$ variables, with individual degrees of its variables bounded by $d$, then the sparsity of each factor of $f$ is bounded by $s^{O({d^2\log{n}})}$. This is the first nontrivial bound on factor sparsity for $d>2$. Our sparsity bound uses techniques from convex geometry, such as the theory of Newton polytopes and an approximate version of the classical Carath\'eodory's Theorem. Our work addresses and partially answers a question of von zur Gathen and Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the sparsity of factors of sparse polynomials

Complete Derandomization of Identity Testing and Reconstruction of Read-Once Formulas

In this paper we study the identity testing problem of arithmetic read-once formulas (ROF) and some related models. A read-once formula is formula (a circuit whose underlying graph is a tree) in which the operations are {+,x} and such that every input variable labels at most one leaf. We obtain the first polynomial-time deterministic identity testing algorithm that operates in the black-box setting for read-once formulas, as well as some other related models. As an application, we obtain the first polynomial-time deterministic reconstruction algorithm for such formulas. Our results are obtained by improving and extending the analysis of the algorithm of [Shpilka-Volkovich, 2015

Synergy Between Circuit Obfuscation and Circuit Minimization

We study close connections between Indistinguishability Obfuscation (IO) and the Minimum Circuit Size Problem (MCSP), and argue that efficient algorithms/construction for MCSP and IO create a synergy. Some of our main results are: - If there exists a perfect (imperfect) IO that is computationally secure against nonuniform polynomial-size circuits, then for all k ? ?: NP ? ZPP^{MCSP} ? SIZE[n^k] (MA ? ZPP^{MCSP} ? SIZE[n^k]). - In addition, if there exists a perfect IO that is computationally secure against nonuniform polynomial-size circuits, then NEXP ? ZPEXP^{MCSP} ? P/poly. - If MCSP ? BPP, then statistical security and computational security for IO are equivalent. - If computationally-secure perfect IO exists, then MCSP ? BPP iff NP = ZPP. - If computationally-secure perfect IO exists, then ZPEXP ? BPP. To the best of our knowledge, this is the first consequence of strong circuit lower bounds from the existence of an IO. The results are obtained via a construction of an optimal universal distinguisher, computable in randomized polynomial time with access to the MCSP oracle, that will distinguish any two circuit-samplable distributions with the advantage that is the statistical distance between these two distributions minus some negligible error term. This is our main technical contribution. As another immediate application, we get a simple proof of the result by Allender and Das (Inf. Comput., 2017) that SZK ? BPP^{MCSP}

The Power of Natural Properties as Oracles

We study the power of randomized complexity classes that are given oracle access to a natural property of Razborov and Rudich (JCSS, 1997) or its special case, the Minimal Circuit Size Problem (MCSP). We show that in a number of complexity-theoretic results that use the SAT oracle, one can use the MCSP oracle instead. For example, we show that ZPEXP^{MCSP} !subseteq P/poly, which should be contrasted with the previously known circuit lower bound ZPEXP^{NP} !subseteq P/poly. We also show that, assuming the existence of Indistinguishability Obfuscators (IO), SAT and MCSP are equivalent in the sense that one has a ZPP algorithm if and only the other one does. We interpret our results as providing some evidence that MCSP may be NP-hard under randomized polynomial-time reductions

The Complexity of Finding S-Factors in Regular Graphs

A graph G has an S-factor if there exists a spanning subgraph F of G such that for all v in V: deg_F(v) in S. The simplest example of such factor is a 1-factor, which corresponds to a perfect matching in a graph. In this paper we study the computational complexity of finding S-factors in regular graphs. Our techniques combine some classical as well as recent tools from graph theory

One-Way Functions and a Conditional Variant of MKTP

One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows. 1) First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs. 2) Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions. 3) Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recently-announced results of Ren and Santhanam [Rahul Ilango et al., 2021], we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs