Counting Solutions to Polynomial Systems via Reductions

This paper provides both positive and negative results for counting solutions to systems of polynomial equations over a finite field. The general idea is to try to reduce the problem to counting solutions to a single polynomial, where the task is easier. In both cases, simple methods are utilized that we expect will have wider applicability (far beyond algebra). First, we give an efficient deterministic reduction from approximate counting for a system of (arbitrary) polynomial equations to approximate counting for one equation, over any finite field. We apply this reduction to give a deterministic poly(n,s,log p)/eps^2 time algorithm for approximately counting the fraction of solutions to a system of s quadratic n-variate polynomials over F_p (the finite field of prime order p) to within an additive eps factor, for any prime p. Note that uniform random sampling would already require Omega(s/eps^2) time, so our algorithm behaves as a full derandomization of uniform sampling. The approximate-counting algorithm yields efficient approximate counting for other well-known problems, such as 2-SAT, NAE-3SAT, and 3-Coloring. As a corollary, there is a deterministic algorithm (with analogous running time) for producing solutions to such systems which have at least eps p^n solutions. Second, we consider the difficulty of exactly counting solutions to a single polynomial of constant degree, over a finite field. (Note that finding a solution in this case is easy.) It has been known for over 20 years that this counting problem is already NP-hard for degree-three polynomials over F_2; however, all known reductions increased the number of variables by a considerable amount. We give a subexponential-time reduction from counting solutions to k-CNF formulas to counting solutions to a degree-k^{O(k)} polynomial (over any finite field of O(1) order) which exactly preserves the number of variables. As a corollary, the Strong Exponential Time Hypothesis (even its weak counting variant #SETH) implies that counting solutions to constant-degree polynomials (even over F_2) requires essentially 2^n time. Similar results hold for counting orthogonal pairs of vectors over F_p

LIPIcs

We revisit the problem of estimating entropy of discrete distributions from independent samples, studied recently by Acharya, Orlitsky, Suresh and Tyagi (SODA 2015), improving their upper and lower bounds on the necessary sample size n. For estimating Renyi entropy of order alpha, up to constant accuracy and error probability, we show the following * Upper bounds n = O(1) 2^{(1-1/alpha)H_alpha} for integer alpha>1, as the worst case over distributions with Renyi entropy equal to H_alpha. * Lower bounds n = Omega(1) K^{1-1/alpha} for any real alpha>1, with the constant being an inverse polynomial of the accuracy, as the worst case over all distributions on K elements. Our upper bounds essentially replace the alphabet size by a factor exponential in the entropy, which offers improvements especially in low or medium entropy regimes (interesting for example in anomaly detection). As for the lower bounds, our proof explicitly shows how the complexity depends on both alphabet and accuracy, partially solving the open problem posted in previous works. The argument for upper bounds derives a clean identity for the variance of falling-power sum of a multinomial distribution. Our approach for lower bounds utilizes convex optimization to find a distribution with possibly worse estimation performance, and may be of independent interest as a tool to work with Le Cam’s two point method

Low-Complexity Cryptographic Hash Functions

Cryptographic hash functions are efficiently computable functions that shrink a long input into a shorter output while achieving some of the useful security properties of a random function. The most common type of such hash functions is collision resistant hash functions (CRH), which prevent an efficient attacker from finding a pair of inputs on which the function has the same output