On Approximating the Entropy of Polynomial Mappings

Abstract

Abstract: We investigate the complexity of the following computational problem: Polynomial Entropy Approximation (PEA): Given a low-degree polynomial mapping p: Fn → Fm, where F is a finite field, approximate the output entropy H(p(Un)), where Un is the uniform distribution on Fn and H may be any of several entropy measures. We show: • Approximating the Shannon entropy of degree 3 polynomials p: Fn 2 → Fm 2 over F2 to within an additive constant (or even n.9) is complete for SZKPL, the class of problems having statistical zero-knowledge proofs where the honest verifier and its simulator are computable in logarithmic space. (SZKPL contains most of the natural problems known to be in the full class SZKP.) • For prime fields F = F2 and homogeneous quadratic polynomials p: Fn → Fm, there is a probabilistic polynomial-time algorithm that distinguishes the case that p(Un) has entropy smaller than k from the case that p(Un) has min-entropy (or even Renyi entropy) greater than (2 + o(1))k. • For degree d polynomials p: Fn 2 → Fm 2, there is a polynomial-time algorithm that distinguishes the case that p(Un) has max-entropy smaller than k (where the max-entropy of a random variable is the logarithm of its support size) from the case that p(Un) has max-entropy at least (1 + o(1)) · kd (for fixed d and large k).