We investigate the complexity of the following computational problem:

Polynomial Entropy Approximation (PEA): Given a low-degree polynomial mapping , where is a finite field, approximate the output entropy where is the uniform distribution on and may be any of several entropy measures.

We show:

- Approximating the Shannon entropy of degree 3 polynomials to within an additive constant (or even n
^{.9}) is complete for**SZKP**, the class of problems having statistical zero-knowledge proofs where the honest verifier and its simulator are computable in logarithmic space. (_{L}**SZKP**contains most of the natural problems known to be in the full class_{L}**SZKP**.) - For prime fields and
*homogeneous*quadratic polynomials , there is a probabilistic polynomial-time algorithm that distinguishes the case that has entropy smaller than*k*from the case that has min-entropy (or even Renyi entropy) greater than (2 +*o*(1))*k*. - For degree
*d*polynomials , there is a polynomial-time algorithm that distinguishes the case that 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 has max-entropy at least (1 +*o*(1))*k*(for fixed^{d}*d*and large*k*).

By:* Zeev Dvir; Dan Gutfreund; Guy N. Rothblum; Salil Vadhan*

Published in: H-0293 in 2010

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