Nearly linear-size holographic proofs

Authors: Alexander Polishchuk and Daniel A. Spielman.

Bibliographic Information: First appeared in the Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, 1994, pp. 194-203.

Abstract

We show how to construct holographic (or transparent) proofs of size $n^{1+\epsilon }$ that can be checked by a verifier that is allowed to read only $\O{1}$ bits of the proof and has access to $\O{\log n}$ random bits, for all $\epsilon >0$. In general, we construct proofs of size $n^{1+2^{-\O{q(n)}}}(\log n)^{\O{q(n)}}$ checkable by the query of $2^{\O{q(n)}}$ bits, for any $q(n) = \O{\log \log n}$. An essential element of our construction is a proof that the low-degree test used by Arora and Safra is effective on domains of size linear in the degree of the encoded polynomial.
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Daniel A. Spielman