Proof: Let

The PCP theorem states that NP PCP(log n; (log n) c). We build gradually towards its proof. As an intermediate step we show that NP PCP(log n; (log n) c) for some constant c > 1. The presentation here is taken in most parts from 4] (which is heavily based on 2], where NP PCP(log n log log n; log n log log n) was proved. We save a log log n factor in the number of random bits (at a cost of a polylogarithmic factor in the number of query bits) by making the following change to the FGLSS protocol: we use low degree extensions rather than multilinear extensions. These low degree extensions were used in 1]. So as to save myself some rewriting, I kept the original notation of the FGLSS paper. Hence the veriier will be denoted by M, and it has access to an \oracle" O (the PCP witness). The whole PCP system together is referred to as a probabilistic oracle-machine. In this notation, we prove: Theorem 1 Any language L 2 NP is accepted by a probabilistic oracle-machine M such that that uses O(log n) random bits and queries O((log n) c) bit locations, where c > 1 is some universal constant. M takes polynomial time to decide upon the query locations and to write down a list of tests to be made on the replies to the queries. Evaluating these tests takes polylogarithmic time. We now proceed to describe the ingredients of our protocol. 1.1 Polynomials We recall here some simple facts regarding polynomials and multinomials over any ((nite) eld F. Lemma 2 For every set of d + 1 (point,value) pairs f(a i ; b i) : 1 i d + 1g (where a i ; b i 2 F and the a i s are distinct), there is a unique polynomial p(x) of degree d such that p(a i) = b i for every i. L i (x) = j6 =i x ? a j a i ? a j be the polynomial that is 1 at a i and 0 at all a j for j 6 = i. Set p(x) = X b i L i (x): Uniqueness follows the fact that over a eld, every two degree d polynomials either agree on at most d points or agree on all points.