Cs221: Computational Complexity Lecture 27: Average-case Complexity, Low-degree Extensions

Theorem 1 If there exists a probabilistic polynomial time (PPT) algorithm A such that PrM←μk,p [A(1 k, 1p,M) 6= Perm(M) mod p ] ≤ 1 3(k+1) , then there exists a PPT algorithm B such that for every M ∈ Zk×k p , Prcoins of B [B(1k, 1p,M) 6= Perm(M) mod p] ≤ 1/3 In particular, we’d have that ModPerm ∈ BPP, and thus P#P = BPP. (We proved ModPerm was #P-hard last class). Note that this theorem is usually read negatively—since we don’t believe these consequences hold, it is unlikely such an A exists. The proof of this theorem follows (as shown in the last class), from this more general one, which we will now prove: