A Dual Version of Reimer's Inequality and a Proof of Rudich's Conjecture

We prove a dual version of the celebrated inequality of D. Reimer (a.k.a. the van den Berg-Kesten conjecture). We use the dual inequality to prove a combinatorial conjecture of S. Rudich motivated by questions in cryptographic complexity. One consequence of Rudich’s Conjecture is that there is an oracle relative to which one-way functions exist but one-way permutations do not. The dual inequality has another combinatorial consequence which allows R. Impagliazzo and S. Rudich to prove that if P = NP then NP ∩ coNP ⊆ i.o.AvgP relative to a random oracle.