Limits on the provable consequences of one-way functions

My rst intellectual companion ii Acknowledgements. I am extremely grateful to Manuel Blum for everything. He has been a fantastic advisor. From the very beginning, he boosted my mathematical self-conndence by treating me as a peer. He also took the time to show me more than his perfected and polished ideas. He showed me the guts of research|false starts, fallacies, inchoate intuitions, and all. This exceptional honesty gave me a generous look at the inner working of a truly creative mind. Knowing Manuel has been, without any doubt, the high point of my intellectual life. As far as this thesis is concerned, I owe Manuel more than a methodological debt. He was thinking about problems and issues treated here long before I came to graduate school. In particular, he asked if P = NP implies that secret-key agreement is impossible with a random oracle, and conjectured that P = NP implies NP R T Co ? NP R = P R. He pointed out that a random oracle provides an ideal one-way function, giving his conjectures a natural interpretation. He also detected the subtle possibility that we might simulate a random permutation oracle in such a way that it is invertible. As with all of Manuel's problems, I learned a great deal from these, not realizing the full depth of some of the issues for several years. Russell Impagliazzo is my dear, dear friend and colleague. He never fails to have unique (often deeply uniied) views on an issue. It is always wonderful to talk to him. Like Manuel, his intellectual honesty has always made it possible for me to learn from him. This thesis is joint work with Russell. We will co-author all papers which result. We have discussed these questions with each other for about two years. The history of the secret-key agreement result includes about forty fallacious proofs; I could never have survived this cruel form of torture if I had been going it alone. A lesser co-author would have given up. Russell was always tenacious and insightful. If he were a sword, I would call him Theorem Killer. iii I am very grateful for discussions with Amos Fiat, Mario Szegedy, and Gabor Tardos. I want to thank Noam and Umesh for helping me check the details of the main result. I am also grateful to my proofreaders,s feedback clearly demonstrated the subtle relation between syntax and …