[Show abstract][Hide abstract] ABSTRACT: Using known results regarding PCP, we present simple proofs of the inapproximability of vertex cover for hypergraphs. Specifically, we show that
Approximating the size of the minimum vertex cover in O(1)-regular hypergraphs to within a factor of 1.99999 is NP-hard.
Approximating the size of the minimum vertex cover in 4-regular hypergraphs to within a factor of 1.49999 is NP-hard.
Both results are inferior to known results (by Trevisan (2001) and Holmerin (2001)), but they are derived using much simpler proofs. Furthermore, these proofs demonstrate the applicability of the FGLSS-reduction in the context of reductions among combinatorial optimization problems.
[Show abstract][Hide abstract] ABSTRACT: Recent works by Ajtai and by Ajtai and Dwork bring to light the old (general) question of whether it is at all possible to base the security of cryptosystems on the assumption that P 6= NP . We discuss this question and in particular review and extend a two-decade old result of Brassard regarding this question. Our conclusion is that the question remains open. Keywords: Cryptography, P 6= NP, promise problems, smart reductions. Work done while visiting LCS, MIT. y DARPA grant DABT63-96-C-0018. 0 1
[Show abstract][Hide abstract] ABSTRACT: This document is written to complement my 1989 lecture notes on Encryption, Signatures and Cryptographic Protocols. In it I sketch what I believe should be done when trying to use these notes as part of a course on Foundations of Cryptography. In addition, I also indicate what I believe should be done in order to augment the material so that it can fit into a comprehensive book on Foundations of Cryptography. 0 1 Introduction I've recently put on the public domain two incomplete manuscripts 1. Lecture notes from a course I gave in 1989 on Foundations of Cryptography . 2. Fragments of a Book on Foundations of Cryptography . In my opinion, the fragments provide a good draft covering three major topics: One-Way Functions, Pseudorandom Generators and Zero-Knowledge Proofs. These topics are central to Cryptography as well as of interest from a Complexity Theoretic point of view. Yet, the fragments do not provide any material on three (arguably more) central topics of crypto...