Pseudo Random Synthesizers Functions and Permutations

The research re ected in this dissertation is a study of computational pseudo randomness More speci cally the main objective of this research is the e cient and simple construction of pseudo random functions and permutations where e ciency refers both to the sequential and parallel time complexity of the computation Pseudo random functions and permutations are fundamental cryptographic primitives with many applications in cryptog raphy and more generally in computational complexity Constructions of Pseudo Random Functions For our constructions of pseudo random functions we introduce and study a new crypto graphic primitive which we call a pseudo random synthesizer and a generalization of this primitive which we call a k dimensional pseudo random synthesizer These primitives are of independent interest as well In addition we consider various applications of our construc tions and study some of the underlying cryptographic assumptions used in these construc tions The main results obtained by this research are Introducing new cryptographic primitives called pseudo random synthesizer and k dimensional pseudo random synthesizer Using pseudo random synthesizers for a parallel construction of a pseudo random func tion the depth of the functions is larger by a logarithmic factor than the depth of the synthesizers Showing several NC implementations of synthesizers based on concrete intractability assumptions such as factoring and the computational Di e Hellman assumption Showing a very simple parallel construction of synthesizers based on what we call weak pseudo random functions which implies simple constructions of synthesizers based on trapdoor one way permutations and based on any hard to learn problem under the de nition of These results yield the rst parallel pseudo random functions based on computational in tractability assumptions and the rst alternative to the original construction of Goldre ich Goldwasser and Micali In addition we show two new constructions of pseudo random functions that are related to the construction based on synthesizers The pseudo randomness of one construction is proven under the assumption that factoring is hard while