# Foundations of Cryptography, Volume 1: Basic Techniques

## Oded Goldreich

Language: English

Pages: 394

ISBN: 2:00220483

Format: PDF / Kindle (mobi) / ePub

Cryptography is concerned with the conceptualization, definition and construction of computing systems that address security concerns. This book presents a rigorous and systematic treatment of the foundational issues: defining cryptographic tasks and solving new cryptographic problems using existing tools. It focuses on the basic mathematical tools: computational difficulty (one-way functions), pseudorandomness and zero-knowledge proofs. Rather than describing ad-hoc approaches, this book emphasizes the clarification of fundamental concepts and the demonstration of the feasibility of solving cryptographic problems. It is suitable for use in a graduate course on cryptography and as a reference book for experts.

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S| · δ− 2 | S| · | B| 2m Guideline: Define a random variable Xn that is uniformly distributed over B. Exercise 26: Failure of an alternative construction of pseudorandom functions: Consider a construction of a function ensemble where the functions in Fn are defined as 178

those discussions have been. Next, I would like to thank a few colleagues and friends with whom I have had significant interactions regarding cryptography and related topics. These include Noga Alon, Boaz Barak, Mihir Bellare, Ran Canetti, Ivan Damgard, Uri Feige, Shai Halevi, Johan Hastad, Amir Herzberg, Russell Impagliazzo, Joe Kilian, Hugo Krawcyzk, Eyal Kushilevitz, Yehuda Lindell, Mike Luby, Daniele Micciancio, Moni Naor, Noam Nisan, Andrew Odlyzko, Yair Oren, Rafail Ostrovsky, Erez Petrank,

functions: In continuation of the discussion in Section 2.4.6, consider another suggestion to base one-way functions on the conjectured difficulty of the Graph Isomorphism problem. This time we present a collection of functions defined by the algorithmic triplet ( I GI , DGI , FGI ). On input 1n , algorithm I GI selects uniformly a d(n)-regular graph on n vertices (i.e., each of the n vertices in the graph has degree d(n)). On input a graph on n vertices, algorithm DGI randomly selects a

signature. The difference between the two is that in the setting of message authentication it is not required that third parties (who may be dishonest) be able to verify the validity of authentication tags produced by the designated users, whereas in the setting of signature schemes it is required that such third parties be able to verify the validity of signatures produced by other users. Hence, digital signatures provide a solution to the message-authentication problem. On the other hand, a

on the entire history of the interaction, and the second process sets the value of this bit (i.e., the ith bit) by flipping an unbiased coin. It is intuitively clear that the resulting string is uniformly distributed; still, a proof is required (since randomized processes are subtle entities that often lead to mistakes). In our setting, the situation is slightly more involved. The process of determining the string is terminated after T < N iterations, and statements are made regarding the