How to generate cryptographically strong sequences of pseudo-random bits
SIAM Journal on Computing
How to construct random functions
Journal of the ACM (JACM)
Efficiency considerations in using semi-random sources
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
One-way functions and Pseudorandom generators
Combinatorica - Theory of Computing
RSA and Rabin functions: certain parts are as hard as the whole
SIAM Journal on Computing - Special issue on cryptography
Multiparty protocols and logspace-hard pseudorandom sequences
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
A hard-core predicate for all one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Pseudorandom generators for space-bounded computations
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Small-bias probability spaces: efficient constructions and applications
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Learning Polynomials with Queries: The Highly Noisy Case
SIAM Journal on Discrete Mathematics
Foundations of Cryptography: Basic Tools
Foundations of Cryptography: Basic Tools
DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION
DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION
Randomness, adversaries and computation (random polynomial time)
Randomness, adversaries and computation (random polynomial time)
Foundations of Cryptography: Volume 2, Basic Applications
Foundations of Cryptography: Volume 2, Basic Applications
Theory and application of trapdoor functions
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
The bit extraction problem or t-resilient functions
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Efficient And Secure Pseudo-Random Number Generation
SFCS '84 Proceedings of the 25th Annual Symposium onFoundations of Computer Science, 1984
Studies in complexity and cryptography
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We provide an exposition of three lemmas that relate general properties of distributions over bit strings to the exclusive-or (xor) of values of certain bit locations. The first XOR-Lemma, commonly attributed to Umesh Vazirani (1986), relates the statistical distance of a distribution from the uniform distribution over bit strings to the maximum bias of the xor of certain bit positions. The second XOR-Lemma, due to Umesh and Vijay Vazirani (19th STOC, 1987), is a computational analogue of the first. It relates the pseudorandomness of a distribution to the difficulty of predicting the xor of bits in particular or random positions. The third Lemma, due to Goldreich and Levin (21st STOC, 1989), relates the difficulty of retrieving a string and the unpredictability of the xor of random bit positions. The most notable XOR Lemma - that is the so-called Yao XOR Lemma - is not discussed here. We focus on the proofs of the aforementioned three lemma. Our exposition deviates from the original proofs, yielding proofs that are believed to be simpler, of wider applicability, and establishing somewhat stronger quantitative results. Credits for these improved proofs are due to several researchers.