How does the generator check whether a set S is pseudoran?

Once a (T(k(n)), E(k(n)))-pseudorandom subset S, is found, the generator checks whether S, is $-approximated by all of the first n Turing machines, in some canonical enumeration, by running each of them as a sampling algorithm for +(k(n)) steps.

What is the probability of a set of S C_ ZL?

The authors say that a set S C_ ZL is r(k)-distinguished by the circuit C ifwhere p c ( S ) (resp. pc(Z,)) denotes the probability that C outputs 1 when given elements of S (resp. I,), chosen with uniform probability.

How is it possible to distinguish between a uniform ensemble?

Notice that since probabilistic algorithms are fed with random bits chosen according to the uniform distribution, it is trivial for them to output uniform ensembles.

How is the probability that a set S is $-approximated by n?

On the other hand, the probability that a set S is $-approximated by n sampling machines is, using Lemma 7, less than r1/2'(")-"-~ .

What is the probability of a random set being distinguished by a given circuit?

Lemma 2. S C Ik of size N b e(k)-distinguished by C is at most 2e For any k-input Boolean circuit C , the probability that a random set - 2Ne 2 ( k )Proof.

Open AccessJournal Article10.1002/RSA.3240030206

Sparse pseudorandom distributions

Oded Goldreich, +1 more

- 01 Mar 1992

- Random Structures and Algorithms

- Vol. 3, Iss: 2, pp 163-174

TL;DR: The existence of sparse pseudorandom distributions which are not only sparse, but also have the property that no polynomial‐time algorithm may find an element in their support, except for a negligible probability, are proved independently of any intractability assumption.

Abstract: The existence of sparse pseudorandom distributions is proved. These are probability distributions concentrated in a very small set of strings, yet it is infeasible for any polynomial‐time algorithm to distinguish between truly random coins and coins selected according to these distributions. It is shown that such distributions can be generated by (nonpolynomial) probabilistic algorithms, while probabilistic polynomial‐time algorithms cannot even approximate all the pseudorandom distributions. Moreover, we show the existence of evasive pseudorandom distributions which are not only sparse, but also have the property that no polynomial‐time algorithm may find an element in their support, except for a negligible probability. All these results are proved independently of any intractability assumption. © 1992 Wiley Periodicals, Inc.

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1. What are the contributions in this paper?

Moreover, the authors show the existence of evasive pseudorandom distributions which are not only sparse, but also have the property that no polynomial-time algorithm may find an element in their support, except for a negligible probability.

2. What is the importance of randomness in cryptography?

Randomness is used in the design of algorithms in fields such as computational number theory, computational geometry and parallel and distributed computing, and is crucial to cryptography.

3. What is the definition of a pseudorandom generator?

pseudorandom generators are defined as efficient deterministic algorithms expanding short seeds into longer bit strings.

4. What is the fundamental notion of pseudorandomness?

Since in most cases the interest is in the behavior of efficient algorithms (modeled by polynomial-time computations), the fundamental notion of pseudorandomness arises.

Citations

Journal Article•10.1137/S0097539791220688

On the Composition of Zero-Knowledge Proof Systems

Oded Goldreich, +1 more

- 01 Feb 1996

- SIAM Journal on Computing

TL;DR: It is proved that three-round interactive proofs and constant-round Arthur--Merlin proofs that are black-box simulation zero-knowledge exist only for languages in BPP, and it follows that the "parallel versions" of the first interactive proofs systems presented for quadratic residuosity, graph isomorphism, and any language in NP, are not black- box simulationzero-knowledge, unless the corresponding languages are in B PP.

...read moreread less

558

Book Chapter•10.1007/BFB0032038

On the Composition of Zero-Knowledge Proof Systems

Oded Goldreich, +1 more

- 01 Jul 1990

TL;DR: A basic question concerning zero-knowledge proof systems is whether their (sequential and/or parallel) composition is zero- knowledge too.

...read moreread less

226

•Journal Article•10.1137/S0097539705447207

An Unconditional Study of Computational Zero Knowledge

Salil Vadhan

- 01 Dec 2006

- SIAM Journal on Computing

TL;DR: To enable this combination, it is proved that every problem in CZK can be decomposed into a problem in SZK together with a set of instances from which a one-way function can be constructed.

...read moreread less

•Journal Article•10.1007/S00037-016-0136-9

Non-interactive proofs of proximity

Tom Gur, +1 more

- 01 Mar 2018

- Computational Complexity

TL;DR: In this article, a study of non-interactive proofs of proximity is presented, where the verifier is only assured of the proximity of a given statement to a correct one by rejecting inputs that are far from being valid.

...read moreread less

•Journal Article•10.1137/080722771

On the Implementation of Huge Random Objects

Oded Goldreich, +2 more

- 01 May 2010

- SIAM Journal on Computing

TL;DR: In this paper, the feasibility of implementing (huge) random objects is studied, and two types of measures of the quality of the implementation are proposed: indistinguishability and truthfulness.

...read moreread less

...

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References

•Book Chapter•10.1007/978-1-4612-0865-5_26

Probability Inequalities for sums of Bounded Random Variables

Wassily Hoeffding

- 01 Mar 1963

- Journal of the American Statistical Asso...

TL;DR: In this article, upper bounds for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt are derived for certain sums of dependent random variables such as U statistics.

...read moreread less

•Journal Article•10.1214/AOMS/1177729330

A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

Herman Chernoff

- 01 Dec 1952

- Annals of Mathematical Statistics

TL;DR: In this paper, it was shown that the likelihood ratio test for fixed sample size can be reduced to this form, and that for large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample with the second test.

...read moreread less

4.1K

•Journal Article•10.1145/6490.6503

How to construct random functions

Oded Goldreich, +2 more

- 10 Aug 1986

- Journal of the ACM

TL;DR: In this paper, a constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented, which is a deterministic polynomial-time algorithm that transforms pairs (g, r), where g is any one-way function and r is a random k-bit string, to computable functions.

...read moreread less

2.1K

Book Chapter•10.1145/3335741.3335752

How to construct random functions

Oded Goldreich, +2 more

- 04 Oct 2019

TL;DR: A constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented that has applications in cryptography, random constructions, and complexity theory.

...read moreread less

Book Chapter•10.1145/3335741.3335751

How to generate cryptographically strong sequences of pseudo random bits

Manuel Blum, +1 more

- 04 Oct 2019

TL;DR: A general algorithmic scheme for constructing polynomial-time deterministic algorithms that stretch a short secret random input into a long sequence of unpredictable pseudo-random bits is presented.

...read moreread less

1.4K

...

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Sparse pseudorandom distributions

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Most frequently asked questions

1. What are the contributions in this paper?

2. What is the importance of randomness in cryptography?

3. What is the definition of a pseudorandom generator?

4. What is the fundamental notion of pseudorandomness?

5. How does the generator check whether a set S is pseudoran?

6. What is the probability of a set of S C_ ZL?

7. How is it possible to distinguish between a uniform ensemble?

8. How is the probability that a set S is $-approximated by n?

9. What is the probability of a random set being distinguished by a given circuit?

Citations

On the Composition of Zero-Knowledge Proof Systems

On the Composition of Zero-Knowledge Proof Systems

An Unconditional Study of Computational Zero Knowledge

Non-interactive proofs of proximity

On the Implementation of Huge Random Objects

References

Probability Inequalities for sums of Bounded Random Variables

A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

How to construct random functions

How to construct random functions

How to generate cryptographically strong sequences of pseudo random bits

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