# P = BPP if E requires exponential circuits: derandomizing the XOR lemma

@inproceedings{Impagliazzo1997PB, title={P = BPP if E requires exponential circuits: derandomizing the XOR lemma}, author={Russell Impagliazzo and Avi Wigderson}, booktitle={Symposium on the Theory of Computing}, year={1997} }

Russell Impagliazzo* Avi Wigdersont Department of Computer Science Institute of Computer Science University of California Hebrew University San Diego, CA 91097-0114 Jerusalem, Israel russell@cs .ucsd. edu avi@cs .huj i. ac. il Yao showed that the XOR of independent random instances of a somewhat hard Boolean problem becomes almost completely unpredictable. In this paper we show that, in non-uniform settings, total independence is not necessary for this result to hold. We give a pseudo-random…

## 559 Citations

### Pseudorandom generators without the XOR Lemma

- MathematicsElectron. Colloquium Comput. Complex.
- 1998

Two different approaches are presented to proving the main result of Impagliazzo and Wigderson that if there exists a decision problem solvable in time 2/sup O(n)/ and having circuit complexity 2/Sup /spl Omega/(n)/ then P=BPP.

### Pseudorandom generators without the XOR lemma

- MathematicsProceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)
- 1999

Two different approaches are presented to proving the main result of Impagliazzo and Wigderson that if there exists a decision problem solvable in time 2/sup O(n)/ and having circuit complexity 2/Sup /spl Omega/(n)/ then P=BPP.

### Holographic proofs and derandomization

- Mathematics18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings.
- 2003

We derive a stronger consequence of EXP having polynomial-size circuits than was known previously, namely that there is a simulation of P in MAPOLYLOG that fools all deterministic polynomial-time…

### Derandomizing Arthur–Merlin Games using Hitting Sets

- Mathematics40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
- 1999

Abstract.We prove that AM (and hence Graph Nonisomorphism) is in NP if for some ε > 0, some language in NE ∩ coNE requires nondeterministic circuits of size 2ε n This improves results of Arvind and…

### Derandomization from Hardness CS 254 Final Paper March 2019

- Mathematics, Computer Science
- 2019

These results gave derandomizations of BPP, the class of languages decidable with randomized algorithms in polynomial time, into various deterministic computation classes based on various assumptions about both uniform and non-uniform hardness.

### On the Derandomization of Constant Depth Circuits

- MathematicsRANDOM-APPROX
- 2001

This appears to be the first use of the XOR lemma to prove an unconditional inapproximability result for an explicit function (in this case parity) and it is proved that BPAC0 can be simulated by uniform quasipolynomial size constant depth circuits.

### Pseudorandom generators without the XOR Lemma (extended abstract)

- Computer ScienceSTOC '99
- 1999

Two different approaches are presented to proving the main result of lmpagliazzo and Wigderso: the first construction of a pseudorandomgenerator that works with a mildly hard predicate without doing hardness amplification and a list-decoding algorithm that improves and simplifies a previous one by Arora and Sudan.

### On Resource-Bounded Measure and Pseudorandomness

- Mathematics, Computer ScienceFSTTCS
- 1997

This paper unconditionally shows the inclusion MA ⊑ ZPPNPNP and that MA∩ coMA is low for ZPPNP and uses the Nisan-Wigderson design of a pseudorandom generator to answer some open questions regarding the derandomization of the classes BP·σ k P and BP·θ k P under plausible measure theoretic assumptions.

### Derandomization and interlacing polynomials Constructive and nonconstructive methods in combinatorics and TCS Lecturer

- Mathematics, Computer Science
- 2018

The Nisan-Widgerson PRG is looked at, which, in tandem with hardness amplification results, is strong enough to prove that BPP = P under the assumption E has a language whose decision problem requires 2-sized circuits.

### 1 Direct Product Theorems

- Computer Science, Mathematics
- 2009

Such theorems are useful in the following settings: (i) Cryptography: Much of Cryptography is based on existence of problems which are hard to solve on average. Direct Product Theorems provide a…

## References

SHOWING 1-10 OF 28 REFERENCES

### Hitting Sets Derandomize BPP

- Computer Science, MathematicsICALP
- 1995

A polynomial time deterministic algorithm is presented which uses any given hitting set to approximate the fractions of 1's in the output of any boolean circuit ofPolynomial size.

### BPP has subexponential time simulations unlessEXPTIME has publishable proofs

- Mathematics, Computer Science[1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference
- 1991

AbstractWe show thatBPP can be simulated in subexponential time for infinitely many input lengths unless exponential timeℴ collapses to the second level of the polynomial-time hierarchy.ℴ has…

### Hard-core distributions for somewhat hard problems

- Mathematics, Computer ScienceProceedings of IEEE 36th Annual Foundations of Computer Science
- 1995

It is shown that for any decision problem that cannot be 1-/spl delta/ approximated by circuits of a given size, there is a specific "hard core" set of inputs which is at least a /splDelta/ fraction of all inputs and on which no circuit of a slightly smaller size can get even a small advantage over a random guess.

### On Yao's XOR-Lemma

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 1995

The XOR-Lemma is derived from Yao's Lemma by first proving that a function constructed by concatenating the values of the original function on several independent instances is much more unpredictable, with respect to specified complexity bounds, than the originalfunction.

### Direct product results and the GCD problem, in old and new communication models

- Computer Science, MathematicsSTOC '97
- 1997

The improved parallel repetition theorem of 2-prover games is applied to derive, for the first time, a direct product theorem for communication complexity, based on interaction between the two models.

### How to recycle random bits

- Computer Science, Mathematics30th Annual Symposium on Foundations of Computer Science
- 1989

It is shown that modified versions of the linear congruential generator and the shift register generator are provably good for amplifying the correctness of a probabilistic algorithm. More precisely,…

### A parallel repetition theorem

- Computer Science, MathematicsSTOC '95
- 1995

We show that a parallel repetition of any two-prover one-round proof system (MIP(2,1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The…

### How to generate cryptographically strong sequences of pseudo random bits

- Computer Science, Mathematics23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
- 1982

A more operative definition of Randomness should be pursued in the light of modern Complexity Theory.

### Security preserving amplification of hardness

- Computer Science, MathematicsProceedings [1990] 31st Annual Symposium on Foundations of Computer Science
- 1990

The task of transforming a weak one-way function (which may be easily inverted on all but a polynomial fraction of the range) into a strong one-way function (which can be easily inverted only on a…