NDLI: Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterministic Reductions

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Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterministic Reductions

Content Provider	Springer Nature Link
Author	Applebaum, Benny Shaltiel, Ronen Yang, Guang Artemenko, Sergei
Copyright Year	2016
Abstract	A circuit C compresses a function $${f : \{0,1\}^n\rightarrow \{0,1\}^m}$$ if given an input $${x\in \{0,1\}^n}$$ , the circuit C can shrink x to a shorter ℓ-bit string x′ such that later, a computationally unbounded solver D will be able to compute f(x) based on x′. In this paper we study the existence of functions which are incompressible by circuits of some fixed polynomial size $${s=n^c}$$ . Motivated by cryptographic applications, we focus on average-case $${(\ell,\epsilon)}$$ incompressibility, which guarantees that on a random input $${x\in \{0,1\}^n}$$ , for every size s circuit $${C:\{0,1\}^n\rightarrow \{0,1\}^{\ell}}$$ and any unbounded solver D, the success probability $${\Pr_x[D(C(x))=f(x)]}$$ is upper-bounded by $${2^{-m}+\epsilon}$$ . While this notion of incompressibility appeared in several works (e.g., Dubrov and Ishai, STOC 06), so far no explicit constructions of efficiently computable incompressible functions were known. In this work, we present the following results: (1) Assuming that E is hard for exponential size nondeterministic circuits, we construct a polynomial time computable boolean function $${f:\{0,1\}^n\rightarrow \{0,1\}}$$ f : { 0 , 1 } n → { 0 , 1 } which is incompressible by size n c circuits with communication $${\ell=(1-o(1)) \cdot n}$$ ℓ = ( 1 - o ( 1 ) ) · n and error $${\epsilon=n^{-c}}$$ ϵ = n - c . Our technique generalizes to the case of PRGs against nonboolean circuits, improving and simplifying the previous construction of Shaltiel and Artemenko (STOC 14). (2) We show that it is possible to achieve negligible error parameter $${\epsilon=n^{-\omega(1)}}$$ ϵ = n - ω ( 1 ) for nonboolean functions. Specifically, assuming that E is hard for exponential size $${\Sigma_3}$$ Σ 3 -circuits, we construct a nonboolean function $${f:\{0,1\}^n\rightarrow \{0,1\}^m}$$ f : { 0 , 1 } n → { 0 , 1 } m which is incompressible by size n c circuits with $${\ell=\Omega(n)}$$ ℓ = Ω ( n ) and extremely small $${\epsilon=n^{-c} \cdot 2^{-m}}$$ ϵ = n - c · 2 - m . Our construction combines the techniques of Trevisan and Vadhan (FOCS 00) with a new notion of relative error deterministic extractor which may be of independent interest. (3) We show that the task of constructing an incompressible boolean function $${f:\{0,1\}^n\rightarrow \{0,1\}}$$ f : { 0 , 1 } n → { 0 , 1 } with negligible error parameter $${\epsilon}$$ ϵ cannot be achieved by “existing proof techniques”. Namely, nondeterministic reductions (or even $${\Sigma_i}$$ Σ i reductions) cannot get $${\epsilon=n^{-\omega(1)}}$$ ϵ = n - ω ( 1 ) for boolean incompressible functions. Our results also apply to constructions of standard Nisan-Wigderson type PRGs and (standard) boolean functions that are hard on average, explaining, in retrospect, the limitations of existing constructions. Our impossibility result builds on an approach of Shaltiel and Viola (STOC 08).
Ending Page	418
Page Count	70
Starting Page	349
File Format	PDF
ISSN	10163328
e-ISSN	14208954
Journal	Computational complexity
Issue Number	2
Volume Number	25
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-04-11
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Computational Mathematics and Numerical Analysis extractors pseudorandomness nondeterministic reductions compression Algorithm Analysis and Problem Complexity
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Computational Theory and Mathematics Computational Mathematics

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