Lower bounds for non-black-box zero knowledge

The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where ‘pathetic’ lower bounds of the form n 1+ ϵ would suffice to derandomize interesting classes of probabilistic algorithms. We show the following: — If the word problem over S 5 requires constant-depth threshold circuits of size n 1+ ϵ for some ϵ >0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size). — If there are no constant-depth arithmetic circuits of size n 1+ ϵ for the problem of multiplying a sequence of n 3×3 matrices, then, for every constant d , black-box identity testing for depth- d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

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The Unrestricted Black-Box Complexity of Jump Functions

Evolutionary Computation ◽

10.1162/evco_a_00185 ◽

2016 ◽

Vol 24 (4) ◽

pp. 719-744 ◽

Cited By ~ 7

Author(s):

Maxim Buzdalov ◽

Benjamin Doerr ◽

Mikhail Kever

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Upper Bounds ◽

Black Box ◽

Problem Instance ◽

Theory Approach ◽

Function Classes ◽

Fitness Value ◽

Jump Function ◽

Jump Sizes

We analyze the unrestricted black-box complexity of the Jump function classes for different jump sizes. For upper bounds, we present three algorithms for small, medium, and extreme jump sizes. We prove a matrix lower bound theorem which is capable of giving better lower bounds than the classic information theory approach. Using this theorem, we prove lower bounds that almost match the upper bounds. For the case of extreme jump functions, which apart from the optimum reveal only the middle fitness value(s), we use an additional lower bound argument to show that any black-box algorithm does not gain significant insight about the problem instance from the first [Formula: see text] fitness evaluations. This, together with our upper bound, shows that the black-box complexity of extreme jump functions is [Formula: see text].

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A Black-Box Approach to Post-Quantum Zero-Knowledge in Constant Rounds

Advances in Cryptology – CRYPTO 2021 - Lecture Notes in Computer Science ◽

10.1007/978-3-030-84242-0_12 ◽

2021 ◽

pp. 315-345

Author(s):

Nai-Hui Chia ◽

Kai-Min Chung ◽

Takashi Yamakawa

Keyword(s):

Black Box ◽

Zero Knowledge

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Upper and Lower Bounds on Black-Box Steganography

Journal of Cryptology ◽

10.1007/s00145-008-9020-3 ◽

2008 ◽

Vol 22 (3) ◽

pp. 365-394 ◽

Cited By ~ 13

Author(s):

Nenad Dedić ◽

Gene Itkis ◽

Leonid Reyzin ◽

Scott Russell

Keyword(s):

Lower Bounds ◽

Black Box ◽

Upper And Lower Bounds

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Black-Box Concurrent Zero-Knowledge Requires (Almost) Logarithmically Many Rounds

SIAM Journal on Computing ◽

10.1137/s0097539701392949 ◽

2002 ◽

Vol 32 (1) ◽

pp. 1-47 ◽

Cited By ~ 34

Author(s):

Ran Canetti ◽

Joe Kilian ◽

Erez Petrank ◽

Alon Rosen

Keyword(s):

Black Box ◽

Zero Knowledge

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