Lower bounds for the complexity of reliable Boolean circuits with noisy gates

Gács, Péter; Gál, Anna

doi:10.1109/18.312190

Cited by 61 publications

(36 citation statements)

References 9 publications

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“…In general, this result cannot be improved, and logarithmic redundancy is inevitable [4,8,6,7]. But this lower bound is caused by the need to encode the input with some error-correcting code and then to decode the answer.…”

Section: Introductionmentioning

confidence: 99%

Reliable Computations Based on Locally Decodable Codes

Romashchenko¹

2006

Stacs 2006

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We investigate the coded model of fault-tolerant computations introduced by D. Spielman. In this model the input and the output of a computation circuit is treated as words in some error-correcting code. A circuit is said to compute some function correctly if for an input which is a encoded argument of the function, the output, been decoded, is the value of the function on the given argument.We consider two models of faults. In the first one we suppose that an elementary processor at each step can be corrupted with some small probability, and faults of different processors are independent. For this model, we prove that a parallel computation running on n elementary non-faulty processors in time t = poly(n) can be simulated on O(n log n/ log log n) faulty processors in time O(t log log n). Note that we get sub-logarithmic blow-up of the memory, which cannot be achieved in the classic model of faulty boolean circuit, where the input is not encoded.In the second model, we assume that at each step some fixed fraction of elementary processors can be corrupted by an adversary, who is free to chose these processors arbitrarily. We show that in this model any computation can be made reliable with exponential blow-up of the memory.Our method employs a sort of mixing mappings, which enjoy some properties of expanders. Based on mixing mappings, we implement an effective selfcorrecting procedure for an array of faulty processors.

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Section: Introductionmentioning

confidence: 99%

Reliable Computations Based on Locally Decodable Codes

Romashchenko¹

2006

Stacs 2006

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“…Under the assumption that all gates fail independently with probability ǫ, the size and depth required for repetition-based fault tolerance has been shown to be within a constant factor of optimal for many basic functions (XOR for example) [5], [6], [7]. The derivation of these bounds highlight a shortcoming of the von Neumann model.…”

Section: Introductionmentioning

confidence: 99%

A framework for coded computation

Rachlin

Savage

2008

2008 IEEE International Symposium on Information Theory

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Abstract-Error-correcting codes have been very successful in protecting against errors in data transmission. Computing on encoded data, however, has proved more difficult. In this paper we extend a framework introduced by Spielman [14] for computing on encoded data. This new formulation offers significantly more design flexibility, reduced overhead, and simplicity. It allows for a larger variety of codes to be used in computation and makes explicit conditions on codes that are compatible with computation. We also provide a lower bound on the overhead required for a single step of coded computation.

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“…Von Neumann studied a model of random errors, where each gate has an (arbitrary) error independently with small fixed probability, and his goal was to obtain correctness (as opposed to privacy). There have been numerous follow up papers to this seminal work, including [16,53,52,27,23,36,28,22], who considered the same noise model, ultimately showing that any circuit of size σ can be encoded into a circuit of size O(σ log σ) that tolerates a fixed constant noise rate, and that any such encoding must have size Ω(σ log σ).…”

Section: Related Workmentioning

confidence: 99%

Securing Circuits against Constant-Rate Tampering

Dachman-Soled

Kalai

2012

Lecture Notes in Computer Science

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Abstract. We present a compiler that converts any circuit into one that remains secure even if a constant fraction of its wires are tampered with. Following the seminal work of Ishai et al. (Eurocrypt 2006), we consider adversaries who may choose an arbitrary set of wires to corrupt, and may set each such wire to 0 or to 1, or may toggle with the wire. We prove that such adversaries, who continuously tamper with the circuit, can learn at most logarithmically many bits of secret information (in addition to black-box access to the circuit). Our results are information theoretic.

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Lower bounds for the complexity of reliable Boolean circuits with noisy gates

Cited by 61 publications

References 9 publications

Reliable Computations Based on Locally Decodable Codes

Reliable Computations Based on Locally Decodable Codes

A framework for coded computation

Securing Circuits against Constant-Rate Tampering

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