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17
Lower bounds for the noisy broadcast problem
 In Proceedings of the 46 th IEEE Symposium on Foundations of Computer Science (FOCS 2005
, 2005
"... We prove the first nontrivial (super linear) lower bound in the noisy broadcast model, defined by El Gamal in [6]. In this model there are n + 1 processors P0, P1,..., Pn, each of which is initially given a private input bit xi. The goal is for P0 to learn the value of f(x1,..., xn), for some speci ..."
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We prove the first nontrivial (super linear) lower bound in the noisy broadcast model, defined by El Gamal in [6]. In this model there are n + 1 processors P0, P1,..., Pn, each of which is initially given a private input bit xi. The goal is for P0 to learn the value of f(x1,..., xn), for some specified function f, using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [16] gave a noiseresistant protocol that allows P0 to learn the entire input with constant probability in O(n log log n) broadcasts. We prove that Gallager’s protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of Ω(n log n) on the depth of a tree that learns the entire input. We also show an Ω(n log log n) lower bound for the number of broadcasts required to compute certain explicit booleanvalued functions, when the correct output must be attained with probability at least 1 − n −α for a constant parameter α> 0 (this bound applies to all threshold functions, as well as any other booleanvalued function with linear sensitivity). This bound also follows by reduction from a lower bound of Ω(n log n) on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) Ω(n) lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts.
Digital Sums And DivideAndConquer Recurrences: Fourier Expansions And Absolute Convergence
, 2002
"... We propose means of computing the Fourier expansions of periodic functions appearing in higher moments of the sumofdigits function and in the solutions of some divideandconquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to eciently computing nu ..."
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We propose means of computing the Fourier expansions of periodic functions appearing in higher moments of the sumofdigits function and in the solutions of some divideandconquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to eciently computing numerically the coecients involved to high precision.
Fault Tolerant Circuits and Probabilistically Checkable Proofs
 IN PROCEEDINGS OF THE 10TH ANNUAL STRUCTURE IN COMPLEXITY THEORY
, 1995
"... We introduce a new model of fault tolerant Boolean circuits. We allow an adversary to choose some gates to be faulty, unlike the model considered by von Neumann and Pippenger where the errors are randomly distributed. Our model also differs from previous models that considered nonrandom faults. Our ..."
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We introduce a new model of fault tolerant Boolean circuits. We allow an adversary to choose some gates to be faulty, unlike the model considered by von Neumann and Pippenger where the errors are randomly distributed. Our model also differs from previous models that considered nonrandom faults. Our main result is that every symmetric function has a small (size O(n), depth O(logn)) fault tolerant circuit that will compute the function adequately, even if a small constant fraction of the gates is modified by an adversary. We also show a perhaps unexpected relation between our model and probabilistically checkable proofs.
Making Polynomials Robust to Noise
, 2009
"... A basic question in any model of computation is how to reliably compute a given function when its inputs are subject to noise. Buhrman, Newman, Röhrig, and de Wolf (2003) posed the noisy computation problem for real polynomials. We give a complete solution to this problem. For any δ> 0 and any p ..."
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A basic question in any model of computation is how to reliably compute a given function when its inputs are subject to noise. Buhrman, Newman, Röhrig, and de Wolf (2003) posed the noisy computation problem for real polynomials. We give a complete solution to this problem. For any δ> 0 and any polynomial p: {0,1} n → [−1,1], we construct a corresponding polynomial probust: R n → R of degree O(deg p + log1/δ) that is robust to noise in the inputs: p(x) − probust(x + ε)  < δ for all x ∈ {0,1} n and all ε ∈ [−1/3,1/3] n. This result is optimal with respect to all parameters. We construct probust explicitly for each p. Previously, it was open to give such a construction even for p = x1 ⊕ x2 ⊕ ·· · ⊕ xn (Buhrman et al., 2003). The proof contributes a technique of independent interest, which allows one to force partial cancellation of error terms in a polynomial.
Nondeterministic combination of connectives
, 2011
"... Combined connectives arise when combining logics [12] and are also useful for analyzing the common properties of two connectives within a given logic [11]. A nondeterministic semantics and a Hilbert calculus are proposed for the meetcombination of connectives (and other language constructors) of a ..."
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Combined connectives arise when combining logics [12] and are also useful for analyzing the common properties of two connectives within a given logic [11]. A nondeterministic semantics and a Hilbert calculus are proposed for the meetcombination of connectives (and other language constructors) of any matrix logic endowed with a Hilbert calculus. The logic enriched with such combined connectives is shown to be a conservative extension of the original logic. It is also proved that both soundness and completeness are preserved. Illustrations are provided for classical propositional logic.
Rounds vs Queries Tradeoff in Noisy Computation
, 2006
"... We show that a noisy parallel decision tree making O(n) queries needs Ω(log ∗ n) rounds to compute OR of n bits. This answers a question of Newman [IEEE Conference on Computational Complexity, 2004, 113–124]. We prove more general tradeoffs between the number of queries and rounds. We also settle a ..."
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We show that a noisy parallel decision tree making O(n) queries needs Ω(log ∗ n) rounds to compute OR of n bits. This answers a question of Newman [IEEE Conference on Computational Complexity, 2004, 113–124]. We prove more general tradeoffs between the number of queries and rounds. We also settle a similar question for computing MAX in the noisy comparison tree model; these results bring out interesting differences among the noise models. 1
Optimal Boolean Circuits with Unreliable Gates (Extended Abstract)
, 1998
"... Given a box of AND gates, each of fanin 2, it is known that k OR gates are mistakenly put into a box of AND gates. Suppose that there is no tool to test which are OR gates. In order to compute the AND function of two values correctly, how many gates are necessary to construct an errorfree circu ..."
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Given a box of AND gates, each of fanin 2, it is known that k OR gates are mistakenly put into a box of AND gates. Suppose that there is no tool to test which are OR gates. In order to compute the AND function of two values correctly, how many gates are necessary to construct an errorfree circuit? This is the socalled AND/OR problem [3]. We consider modular circuits, which are leveled circuits and whose lowest level with exactly two gates has the roots of two disjoint subcircuits. Kleitman et al. [3] showed that it needs at most O((k + 1) log 2 3 ) gates to tolerate k OR gates, where they gave a construction achieving this bound. By combinatorial approach, we prove that for any k the optimal modular circuit needs 1+2 P k i=1 2 z(i) gates, where z(i) is the number of zero in the binary representation of i. Also we characterize the fault tolerant circuit (not necessarily modular) with polynomial over a field. 1 Introduction Designing Boolean circuits that can tolerat...
Securing Circuits Against ConstantRate Tampering
"... 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 eac ..."
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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 blackbox access to the circuit). Our results are information theoretic. Key words: sidechannel attacks, tampering, circuit compiler, PCP of proximity
Formulas Resilient to ShortCircuit Errors
"... We show how to efficiently convert any boolean formula F into a boolean formula E that is resilient to shortcircuit errors (as introduced by Kleitman et al. [KLM94]). A gate has a shortcircuit error when the value it computes is replaced by the value of one of its inputs. We guarantee that E compu ..."
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We show how to efficiently convert any boolean formula F into a boolean formula E that is resilient to shortcircuit errors (as introduced by Kleitman et al. [KLM94]). A gate has a shortcircuit error when the value it computes is replaced by the value of one of its inputs. We guarantee that E computes the same function as F, as long as at most (1/10 − ε) of the gates on each path from the output to an input have been corrupted in E. The corruptions may be chosen adversarially, and may depend on the formula E and even on the input. We obtain our result by extending the KarchmerWigderson connection between formulas and communication protocols to the setting of adversarial error. This enables us to obtain errorresilient formulas from errorresilient communication protocols.