Results 11  20
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520
Software reliability via runtime resultchecking
 J. ACM
, 1997
"... We review the eld of resultchecking, discussing simple checkers and selfcorrectors. We argue that such checkers could protably be incorporated in software as an aid to ecient debugging and enhanced reliability. We consider how to modify traditional checking methodologies to make them more appropr ..."
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Cited by 124 (2 self)
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We review the eld of resultchecking, discussing simple checkers and selfcorrectors. We argue that such checkers could protably be incorporated in software as an aid to ecient debugging and enhanced reliability. We consider how to modify traditional checking methodologies to make them more appropriate for use in realtime, realnumber computer systems. In particular, we suggest that checkers should be allowed to use stored randomness: i.e., that they should be allowed to generate, preprocess, and store random bits prior to runtime, and then to use this information repeatedly in a series of runtime checks. In a case study of checking a general realnumber linear transformation (for example, a Fourier Transform), we present a simple checker which uses stored randomness, and a selfcorrector which is particularly ecient if stored
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic ..."
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Cited by 108 (19 self)
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Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation
Learning polynomials with queries: The highly noisy case
, 1995
"... Given a function f mapping nvariate inputs from a finite Kearns et. al. [21] (see also [27, 28, 22]). In the setting of agfieldFintoF, we consider the task of reconstructing a list nostic learning, the learner is to make no assumptions regarding of allnvariate degreedpolynomials which agree withf ..."
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Cited by 97 (17 self)
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Given a function f mapping nvariate inputs from a finite Kearns et. al. [21] (see also [27, 28, 22]). In the setting of agfieldFintoF, we consider the task of reconstructing a list nostic learning, the learner is to make no assumptions regarding of allnvariate degreedpolynomials which agree withfon a the natural phenomena underlying the input/output relationship tiny but nonnegligible fraction, , of the input space. We give a of the function, and the goal of the learner is to come up with a randomized algorithm for solving this task which accessesfas a simple explanation which best fits the examples. Therefore the black box and runs in time polynomial in1;nand exponential in best explanation may account for only part of the phenomena. d, provided is(pd=jFj). For the special case whend=1, In some situations, when the phenomena appears very irregular, we solve this problem for jFj>0. In this case the providing an explanation which fits only part of it is better than nothing. Interestingly, Kearns et. al. did not consider the use of running time of our algorithm is bounded by a polynomial queries (but rather examples drawn from an arbitrary distribuand exponential ind. Our algorithm generalizes a previously tion) as they were skeptical that queries could be of any help. known algorithm, due to Goldreich and Levin, that solves this We show that queries do seem to help (see below). task for the case whenF=GF(2)(andd=1).
An efficient algorithm for the sparse mixed resultant
 In Proceedings of AAECC
, 1993
"... We propose a compact formula for the mixed resultant ofa system of n+1 sparse Laurent polynomials in n variables. Our approach is conceptually simple and geometric, in that it applies a mixed subdivision to the Minkowski Sum of the input Newton polytopes. It constructs a matrix whose determinant is ..."
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Cited by 79 (23 self)
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We propose a compact formula for the mixed resultant ofa system of n+1 sparse Laurent polynomials in n variables. Our approach is conceptually simple and geometric, in that it applies a mixed subdivision to the Minkowski Sum of the input Newton polytopes. It constructs a matrix whose determinant is a nonzero multiple of the resultant so that the latter can be defined as the GCD of n + 1 such determinants. For any specialization of the coe cients there are two methods which use one extra perturbation variable and return the resultant. Our algorithm is the first to present a determinantal formula for arbitrary systems; moreover, its complexity for unmixed systems is polynomial in the resultant degree. Further empirical results suggest that this is the most efficient method to date for sparse elimination.
A verifiable random function with short proofs and keys
 PKC 2005, LNCS
, 2005
"... Abstract. We give a simple and efficient construction of a verifiable random function (VRF) on bilinear groups. Our construction is direct. In contrast to prior VRF constructions [14, 15], it avoids using an inefficient GoldreichLevin transformation, thereby saving several factors in security. Our ..."
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Cited by 76 (3 self)
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Abstract. We give a simple and efficient construction of a verifiable random function (VRF) on bilinear groups. Our construction is direct. In contrast to prior VRF constructions [14, 15], it avoids using an inefficient GoldreichLevin transformation, thereby saving several factors in security. Our proofs of security are based on a decisional bilinear DiffieHellman inversion assumption, which seems reasonable given current state of knowledge. For small message spaces, our VRF’s proofs and keys have constant size. By utilizing a collisionresistant hash function, our VRF can also be used with arbitrary message spaces. We show that our scheme can be instantiated with an elliptic group of very reasonable size. Furthermore, it can be made distributed and proactive. 1
On the size of Kakeya sets in finite fields
 J. AMS
, 2008
"... Abstract. A Kakeya set is a subset of � n, where � is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least Cn · q n, where Cn depends only on n. This answers a question of Wolff [Wol99]. 1. ..."
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Cited by 76 (5 self)
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Abstract. A Kakeya set is a subset of � n, where � is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least Cn · q n, where Cn depends only on n. This answers a question of Wolff [Wol99]. 1.
Set Reconciliation with Nearly Optimal Communication Complexity
 in International Symposium on Information Theory
, 2000
"... We consider the problem of efficiently reconciling two similar sets held by different hosts while minimizing the communication complexity. This type of problem arises naturally from gossip protocols used for the distribution of information. We describe an approach to set reconciliation based on the ..."
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Cited by 76 (16 self)
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We consider the problem of efficiently reconciling two similar sets held by different hosts while minimizing the communication complexity. This type of problem arises naturally from gossip protocols used for the distribution of information. We describe an approach to set reconciliation based on the encoding of sets as polynomials. The resulting protocols exhibit tractable computational complexity and nearly optimal communication complexity. Also, these protocols can be adapted to work over a broadcast channel, allowing many clients to reconcile with one host based on a single broadcast, even if each client is missing a different subset.
On the complexity of numerical analysis
 IN PROC. 21ST ANN. IEEE CONF. ON COMPUTATIONAL COMPLEXITY (CCC ’06
, 2006
"... We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation ..."
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Cited by 73 (5 self)
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a divisionfree straightline program producing an integer N, decide whether N> 0. • In the BlumShubSmale model, polynomial time computation over the reals (on discrete inputs) is polynomialtime equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomialtime equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.
Efficient incremental algorithms for the sparse resultant and the mixed volume
 J. Symbolic Computation
, 1995
"... We propose a new and efficient algorithm for computing the sparse resultant of a system of n + 1 polynomial equations in n unknowns. This algorithm produces a matrix whose entries are coefficients of the given polynomials and is typically smaller than the matrices obtained by previous approaches. Th ..."
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Cited by 66 (11 self)
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We propose a new and efficient algorithm for computing the sparse resultant of a system of n + 1 polynomial equations in n unknowns. This algorithm produces a matrix whose entries are coefficients of the given polynomials and is typically smaller than the matrices obtained by previous approaches. The matrix determinant is a nontrivial multiple of the sparse resultant from which the sparse resultant itself can be recovered. The algorithm is incremental in the sense that successively larger matrices are constructed until one is found with the above properties. For multigraded systems, the new algorithm produces optimal matrices, i.e., expresses the sparse resultant as a single determinant. An implementation of the algorithm is described and experimental results are presented. In addition, we propose an efficient algorithm for computing the mixed volume of n polynomials in n variables. This computation provides an upper bound on the number of common isolated roots. A publicly available implementation of the algorithm is presented and empirical results are reported which suggest that it is the fastest mixed volume code to date.
Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems
, 1992
"... The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and acce ..."
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Cited by 65 (8 self)
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The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and accepting probabilistic guarantees from the verifier [BFL91, BFLS91, FGL + 91, AS92]. We improve upon the efficiency of the proof systems developed above and obtain proofs which can be verified probabilistically by examining only a constant number of (randomly chosen) bits of the proof. The efficiently verifiable proofs constructed here rely on the structural properties of lowdegree polynomials. We explore the properties of these functions by examining some simple and basic questions about them. We consider questions of the form: • (testing) Given an oracle for a function f, is f close to a lowdegree polynomial? • (correcting) Let f be close to a lowdegree polynomial g, is it possible to efficiently reconstruct the value of g on any given input using an oracle for f? 2 The questions described above have been raised before in the context of coding theory as the problems of errordetecting and errorcorrecting of codes. More recently