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84
Simulating Physics with Computers
 SIAM Journal on Computing
, 1982
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 393 (1 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored. AMS subject classifications: 82P10, 11Y05, 68Q10. 1 Introduction One of the first results in the mathematics of computation, which underlies the subsequent development of much of theoretical computer science, was the distinction between computable and ...
SelfTesting/Correcting with Applications to Numerical Problems
, 1990
"... Suppose someone gives us an extremely fast program P that we can call as a black box to compute a function f . Should we trust that P works correctly? A selftesting/correcting pair allows us to: (1) estimate the probability that P (x) 6= f(x) when x is randomly chosen; (2) on any input x, compute ..."
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Cited by 340 (26 self)
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Suppose someone gives us an extremely fast program P that we can call as a black box to compute a function f . Should we trust that P works correctly? A selftesting/correcting pair allows us to: (1) estimate the probability that P (x) 6= f(x) when x is randomly chosen; (2) on any input x, compute f(x) correctly as long as P is not too faulty on average. Furthermore, both (1) and (2) take time only slightly more than Computer Science Division, U.C. Berkeley, Berkeley, California 94720, Supported by NSF Grant No. CCR 8813632. y International Computer Science Institute, Berkeley, California 94704 z Computer Science Division, U.C. Berkeley, Berkeley, California 94720, Supported by an IBM Graduate Fellowship and NSF Grant No. CCR 8813632. the original running time of P . We present general techniques for constructing simple to program selftesting /correcting pairs for a variety of numerical problems, including integer multiplication, modular multiplication, matrix multiplicatio...
On Fast Multiplication of Polynomials Over Arbitrary Algebras
 Acta Informatica
, 1991
"... this paper we generalize the wellknown SchonhageStrassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra A. Our main result is an algorithm to multiply polyno ..."
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Cited by 156 (6 self)
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this paper we generalize the wellknown SchonhageStrassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra A. Our main result is an algorithm to multiply polynomials of degree ! n in
Robust PCPs of Proximity, Shorter PCPs and Applications to Coding
 in Proc. 36th ACM Symp. on Theory of Computing
, 2004
"... We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) ..."
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Cited by 80 (25 self)
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We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) Boolean queries.
All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication
 Journal of the ACM
, 2000
"... We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves... ..."
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Cited by 60 (6 self)
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We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves...
Exact and Approximate Distances in Graphs  a survey
 In ESA
, 2001
"... We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems. ..."
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Cited by 57 (0 self)
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We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.
HighSpeed RSA Implementation
, 1994
"... Introduction to Arithmetic for Digital System Designers. New York, NY: Holt, Rinehart and Winston, 1982. #52# Y. Yacobi. Exponentiating faster with addition chains. In I. B. Damg#ard, editor, Advances in Cryptology  EUROCRYPT 90, Lecture Notes in Computer Science, No. 473, pages 222#229. New York ..."
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Cited by 48 (7 self)
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Introduction to Arithmetic for Digital System Designers. New York, NY: Holt, Rinehart and Winston, 1982. #52# Y. Yacobi. Exponentiating faster with addition chains. In I. B. Damg#ard, editor, Advances in Cryptology  EUROCRYPT 90, Lecture Notes in Computer Science, No. 473, pages 222#229. New York, NY: SpringerVerlag, 1990. #53# A. C.C. Yao. On the evaluation of powers. SIAM Journal on Computing, 5#1#:100#103, March 1976. Bibliography 69 #25# C#. K. Ko#c and C. Y. Hung. Multioperand modulo addition using carry save adders. Electronics Letters, 26#6#:361#363, 15th March 1990. #26# C# . K. Ko#c and C. Y. Hung. Bitlevel systolic arrays for modular multiplication. Journal of VLSI Signal Processing, 3#3#:215#223, 1991. #27# C#. K. Ko#c and C. Y. Hung. Adaptive mary segmentation and canonical recoding algorithms for multiplication of large binary numbers. Computers and Mathematics with Ap
Improved Sparse Multivariate Polynomial Interpolation Algorithms*
, 1988
"... . We consider the problem of interpolating sparse multivariate polynomials from their values. We discuss two algorithms for sparse interpolation, one due to BenOr and Tiwari (1988) and the other due to Zippel (1988). We present efficient algorithms for finding the rank of certain special Toeplitz s ..."
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Cited by 48 (11 self)
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. We consider the problem of interpolating sparse multivariate polynomials from their values. We discuss two algorithms for sparse interpolation, one due to BenOr and Tiwari (1988) and the other due to Zippel (1988). We present efficient algorithms for finding the rank of certain special Toeplitz systems arising in the BenOr and Tiwari algorithm and for solving transposed Vandermonde systems of equations, the use of which greatly improves the time complexities of the two interpolation algorithms. 1. Introduction We consider the problem of interpolating a multivariate polynomial over a field of characteristic zero from its values at several points. While techniques for interpolating dense polynomials have been known for a long time (e.g., Lagrangian interpolation formula for univariate polynomials), and probabilistic algorithms for interpolating sparse multivariate polynomials have existed since 1979 (Zippel 1979, 1988), until recently no algorithm was known to interpolate sparse mul...
Faster Integer Multiplication
 STOC'07
, 2007
"... For more than 35 years, the fastest known method for integer multiplication has been the SchönhageStrassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complex ..."
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Cited by 41 (0 self)
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For more than 35 years, the fastest known method for integer multiplication has been the SchönhageStrassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complexity of an optimal algorithm is Θ(n log n). We present a major step towards closing the gap from above by presenting an algorithm running in time n log n 2 O(log ∗ n). The main result is for boolean circuits as well as for multitape Turing machines, but it has consequences to other models of computation as well.