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Cacheoblivious algorithms
, 1999
"... requirements for the degree of Master of Science. This thesis presents "cacheoblivious " algorithms that use asymptotically optimal amounts of work, and move data asymptotically optimally among multiple levels of cache. An algorithm is cache oblivious if no program variables dependent on ..."
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Cited by 90 (1 self)
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requirements for the degree of Master of Science. This thesis presents "cacheoblivious " algorithms that use asymptotically optimal amounts of work, and move data asymptotically optimally among multiple levels of cache. An algorithm is cache oblivious if no program variables dependent on hardware configuration parameters, such as cache size and cacheline length need to be tuned to minimize the number of cache misses. We show that the ordinary algorithms for matrix transposition, matrix multiplication, sorting, and Jacobistyle multipass filtering are not cache optimal. We present algorithms for rectangular matrix transposition, FFT, sorting, and multipass filters, which are asymptotically optimal on computers with multiple levels of caches. For a cache with size Z and cacheline length L, where Z = (L2), the number of cache misses for an m x n matrix transpose is E(1 + mn/L). The number of cache misses for either an npoint FFT or the sorting of n numbers is 0(1 + (n/L)(1 + logzn)). The cache complexity of computing n time steps of a Jacobistyle multipass filter on an array of size n is E(1 + n/L + n2 /ZL). We also give an 8(mnp)work algorithm to multiply an m x n matrix by an n x p matrix
Higher Order Correlation Attacks, XL algorithm and Cryptanalysis of Toyocrypt
, 2002
"... Abstract. A popular technique to construct stream ciphers is to use a linear sequence generator with a very large period and good statistical properties and a nonlinear filter. There is abundant literature on how to use linear approximations of this nonlinear function to attack the cipher, which i ..."
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Cited by 67 (8 self)
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Abstract. A popular technique to construct stream ciphers is to use a linear sequence generator with a very large period and good statistical properties and a nonlinear filter. There is abundant literature on how to use linear approximations of this nonlinear function to attack the cipher, which is known as (fast) correlation attacks. In this paper we explore nonlinear approximations, much less well known. We will reduce the cryptanalysis of a stream cipher to solving an overdefined system of multivariate equations. At Eurocrypt 2000, Courtois, Klimov, Patarin and Shamir have introduced the XL algorithm for solving systems of overdefined multivariate quadratic equations over finite fields. The exact complexity of the XL algorithm remains an open problem. and some authors such as T.T.Moh have expressed serious doubts whether it actually works very well. However there is no doubt that such methods work very well for largely overdefined systems (much more equations than variables), and we confirm this by computer simulations. Luckily systems we obtain in cryptanalysis of stream ciphers are precisely very overdefined. In this paper we will show how to break efficiently stream ciphers that are known to be immune to all the previously known attacks. For example, we will be able to break the stream
Arithmetic Circuits: a survey of recent results and open questions
"... A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last fi ..."
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Cited by 65 (5 self)
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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we
2006, Quantum verification of matrix products
 Proceedings of the 17th ACMSIAM Symposium on Discrete Algorithms
"... We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worstcase time O(n 5/3) and expected time O(n 5/3 / min(w, √ n) 1/3), where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in ..."
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Cited by 47 (0 self)
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We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worstcase time O(n 5/3) and expected time O(n 5/3 / min(w, √ n) 1/3), where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in time O(n 7/4). We also present a quantum matrix multiplication algorithm that is efficient when the result has few nonzero entries. 1
The Fastest And Shortest Algorithm For All WellDefined Problems
, 2002
"... An algorithm M is described that solves any welldefined problem p as quickly as the fastest algorithm computing a solution to p, save for a factor of 5 and loworder additive terms. M optimally distributes resources between the execution of provably correct psolving programs and an enumeration of ..."
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Cited by 43 (7 self)
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An algorithm M is described that solves any welldefined problem p as quickly as the fastest algorithm computing a solution to p, save for a factor of 5 and loworder additive terms. M optimally distributes resources between the execution of provably correct psolving programs and an enumeration of all proofs, including relevant proofs of program correctness and of time bounds on program runtimes. M avoids Blum's speedup theorem by ignoring programs without correctness proof. M has broader applicability and can be faster than Levin's universal search, the fastest method for inverting functions save for a large multiplicative constant. An extension of Kolmogorov complexity and two novel natural measures of function complexity are used to show that the most efficient program computing some function f is also among the shortest programs provably computing f.
Allpairs shortest paths with real weights in O(n³ / log n) time
 PROC. OF THE 9TH WADS, LECTURE NOTES IN COMPUTER SCIENCE 3608
, 2005
"... We describe an O(n³ / log n) ..."
Model checking propositional dynamic logic with all extras
 Journal of Applied Logic
, 2005
"... This paper presents a model checking algorithm for Propositional Dynamic Logic (PDL) with looping, repeat, test, intersection, converse, program complementation as well as contextfree programs. The algorithm shows that the model checking problem for PDL remains PTIMEcomplete in the presence of all ..."
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Cited by 32 (6 self)
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This paper presents a model checking algorithm for Propositional Dynamic Logic (PDL) with looping, repeat, test, intersection, converse, program complementation as well as contextfree programs. The algorithm shows that the model checking problem for PDL remains PTIMEcomplete in the presence of all these operators, in contrast to the high increase in complexity that they cause for the satisfiability problem.
Fast ContextFree Grammar Parsing Requires Fast Boolean Matrix Multiplication
, 2002
"... In 1975, Valiant showed that Boolean matrix multiplication can be used for parsing contextfree grammars (CFGs), yielding the asympotically fastest (although not practical) CFG parsing algorithm known. We prove a dual result: any CFG parser with time complexity $O(g n^{3  \epsilson})$, where $g$ is ..."
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Cited by 32 (0 self)
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In 1975, Valiant showed that Boolean matrix multiplication can be used for parsing contextfree grammars (CFGs), yielding the asympotically fastest (although not practical) CFG parsing algorithm known. We prove a dual result: any CFG parser with time complexity $O(g n^{3  \epsilson})$, where $g$ is the size of the grammar and $n$ is the length of the input string, can be efficiently converted into an algorithm to multiply $m \times m$ Boolean matrices in time $O(m^{3  \epsilon/3})$. Given that practical, substantially subcubic Boolean matrix multiplication algorithms have been quite difficult to find, we thus explain why there has been little progress in developing practical, substantially subcubic general CFG parsers. In proving this result, we also develop a formalization of the notion of parsing.
Faster Possibility Detection by Combining Two Approaches
 IN PROCEEDINGS OF THE WORKSHOP ON DISTRIBUTED ALGORITHMS (WDAG
, 1995
"... A new algorithm is presented for detecting whether a particular computation of an asynchronous distributed system satisfies Poss Φ (read “possibly Φ”), meaning the system could have passed through a global state satisfying Φ. Like the algorithm of Cooper and Marzullo, Φ may be any global state pre ..."
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Cited by 21 (2 self)
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A new algorithm is presented for detecting whether a particular computation of an asynchronous distributed system satisfies Poss Φ (read “possibly Φ”), meaning the system could have passed through a global state satisfying Φ. Like the algorithm of Cooper and Marzullo, Φ may be any global state predicate; and like the algorithm of Garg and Waldecker, Poss Φ is detected quite efficiently if Φ has a certain structure. The new algorithm exploits the structure of some predicates Φ not handled by Garg and Waldecker’s algorithm to detect Poss Φ more efficiently than is possible with any algorithm that, like Cooper and Marzullo’s, evaluates Φ on every global state through which the system could have passed. A second algorithm is also presented for offline detection of Poss Φ. It uses Strassen’s scheme for fast matrix multiplication. The intrinsic complexity of offline and online detection of Poss Φ is discussed.