Results 1  10
of
32
Nearly Optimal Algorithms For Canonical Matrix Forms
, 1993
"... A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nea ..."
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Cited by 58 (11 self)
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A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nearly matches the lower bound of \Omega(MM(n)) operations in K for this problem, and improves on the O(n^4) operations in K required by the previously best known algorithms. We also demonstrate a fast parallel implementation of our algorithm for the Frobenius form, which is processorefficient on a PRAM. As an application we give an algorithm to evaluate a polynomial g(x) in K[x] at T which requires only O~(MM(n)) operations in K when deg g < n^2. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known.
Parallel algorithms for matrix normal forms
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 1990
"... Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field. The algorithm has two important advantages over our previous one: the multipliers relating the Smith form to the inpu ..."
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Cited by 33 (2 self)
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Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field. The algorithm has two important advantages over our previous one: the multipliers relating the Smith form to the input matrix are computed, and the algorithm is probabilistic of Las Veg as type, i.e., always finds the correct answer. The Smith form algorithm is also a good sequential algorithm. Our algorithm reduces the problem of Smith form computation to two Hermite form computations. Thus the Smith form problem has complexity asymptotically that of the Hermite form problem. We also construct fast parallel algorithms for Jordan normal form and testing similarity of matrices. Both the similarity and nonsimilarity problems are in the complexity class RNC for the usual coefficient fields, i.e., they can be probabilistically decided in polylogarithmic time using polynomially many processors.
Computing Popov and Hermite forms of polynomial matrices
 In International Symposium on Symbolic and Algebmic Computation, Zutich, .%isse
, 1996
"... For a polynomial matrix P(z) of degree d in M~,~(K[z]) where K is a commutative field, a reduction to the Hermite normal form can be computed in O (ndM(n) + M(nd)) arithmetic operations if M(n) is the time required to multiply two n x n matrices over K. Further, a reduction can be computed using O(l ..."
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Cited by 19 (10 self)
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For a polynomial matrix P(z) of degree d in M~,~(K[z]) where K is a commutative field, a reduction to the Hermite normal form can be computed in O (ndM(n) + M(nd)) arithmetic operations if M(n) is the time required to multiply two n x n matrices over K. Further, a reduction can be computed using O(log~+ ’ (ml)) pamlel arithmetic steps and O(L(nd) ) processors if the same processor bound holds with time O (logX (rid)) for determining the lexicographically first maximal linearly independent subset of the set of the columns of an nd x nd matrix over K. These results are obtamed by applying in the matrix case, the techniques used in the scalar case of the gcd of polynomials.
Intruder deduction for aclike equational theories with homomorphisms
 Research Report LSV0416, LSV, ENS de Cachan
, 2004
"... Abstract. Cryptographic protocols are small programs which involve a high level of concurrency and which are difficult to analyze by hand. The most successful methods to verify such protocols rely on rewriting techniques and automated deduction in order to implement or mimic the process calculus des ..."
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Cited by 19 (3 self)
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Abstract. Cryptographic protocols are small programs which involve a high level of concurrency and which are difficult to analyze by hand. The most successful methods to verify such protocols rely on rewriting techniques and automated deduction in order to implement or mimic the process calculus describing the protocol execution. We focus on the intruder deduction problem, that is the vulnerability to passive attacks, in presence of several variants of AClike axioms (from AC to Abelian groups, including the theory of exclusive or) and homomorphism which are the most frequent axioms arising in cryptographic protocols. Solutions are known for the cases of exclusive or, of Abelian groups, and of homomorphism alone. In this paper we address the combination of these AClike theories with the law of homomorphism which leads to much more complex decision problems. We prove decidability of the intruder deduction problem in all cases considered. Our decision procedure is in EXPTIME, except for a restricted case in which we have been able to get a PTIME decision procedure using a property of onecounter and pushdown automata. 1
Easy intruder deduction problems with homomorphisms
 INFORMATION PROCESSING LETTERS
, 2006
"... We present complexity results for the verification of security protocols. Since the perfect cryptography assumption is unrealistic for cryptographic primitives with visible algebraic properties, we extend the classical DolevYao model by permitting the intruder to exploit these properties. More prec ..."
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Cited by 19 (7 self)
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We present complexity results for the verification of security protocols. Since the perfect cryptography assumption is unrealistic for cryptographic primitives with visible algebraic properties, we extend the classical DolevYao model by permitting the intruder to exploit these properties. More precisely, we are interested in theories such as Exclusive or and Abelian groups in combination with the homomorphism axiom. We show that the intruder deduction problem is in PTIME in both cases, improving the EXPTIME complexity results presented in [10].
Reliable numerical methods for polynomial matrix triangulation
 IEEE Trans. Automat. Contr
, 1999
"... Abstract: Numerical procedures are proposed for triangularizing polynomial matrices over the eld of polynomial fractions and over the ring of polynomials. They are based on two standard polynomial techniques: Sylvester matrices and interpolation. In contrast to other triangularization methods, the a ..."
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Cited by 13 (6 self)
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Abstract: Numerical procedures are proposed for triangularizing polynomial matrices over the eld of polynomial fractions and over the ring of polynomials. They are based on two standard polynomial techniques: Sylvester matrices and interpolation. In contrast to other triangularization methods, the algorithms described in this paper only rely on wellworked numerically reliable tools. They can also be used for greatest common divisor extraction, polynomial rank evaluation or polynomial nullspace computation.
A Fast Las Vegas Algorithm for Computing the Smith Normal Form of a Polynomial Matrix
 ALGEBRA AND APPLICATIONS
, 1997
"... A Las Vegas probabilistic algorithm is presented that finds the Smith normal form S 2 Q[x] n\Thetan of a nonsingular input matrix A 2 ZZ [x] n\Thetan . The algorithm requires an expected number of O~(n 3 d(d + n 2 log jjAjj)) bit operations (where jjAjj bounds the magnitude of all intege ..."
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Cited by 12 (3 self)
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A Las Vegas probabilistic algorithm is presented that finds the Smith normal form S 2 Q[x] n\Thetan of a nonsingular input matrix A 2 ZZ [x] n\Thetan . The algorithm requires an expected number of O~(n 3 d(d + n 2 log jjAjj)) bit operations (where jjAjj bounds the magnitude of all integer coefficients appearing in A and d bounds the degrees of entries of A). In practice, the main cost of the computation is obtaining a nonunimodular triangularization of a polynomial matrix of same dimension and with similar size entries as the input matrix. We show how to accomplish this in O~(n 5 d(d + log jjAjj) log jjAjj) bit operations using standard integer, polynomial and matrix arithmetic. These complexity results improve significantly on previous algorithms in both a theoretical and practical sense.
A Linear Space Algorithm for Computing the Hermite Normal Form
 Proceedings ISSAC 2001, Lecture Notes in Computer Sci., 2146
, 2001
"... Computing the Hermite Normal Form of an n n integer matrix using the best current algorithms typically requires O(n 3 log M) space, where M is a bound on the entries of the input matrix. Although polynomial in the input size (which is O(n 2 log M)), this space blowup can easily become a seriou ..."
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Cited by 11 (2 self)
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Computing the Hermite Normal Form of an n n integer matrix using the best current algorithms typically requires O(n 3 log M) space, where M is a bound on the entries of the input matrix. Although polynomial in the input size (which is O(n 2 log M)), this space blowup can easily become a serious issue in practice when working on big integer matrices. In this paper we present a new algorithm for computing the Hermite Normal Form which uses only O(n 2 log M) space (i.e., essentially the same as the input size). When implemented using standard algorithms for integer and matrix multiplication, our algorithm has the same time complexity of the asymptotically fastest (but space inecient) algorithms. We also present a heuristic algorithm for HNF that achieves a substantial speedup when run on randomly generated input matrices.
Fast Computation Of The Smith Form Of A Sparse Integer Matrix
 Computational Complexity
, 1996
"... . We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A 2 Z m\Thetan . The algorithm treats A as a "blackbox"  A is only used to compute matrixvector products and we don't access individual entries in A directly. The algorithm requ ..."
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Cited by 11 (3 self)
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. We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A 2 Z m\Thetan . The algorithm treats A as a "blackbox"  A is only used to compute matrixvector products and we don't access individual entries in A directly. The algorithm requires about O(m 2 log kAk) black box evaluations w 7! Aw mod p for wordsized primes p and w 2 Z n\Theta1 p , plus O(m 2 n log kAk+ m 3 log 2 kAk) additional bit operations. For sparse matrices this represents a substantial improvement over previously known algorithms. The new algorithm suffers from no "fillin" or intermediate value explosion, and uses very little additional space. We also present an asymptotically fast algorithm for dense matrices which requires about O(n \Delta MM(m) log kAk +m 3 log 2 kAk) bit operations, where O(MM(m)) operations are sufficient to multiply two m \Theta m matrices over a field. Both algorithms are probabilistic of the Monte Carlo type  on any input the...
Unification and Matching modulo Nilpotence
 In Proc. CADE13, volume 1104 of LNCS
, 1996
"... . We consider equational unification and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x;x) = 0 where 0 is a constant. Nilpotent matching and unification are shown to be NPcomplete. In the presence of associativity and commutativity, t ..."
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Cited by 9 (0 self)
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. We consider equational unification and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x;x) = 0 where 0 is a constant. Nilpotent matching and unification are shown to be NPcomplete. In the presence of associativity and commutativity, the problems still remain NPcomplete. But when 0 is also assumed to be the unity for the function f , the problems are solvable in polynomial time. We also show that the problem remains in P even when a homomorphism is added. Secondorder matching modulo nilpotence is shown to be undecidable. Subject area: MECHANISMS: unification 1 Introduction Equational unification is an important computational problem in automated theorem proving. Its usefulness derives from the ability to `build in' many proof steps into the pattern matching algorithm, possibly shortening the search for a proof. As a new practical application, we define a class of set constraints and show that problems in this class ca...