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 processor-efficient 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.

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 non-similarity problems are in the complexity class RNC for the usual coefficient fields, i.e., they can be probabilistically decided in poly-logarithmic time using polynomially many processors. 1. Introduction. The different normal forms of matrices, Hermite, Smith and Jordan Normal Forms are widely used in many different branches of science and engineering. Sequential algorithms for

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.

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 AC-like 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 AC-like 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 one-counter and pushdown automata. 1

: Numerical procedures are proposed for triangularizing polynomial matrices over the field 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 well-worked numerically reliable tools. They can also be used for greatest common divisor extraction, polynomial rank evaluation or polynomial null-space computation. Key Words : Triangularization, Polynomial Matrices, Numerical Methods. y This work is part of the Barrande Project No. 97/005-97/026. It was also supported by the Grant Agency of the Czech Republic under contract No. 102/97/0861, by the Ministry of Education of the Czech Republic under contract No. VS97/034 and by the French Ministry of Education and Research under contract No. 10-INSA-96. z corresponding author. E-mail: henrion@laas.fr. FAX: (33 5) 61 33 69 69. 1 Introduction A com...

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 Dolev-Yao 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].

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 non-unimodular 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.

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 blow-up 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.

. 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 "black-box" -- 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 word-sized 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 "fill-in" 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...

We present a Las Vegas probabalistic algorithm for reducing the computation of Hermite normal forms of rectangular polynomial matrices. In particular, the problem of computing the Hermite normal form of a rectangular m \Theta n matrix (with m ? n) reduces to that of computing the Hermite normal form of a matrix of size (n + 1) \Theta n having entries of similar coefficient size and degree. The main cost of the reduction is the same as the cost of fraction-free Gaussian elimination of an m \Theta n polynomial matrix. As an application, the reduction allows for the efficient computation of one-sided GCD's of two matrix polynomials along with the solution of the matrix diophantine equation associated to such a GCD. 1 Introduction Let A be a matrix in F[x] m\Thetan , F a field, with full column rank. The Hermite normal form of A is a matrix H in F[x] m\Thetan obtainable from A by unimodular row transformations such that H is upper triangular with all diagonal entries monic and such ...