Results 1 
2 of
2
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 ..."
Abstract

Cited by 62 (13 self)
 Add to MetaCart
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.
Exponentiation Using Addition Chains For Finite Fields
, 2000
"... We discuss two different ways to speed up exponentiation in finite fields F q n : on the one hand, reduction of the total number of operations in F q n , and on the other hand, fast computation of a single operation. Two data structures are particularly useful: sparse irreducible polynomials and nor ..."
Abstract
 Add to MetaCart
We discuss two different ways to speed up exponentiation in finite fields F q n : on the one hand, reduction of the total number of operations in F q n , and on the other hand, fast computation of a single operation. Two data structures are particularly useful: sparse irreducible polynomials and normal bases. We introduce weighted qaddition chains to derive efficient algorithms, and report on implementation results for our methods.