On lattice reduction for polynomial matrices

Mulders, Thom; Storjohann, Arne

doi:10.1016/s0747-7171(02)00139-6

Cited by 96 publications

(119 citation statements)

References 4 publications

Supporting

Mentioning

119

Contrasting

Order By: Relevance

“…We refer to [14,23] and the references therein for discussions on previous reduction algorithms and applications of the form especially in linear algebra and in linear control theory. If r is the rank of A, the best previously known cost for reducing A was O(n 2 rd 2 ) operations in K [14].…”

Section: Column Reductionmentioning

confidence: 99%

On the complexity of polynomial matrix computations

Giorgi

Jeannerod

Villard

2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

113

234

View full text Add to dashboard Cite

We study the link between the complexity of polynomial matrix multiplication and the complexity of solving other basic linear algebra problems on polynomial matrices. By polynomial matrices we mean n × n matrices in K[x] of degree bounded by d, with K a commutative field. Under the straight-line program model we show that multiplication is reducible to the problem of computing the coefficient of degree d of the determinant. Conversely, we propose algorithms for minimal approximant computation and column reduction that are based on polynomial matrix multiplication; for the determinant, the straight-line program we give also relies on matrix product over K [x] and provides an alternative to the determinant algorithm of [16,17]. We further show that all these problems can be solved in particular in O˜(n ω d) operations in K. Here the "soft O" notation O˜in-dicates some missing log(nd) factors and ω is the exponent of matrix multiplication over K.

show abstract

Section: Column Reductionmentioning

confidence: 99%

On the complexity of polynomial matrix computations

Giorgi

Jeannerod

Villard

2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

113

234

View full text Add to dashboard Cite

show abstract

“…We would like to add that if the entries are polynomials over a possibly finite field, there are additional new techniques possible (Jeannerod & Villard 2004;Mulders & Storjohann 2003;Storjohann 2002Storjohann , 2003. Storjohann (2004) has extended his 2003 techniques to construct a Las Vegas algorithm that computes det A where A ∈ Z n×n in (n ω log A ) 1+o(1) bit operations, when n × n matrices are multiplied in O(n ω ) algebraic operations.…”

Section: Discussionmentioning

confidence: 99%

On the complexity of computing determinants

Kaltofen

Villard²

2005

comput. complex.

View full text Add to dashboard Cite

Abstract. We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n × n matrix A with integer entries in (n 3.2 log A ) 1+o(1) and (n 2.697263 log A ) 1+o(1) bit operations; here A denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors C 1 (log n) C 2 (loglog A ) C 3 for positive real constants C 1 , C 2 , C 3 . The bit complexity (n 3.2 log A ) 1+o(1) results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n 3.2+o(1) and O(n 2.697263 ) ring additions, subtractions and multiplications.

show abstract

“…The integers in the Coppersmith-Howgrave-GrahamNagaraj algorithm can be replaced by polynomials over a finite field. The LLL algorithm for integer lattice-basis reduction is replaced by simpler algorithms-see, e.g., (Lenstra 1985, Section 1) and (Mulders and Storjohann 2003, Section 2)-for polynomial latticebasis reduction. Many of the cryptanalytic applications of the algorithm are uninteresting for polynomials, since polynomials can be factored efficiently into irreducibles.…”

Section: Review Of Divisors In Arithmetic Progressionsmentioning

confidence: 99%