Certification of minimal approximant bases
Résumé
Considering a given computational problem, a certificate is a piece of
additional data that one attaches to the output in order to help verifying that
this output is correct. Certificates are often used to make the verification
phase significantly more efficient than the whole (re-)computation of the
output. Here, we consider the minimal approximant basis problem, for which the
fastest known algorithms compute a polynomial matrix of dimensions $m\times m$
and average degree $D/m$ using $O\tilde{~}(m^\omega \frac{D}{m})$ field
operations. In the usual setting where the matrix to approximate has $n$
columns with $n\le m$, we provide a certificate of size $m n$, which can be
computed in $O(m^\omega \frac{D}{m})$ operations and which allows us to verify
an approximant basis by a Monte Carlo algorithm with cost bound $O(m^\omega +
mD)$.
Besides theoretical interest, our motivation also comes from the fact that
approximant bases arise in most of the fastest known algorithms for linear
algebra over the univariate polynomials; thus, this work may help in
designing certificates for other polynomial matrix computations.
Furthermore, cryptographic challenges such as breaking records for discrete
logarithm computations or for integer factorization rely in particular on
computing minimal approximant bases for large instances: certificates can
then be used to provide reliable computation on outsourced and error-prone
clusters.
Origine | Fichiers produits par l'(les) auteur(s) |
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