Certification of minimal approximant bases
Résumé
For a given computational problem, a certificate is a piece of data that one
(the prover) attaches to the output with the aim of allowing efficient
verification (by the verifier) that this output is correct. Here, we
consider the minimal approximant basis problem, for which the fastest known
algorithms output a polynomial matrix of dimensions $m \times m$ and average
degree $D/m$ using $O\tilde{~}(m^\omega \frac{D}{m})$ field operations. We
propose a certificate which, for typical instances of the problem, is
computed by the prover using $O(m^\omega \frac{D}{m})$ additional field
operations and allows verification of the approximant basis by a Monte Carlo
algorithm with cost bound $O(m^\omega + m D)$.
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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