Evaluation properties of symmetric polynomials

Pierrick Gaudry (1) , Eric Schost (1) , Nicolas M. Thiéry (2, 3)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 LAPCS - Laboratoire de Probabilités, Combinatoire et Statistique

By the fundamental theorem of symmetric polynomials, if $P \in \Q[X_1,\dots,X_n]$ is symmetric, then it can be written $P=Q(\sigma_1,\dots,\sigma_n)$, where $\sigma_1,\dots,\sigma_n$ are the elementary symmetric polynomials in $n$ variables, and $Q$ is in $\Q[S_1,\dots,S_n]$. We investigate the complexity properties of this construction in the straight-line program model, showing that the complexity of evaluation of $Q$ depends only on $n$ and on the complexity of evaluation of $P$. Similar results are given for the decomposition of a general polynomial in a basis of $\Q[X_1,\dots,X_n]$ seen as a module over the ring of symmetric polynomials, as well as for the computation of the Reynolds operator.

Domaines: Cryptographie et sécurité [cs.CR]

Pierrick Gaudry, Eric Schost, Nicolas M. Thiéry. Evaluation properties of symmetric polynomials. International Journal of Algebra and Computation, 2006, 16 (3), pp.505 - 523. ⟨10.1142/S0218196706003128⟩. ⟨inria-00000629⟩