Oracle-supported drawing of the Groebner {\em escalier}

Maria Emilia Alonso (1) , Maria Grazia Marinari (2) , Teo Mora (3)

1 Departamento de Álgebra [Madrid]
2 DM - Dipartamento di Matematica
3 DSI - DSI

The aim of this note is to discuss the following quite queer Problem: \noindent GIVEN \noindent i) the free non-commutative polynomial ring, ${\Cal P} := {\Bbb F}\langle X_1,\ldots,X_n\rangle$ {\em (public)}, \noindent ii) a bilateral ideal ${\sf I}\subset {\Bbb F}\langle X_1,\ldots,X_n\rangle$ {\em (private)}, \noindent iii) a finite set $G := \{g_1,\ldots,g_l\}\subset{\sf I}$ of elements of the ideal ${\sf I}$ {\em (public)}, \noindent a noetherian semigroup term-ordering $\prec,$ {\rm (private)}, on the word semigroup ${\Cal T} := < X_1,\ldots,X_n>$, \noindent COMPUTE \noindent --a finite subset $H\subset\Gamma({\sf I})$ of the Gröbner basis $\Gamma({\sf I})$ of ${\sf I}$ w.r.t. $\prec$ s.t., for each $g_i\in G$ its {\em normal form} $NF(g_i,H)$ w.r.t. $H$ is zero, \noindent ``by means of a finite number of queries to an oracle'', which, \noindent given a term $\tau\in{\Cal T}$ returns its {\em canonical form} $\Can(\tau,{\sf I},\prec)$ w.r.t. the ideal ${\sf I}$ and the term-ordering $\prec$. \qed This queer problem has been suggested to us by Bulygin (2005) where a similar problem, but with stronger assumptions, is faced in order to set up a chosen-cyphertext attack against the cryptographic system proposed in Rai (2004).

Mots clés: Groebner bases; Cryptography

Domaines: Algèbre commutative [math.AC]

Maria Emilia Alonso, Maria Grazia Marinari, Teo Mora. Oracle-supported drawing of the Groebner {\em escalier}. 2010. ⟨hal-00492673⟩