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 I⊂F⟨X1,…,Xn⟩ {\em
(private)}, \noindent iii) a finite set G:={g1,…,gl}⊂I
of elements of the ideal I {\em (public)}, \noindent a noetherian
semigroup term-ordering ≺, {\rm (private)}, on the word semigroup {\Cal
T} := , \noindent COMPUTE \noindent --a finite subset
H⊂Γ(I) of the Gr\"obner basis Γ(I) of I
w.r.t. ≺ s.t., for each gi∈G its {\em normal form} NF(gi,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 I and the
term-ordering ≺. \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)