D. Boneh and R. Venkatesan have recently proposed an approach to proving
that a reasonably small portions of most significant bits of the
Diffie--Hellman key modulo a prime are as secure the the whole key. Some
further improvements and generalizations have been obtained by
I. M. Gonzales Vasco and I. E. Shparlinski.

E. R. Verheul has obtained certain analogies of these results in the case
of Diffie--Hellman keys in extensions of finite fields, when an oracle is
given to compute a certain polynomial function of the key, for example, the trace in the background field. Here we obtain some new results in this direction concerning the case of so-called "noisy" oracles.