Security of Polynomial Transformations of the Diffie--Hellman Key

D. Boneh and R. Venkatesan have recently proposed an approachto 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 unreliable oracles