Security of the most significant bits of the Shamir message passing scheme

Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a ``hidden'' element $\alpha$ of a finite field $\mathbb{F}_p$ of $p$ elements from rather short strings of the most significant bits of the remainder modulo $p$ of $\alpha t$ for several values of $t$ selected uniformly at random from $\mathbb{F}_p^*$. Unfortunately the applications to the computational security of most significant bits of private keys of some finite field exponentiation based cryptosystems given by Boneh and Venkatesan are not quite correct. For the Diffie-Hellman cryptosystem the result of Boneh and Venkatesan has been corrected and generalized in our recent paper. Here a similar analysis is given for the Shamir message passing scheme. The results depend on some bounds of exponential sums.