A hidden number problem in small subgroups

Boneh and Venkatesan have proposed a polynomial time algorithm for recovering a hidden element $\alpha \in \mathbb{F}_p$, where $p$ is prime, from rather short strings of the most significant bits of the residue of $\alpha t$ modulo $p$ for several randomly chosen $t\in \mathbb{F}_p$. González Vasco and the first author have recently extended this result to subgroups of $\mathbb{F}_p^*$ of order at least $p^{1/3+\varepsilon}$ for all $p$ and to subgroups of order at least $p^\varepsilon$ for almost all $p$. Here we introduce a new modification in the scheme which amplifies the uniformity of distribution of the multipliers $t$ and thus extend this result to subgroups of order at least $(\log p)/(\log \log p)^{1-\varepsilon}$ for all primes $p$. As in the above works, we give applications of our result to the bit security of the Diffie-Hellman secret key starting with subgroups of very small size, thus including all cryptographically interesting subgroups.