**Hidden Number Problem in Small Subgroups**

*Igor Shparlinski and Arne Winterhof*

**Abstract: **Boneh and Venkatesan have proposed a polynomial time algorithm for
recovering a "hidden" element $\alpha \in \F_p$, 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 $t\in \F_p$. Gonz{\'a}lez Vasco and the first author have recently extended this result to subgroups of $\F_p^*$ 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.

**Category / Keywords: **public-key cryptography / Hidden number problem, Exponential sums, Diffie-Hellman scheme,

**Date: **received 13 Mar 2003

**Contact author: **igor at comp mq edu au

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**Version: **20030313:102615 (All versions of this report)

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