A hidden number problem in small subgroups

Boneh and Venkatesan have proposed a polynomial time algorithm for recovering a hidden element it α ∈ F p , where p is prime, from rather short strings of the most significant bits of the residue of at modulo p for several randomly chosen t ∈ F p . Gonzalez Vasco and the first author have recently extended this result to subgroups of F* p of order at least p 1/3+e for all p and to subgroups of order at least p e 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-e 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.

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