Abstract
Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a “hidden” element α 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 αt for several values of t selected uniformly at random from \(\mathbb{F}_p^* \) We use some recent bounds of exponential sums to generalize this algorithm to the case when t is selected from a quite small subgroup of \(\mathbb{F}_p^* \). Namely, our results apply to subgroups of size at least p 1/3+ɛ for all primes p and to subgroups of size at least p ɛ for almost all primes p, for any fixed ɛ > 0. We also use this generalization to improve (and correct) one of the statements of the aforementioned work about the computational security of the most significant bits of the Diffie-Hellman key.
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Vasco, M.I.G., Shparlinski, I.E. (2001). On the Security of Diffie-Hellman Bits. In: Lam, KY., Shparlinski, I., Wang, H., Xing, C. (eds) Cryptography and Computational Number Theory. Progress in Computer Science and Applied Logic, vol 20. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8295-8_19
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DOI: https://doi.org/10.1007/978-3-0348-8295-8_19
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