Abstract
In this paper we revisit the modular inversion hidden number problem, which is to find a hidden number given several integers and partial bits of the corresponding modular inverse integers (in the sense of modulo a prime number) of the sums of the known integers and that unknown integer. Along with another direction different to the previous study, we present a better polynomial time algorithm to solve the problem by utilizing a technique of priority queue computation and by constructing related lattices from algebraically dependent polynomials. Let n be the number of known integers, our algorithm assumes to know about \(\left({{1}\over{2}}+{{1}\over{(n+1)!}}\right)\) of the bits of the modular inverses, which means about \({{2}\over{3}}\) of bits are required to be known in our algorithm when n = 2, while in the case that only \(\frac{2}{3}\) of bits of the modular inverses are required to be known, the result of Boneh et al. and the latest algorithm of Ling et al. in Journal of Symbolic Computation need more samples (i.e., known integers and the corresponding partial bits). Our algorithm is also better for other n.
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Xu, J., Hu, L., Huang, Z., Peng, L. (2014). Modular Inversion Hidden Number Problem Revisited. In: Huang, X., Zhou, J. (eds) Information Security Practice and Experience. ISPEC 2014. Lecture Notes in Computer Science, vol 8434. Springer, Cham. https://doi.org/10.1007/978-3-319-06320-1_39
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DOI: https://doi.org/10.1007/978-3-319-06320-1_39
Publisher Name: Springer, Cham
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