Abstract
In the Hidden Number Problem (HNP), the goal is to find a hidden number s, when given p, g and access to an oracle that on query a returns the k most significant bits of \(s\cdot g^a \bmod p\).
We present an algorithm solving HNP, when given an advice depending only on p and g; the running time and advice length are polynomial in logp. This algorithm improves over prior HNP algorithms in achieving: (1) optimal number of bits k ≥ 1 (compared with k ≥ Ω(loglogp)); (2) robustness to random noise; and (3) handling a wide family of predicates on top of the most significant bit.
As a central tool we present an algorithm that, given oracle access to a function f over \({\mathbb Z}_N\), outputs all the significant Fourier coefficients of f (i.e., those occupying, say, at least 1% of the energy). This algorithm improves over prior works in being:
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Local. Its running time is polynomial in logN and \(L_1(\widehat f)\) (for \(L_1(\widehat f)\) the sum of f’s Fourier coefficients, in absolute value).
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Universal. For any N,t, the same oracle queries are asked for all functions f over \({\mathbb Z}_N\) s.t. \(L_1(\widehat f)\le t\).
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Robust. The algorithm succeeds with high probability even if the oracle to f is corrupted by random noise.
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Akavia, A. (2009). Solving Hidden Number Problem with One Bit Oracle and Advice. In: Halevi, S. (eds) Advances in Cryptology - CRYPTO 2009. CRYPTO 2009. Lecture Notes in Computer Science, vol 5677. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03356-8_20
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