Solving hidden number problem with one bit oracle and advice

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 . 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 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: Local. Its running time is polynomial in logN and L₁(f̂) (for L ₁(f̂) the sum of f's Fourier coefficients, in absolute value). Universal. For any N, t, the same oracle queries are asked for all functions f over ℤ_N s.t. L₁(f̂) ≤ t. Robust. The algorithm succeeds with high probability even if the oracle to f is corrupted by random noise.