Computing the Fourier transform is a basic building block used in numerous applications. For data intensive applications, even the $O(N\log N)$ running time of the Fast Fourier Transform (FFT) algorithm may be too slow, and {\em sub-linear} running time is necessary. Clearly, outputting the entire Fourier transform in sub-linear time is infeasible, nevertheless, in many applications it suffices to find only the {\em $\tau$-significant Fourier transform coefficients}, that is, the Fourier coefficients whose magnitude is at least $\tau$-fraction (say, 1\%) of the energy (\ie, the sum of squared Fourier coefficients). We call algorithms achieving the latter {\em SFT algorithms}.

In this paper we present a {\em deterministic} algorithm that finds the $\tau$-significant Fourier coefficients of functions $f$ over {\em any finite abelian group $G$} in time polynomial in $\log\card G$, $1/\tau$ and $L_1(f)$ (for $L_1(f)$ denoting the sum of absolute values of the Fourier coefficients of $f$). Our algorithm is robust to random noise.

Our algorithm is the first deterministic and efficient (\ie, polynomial in $\log\card G$) SFT algorithm to handle functions over any finite abelian groups, as well as the first such algorithm to handle functions over $\ZZ_N$ that are neither compressible nor Fourier-sparse. Our analysis is the first to show robustness to noise in the context of deterministic SFT algorithms.

Using our SFT algorithm we obtain (1) deterministic (universal and explicit) algorithms for sparse Fourier approximation, compressed sensing and sketching; (2) an algorithm solving the Hidden Number Problem with advice, with cryptographic bit security implications; and (3) an efficient decoding algorithm in the random noise model for polynomial rate variants of Homomorphism codes and any other concentrated \& recoverable codes.