Annihilators of Fast Discrete Fourier Spectra Attacks

Spectra attacks proposed recently are more data efficient than algebraic attacks against stream cipher. They are also time-andspace efficient. A measurement of the security of a stream cipher against spectra attacks is spectral immunity, the lowest spectral weight of the annihilator of the key stream. We study both the annihilator and the spectral immunity. We obtain a necessary and sufficient condition for the existence of low spectral weight annihilator and find it is more difficult to decide the (non)existence of the low weight annihilator for spectra attacks than for algebraic attacks. We also give some basic properties of annihilators and find the probability of a periodic sequence to be the annihilator of another sequence of the same period is low. Finally we prove that the spectral immunity is upper bounded by half of the period of the key stream. As a result, to recover any key stream, the least amount of bits required by spectra attacks is at most half of its period.