Quantum algorithms for learning Walsh spectra of multi-output Boolean functions - Cui, Jingyi; Guo, Jiansheng; Xu, Linhong; Li, Mingming (Quantum information processing, 2019)
In classical cryptography, many cryptographic primitives could be treated as multi-output Boolean functions. The analysis of such functions is of great interest for cryptologists owing to their wide ranges of applications. Since each multi-output Boolean function can be uniquely determined by its Walsh transform, the Walsh spectra could reveal the properties of multi-output Boolean functions. In this paper, several quantum algorithms for learning Walsh spectra of multi-output Boolean functions are proposed. Firstly, with the usage of the amplitude estimation algorithm based on the Monte Carlo method, we present a quantum algorithm that allows one to estimate the Walsh coefficient of a multi-output Boolean function at a specified point with an additive error ϵ and probability at least 1 - δ . The corresponding query complexity is O ( ϵ - 1 log δ - 1 ) . There is an almost quadratic speedup over the classical algorithm. Secondly, we propose a generalized phase kick-back technique for multi-output Boolean functions to encode multiple Walsh coefficients on the amplitudes of states. Based on this generalized technique, a quantum Goldreich–Levin algorithm for arbitrary multi-output Boolean function F : { 0 , 1 } n → { 0 , 1 } m where m , n ∈ Z is proposed to find those Walsh coefficients satisfying the threshold boundary condition τ with probability at least 1 - δ . The whole query complexity is O 2 m + 5 + n / 2 τ 3 log 2 m + 5 n δ τ 2 . Finally, by using the same idea of the swap-test circuit, the query complexity of the modified quantum Goldreich–Levin algorithm could be lowered to O 2 m + 9 n π τ 4 log 2 m + 3 n δ τ 2 achieving a further speedup when τ is no less than O ( 2 - n / 2 + 6 n ) . Those two quantum Goldreich–Levin algorithms have their own advantages in implementation and query complexity.