On the Parity of Power Permutations

Side-channel analysis (SCA) attacks and many countermeasures to foil these attacks have been the subject of a large body of research. Different masking schemes have been proposed as countermeasures, one of which is Threshold Implementation (TI), which carries proof of security against DPA even in the presence of glitches. At the same time, it requires a smaller area and uses much less randomness than the other secure masking methods. One of the methods to have an efficient TI of high degree S-boxes is the decomposition method. Our goal in this paper is to analyze the nonlinear components of symmetric cryptographic algorithms. To minimize the area of the protected implementation of cryptographic algorithms, we show the conditions to decompose the substitutions boxes, which are permutations, of high algebraic degree into the ones of lower degree. To find the conditions, we target the decomposition of permutations into quadratic or cubic permutations by considering the power permutations and their parities, which help us determine whether the higher degree permutations are decomposable power permutations or not. Finally, the decomposition results about the finite fields and corresponding lower degree power permutations are presented.