Generalization of Siegenthaler Inequality and Schnorr-Vaudenay Multipermutations

Siegenthaler inequality shows the existence of a tradeoff between the correlation-immunity order and the nonlinearity order of a Boolean functions. We generalize this result to correlation-immune functions over any finite field. We then construct a family of correlation-immune functions achieving this bound; these functions are notably well-suited for combining linear feedback shift registers. We also apply this result to the cryptanalysis of any cryptographic primitive based on boxes connected by a network. Schnorr and Vaudenay have previously recommended that these boxes should be multipermutations; we here refine this condition since we show that each binary component of these multipermutations, seen as a Boolean function, should have low degree.