Boolean Bent Functions in Impossible Cases: Odd and Plane Dimensions

Bent or perfect nonlinear Boolean functions represent the best resistance against the so-called linear and differential cryptanalysis. But this kind of cryptographic relevant functions only exists when the number of input bits m is an even integer and is larger than the double of the number of output bits n. Unfortunately the non-existence cases, the odd dimension (m is an odd integer) or the plane dimension ( n m = ), are not illegitimate from a cryptographic point of view and even commonly considered. New notions of bentness and perfect nonlinearity are then needed in those impossible cases for the traditional theory. In this paper, by replacing the usual XOR by another kind of bit-strings combination, we explicitly construct new “bent” Boolean functions in traditionally impossible cases.