We first give a proof of an isomorphism between the group of affine equivalent maps and the automorphism group of Sylvester Hada-mard matrices. Secondly, we prove the existence of new nonlinearity preserving bijective mappings without explicit construction. Continuing the study of the group of nonlinearity preserving bijective mappings acting on n -variable Boolean functions, we further give the exact number of those mappings for n ≤ 6 . Moreover, we observe that it is more beneficial to study the automorphism group of bijective mappings as a subgroup of the symmetric group of the 2^n dimensional 𝔽_2 -vector space due to the existence of non-affine mapping classes.
In this paper, we present a function in F-2[X] and prove that several of its properties closely resemble those of Euler's phi function. Additionally, we conjecture another property for this function that can be used as a simple primality test in F-2[X], and we provide numerical evidence to support this conjecture. Finally, we further apply the previous results to design a simple primality test for trinomials.
