"On Finding the Eigenvalues of the Matrix of Rotation Symmetric Boolean Functions

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We consider the action on F n 2 by cyclic permutations (Z/nZ). Two elements x, y ∈ F n 2 are in the same orbit if they are cyclic shifts of each other. Cryptographic properties of rotation symmetric Boolean functions can be efficiently computed using the square matrix nA, the construction of which uses orbit representatives of the cyclic shifting action. In 2018, Ciungu and Iovanov proved that nA2 = 2n · I, the identity matrix of dimension gn × gn where gn is the number of orbits. In this paper, we answer the open question of the precise number of positive and negative eigenvalues of nA.