Research on the Noise Functions Produced by Macrocosm Approximation of XOR with Addition Modulo 2n

This paper proves that the noise functions produced by six kinds of macrocosm approximations of exclusive or with addition modulo 2n have the same probability distribution values. Then, the relations between the six noise functions are given. Moreover, this paper presents the computing formulas for the probability distributions and the square sums of the probability values, and proposes a correct formula for the square sums of the probability values of noise function produced by macrocosm approximations of addition modulo 2n with exclusive or. The results presented in this paper are useful in some distinguishing attacks.