On Inversion in Z_{2^n-1}

In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z_{2^n−1}, and we described the binary weights of the inverses of the Gold and Kasami exponents. We studied the function Inv_d(n), which for a fixed positive integer d maps integers n⩾1 to the least positive residue of the inverse of d modulo 2^n−1, if it exists. In particular, we showed that the function Inv_d is completely determined by its values for 1⩽n⩽θ_d, where θ_d is the order of 2 modulo the largest odd divisor of d.