On inversion in Z2n−1

In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z2n−1, and we described the binary weights of the inverses of the Gold and Kasami exponents. We studied the function Invd(n), which for a fixed positive integer d maps integers n⩾1 to the least positive residue of the inverse of d modulo 2n−1, if it exists. In particular, we showed that the function Invd is completely determined by its values for 1⩽n⩽θd, where θd is the order of 2 modulo the largest odd divisor of d.1