On Inversion in Z_{2^n-1}

TL;DR

This paper explicitly computes inverses of certain APN exponents in Z_{2^n-1} and analyzes the properties of the inverse function \\de(n), providing new insights into their structure and behavior.

Contribution

It explicitly determines inverses of Dobbertin and Welch APN exponents and characterizes the inverse function \\de(n) for various exponents in Z_{2^n-1}.

Findings

Explicit inverses for Dobbertin and Welch APN exponents.

Description of binary weights of inverses of Gold and Kasami exponents.

The function \\de(n) is fully determined by its values up to a certain order.

Abstract

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 \de(n), which for a fixed positive integer d maps integers n\geq 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 \de is completely determined by its values for 1 \leq n \leq \ordb, where \ordb is the order of 2 modulo the largest odd divisor of d.