Algebraic degree of the inverse of linearized polynomials

The inverse of linearized polynomials might be a good candidate of vector Boolean functions for cryptographic applications since it is a generalization of the inverse function that is widely used in cryptographic primitives. In Crypto 2001, a construction method of vector resilient functions was proposed using linearized polynomials and linear codes. Unfortunately, the analysis of the algebraic degree of the inverse of linearized polynomials was wrong. In this paper, we correct the inexact result. More precisely, we give the exact maximal algebraic degree and an upper bound of the minimal algebraic degree.