On cryptographic properties of the cosets of R(1, m)

We introduce a new approach for the study of weight distributions of cosets of the Reed-Muller code of order 1. Our approach is based on the method introduced by Kasami (1968), using Pless (1963) identities. By interpreting some equations, we obtain a necessary condition for a coset to have a "high" minimum weight. Most notably, we are able to distinguish such cosets which have three weights only. We then apply our results to the problem of the nonlinearity of Boolean functions. We particularly study the links between this criterion and the propagation characteristics of a function.