A Note on the Kasami Power Function

This work is motivated by the observation that the function F2m to F2m defined by x + (x + 1) + a for some a ∈ F2m can be used to construct difference sets. A desired condition is, that the functionφd(x) := x+(x+1) is a 2-to-1 mapping. If s = 1, then the function x has to be APN. If s > 1, then there is up to equivalence only one function known: The function φd is a 2-to-1 mapping if d is the Gold parameter d = 2 + 1 with gcd(k,m) = s. We show in this paper, that φd is also a 2-to-1 mapping if d is the Kasami parameter d = 2 −2 +1 with gcd(k,m) = s and m/s odd. We hope, that this observation can be used to construct more difference sets. key words: difference set, finite field, 2-to-1 mapping, APN, Kasami power function, Gold power function