Properties and Applications of Preimage Distributions of Perfect Nonlinear Functions

The preimage distributions of perfect nonlinear functions from an Abelian group of order n to an Abelian group of order 3 or 4, respectively, are studied. Based on the properties of the preimage distributions of perfect nonlinear functions from an Abelian group of order 3r to an Abelian group of order 3, the weight distributions of the ternary linear codes C _Pi from the perfect nonlinear functions Pi(x) from F ₃ r to itself are determined. These results suggest that two open problems, proposed by Carlet, Ding, and Yuan in 2005 and 2006, respectively, are answered.