THE LINEAR SPECTRUM OF QUADRATIC APN FUNCTIONS

Almost perfect nonlinear (APN) functions are studied. We introduce the linear spectrum Lambda(F) = (lambda(F)(0), ..., lambda(F)(2n-1)) of a quadratic APN function F, where lambda(F)(k) equals the number of linear functions L such that vertical bar{a is an element of F-2(n) \ {0} : B-a(F) = B-a(F + L)}vertical bar = k and B-a(F) = {F(x)+F(x+a) : x is an element of F-2(n)}. We prove that lambda(F)(k) = 0 for all even k <= 2(n)-2 and for all k < (2(n)-1)/3, where F is a quadratic APN function in even number of variables n. Linear spectra for APN functions in small number of variables n = 3,4,5,6 are computed and presented. We consider APN Gold functions F(x) = x(2k+1) for (k,n) = 1 and prove that lambda(F)(2n-1) = 2(n+n/2) if n = 4t for some t and k = n/2 +/- 1, and lambda(F)(2n-1) = 2(n) otherwise.