More Differential Properties of the Ness-Helleseth Function

Let n ≥ 3 be an odd integer, d 1 =3 n -1/2-1, d 2 =3 n -2 and u be an element of the finite field F 3n . This paper shows that fu(x)=uxd1+xd2 is an almost perfect nonlinear (APN) function on F 3 n if and only if χ( u +1)= χ( u -1)=χ( u ), where χ(∙) denotes the quadratic character of F3n. This settles the open problem raised by Ness and Helleseth in IEEE Trans. Inf. Theory 53(7): 2581-2586, 2007, where only the sufficiency part of the result was proved. Furthermore, we investigate the differential spectra of fu(x) for elements u satisfying χ( u +1)=χ( u -1) and express them in terms of several quadratic character sums of cubic polynomials.