The differential spectrum of a ternary power mapping

A function f(x) from the finite field GF(p(n)) to itself is said to be differentially delta-uniform when the maximum number of solutions x is an element of GF(p(n)) of f(x + a) - f(x) - b for any a is an element of GF(p(n))* and b is an element of GF(p(n)) is equal to delta. Let p - 3 and d - 3(n)- 3. When n > 1 is odd, the power mapping f (x) - x(d) over GF(3(n)) was proved to be differentially 2-uniform by Helleseth, Rong and Sandberg in 1999. For even n, they showed that the differential uniformity Delta(f) of f(x) satisfies 1 <= Delta(f) <= 5. In this paper, we present more precise results on the differential property of this power mapping. For d - 3(n)-3 with even n > 2, we show that the power mapping x(d) over GF(3(n)) is differentially 4-uniform when n 2 (mod 4) and is differentially 5-uniform when n 0 (mod 4). Furthermore, we determine the differential spectrum of x(d) for any integer n > 1. (C) 2020 Elsevier Inc. All rights reserved.