The differential spectrum of a ternary power mapping

Postponed access: the file will be available after 2022-03-06A function f(x)from the finite field GF(pn)to itself is said to be differentially δ-uniform when the maximum number of solutions x ∈GF(pn)of f(x +a) −f(x) =bfor any a ∈GF(pn)∗and b ∈GF(pn)is equal to δ. Let p =3and d =3n−3. When n >1is odd, the power mapping f(x) =xdover GF(3n)was proved to be differentially 2-uniform by Helleseth, Rong and Sandberg in 1999. Fo r even n, they showed that the differential uniformity Δfof f(x)satisfies 1 ≤Δf≤5. In this paper, we present more precise results on the differential property of this power mapping. Fo r d =3n−3with even n >2, we show that the power mapping xdover GF(3n)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 xdfor any integer n >1.acceptedVersio

The Differential Spectrum of the Power Mapping xpn−3

Let n be a positive integer and p a prime. The power mapping xpn−3 over Fpn has desirable differential properties, and its differential spectra for p=2,3 have been determined. In this paper, for any odd prime p , by investigating certain quadratic character sums and some equations over Fpn , we determine the differential spectrum of xpn−3 with a unified approach. The obtained result shows that for any given odd prime p , the differential spectrum can be expressed explicitly in terms of n . Compared with previous results, a special elliptic curve over Fp plays an important role in our computation for the general case p≥5.acceptedVersio