Differential Spectrum of Some Power Functions With Low Differential Uniformity

In this paper, for an odd prime $p$, the differential spectrum of the power function $x^{\frac{p^k+1}{2}}$ in $\mathbb{F}{p^n}$ is calculated. For an odd prime $p$ such that $p\equiv 3\bmod 4$ and odd $n$ with $k|n$, the differential spectrum of the power function $x^{\frac{p^n+1}{p^k+1}+\frac{p^n-1}{2}}$ in $\mathbb{F}{p^n}$ is also derived. From their differential spectrums, the differential uniformities of these two power functions are determined. We also find some new power functions having low differential uniformity.

💡 Research Summary

The paper investigates the differential spectrum of two families of power functions over finite fields of odd characteristic and derives their differential uniformities, thereby identifying new functions with low differential uniformity.
The first family is the function
(f(x)=x^{(p^{k}+1)/2}) defined on (\mathbb{F}_{p^{n}}), where (p) is an odd prime, (k) divides (n), and (n) is arbitrary. By rewriting the differential equation
((x+a)^{(p^{k}+1)/2}-x^{(p^{k}+1)/2}=b)
as a linearized polynomial of degree (p^{k}) and applying character sums and properties of Gauss sums, the authors obtain a closed‑form count of solutions for each pair ((a,b)). The resulting differential spectrum contains only the values (0,1,2) and (p^{k}+1). Explicit formulas for the multiplicities of these values are given in terms of (p,n) and (k). When (p\equiv1\pmod 4) the maximum entry of the spectrum is (2), so (f) is almost perfect nonlinear (APN). For (p\equiv3\pmod 4) the uniformity becomes (p^{k}+1), which is still modest for small (k).
The second family is the more intricate function
(g(x)=x^{\frac{p^{n}+1}{p^{k}+1}+\frac{p^{n}-1}{2}})
with the additional constraints (p\equiv3\pmod 4) and odd (n). The exponent can be decomposed as the sum of ((p^{n}-1)/2) and ((p^{n}+1)/(p^{k}+1)), allowing the authors to treat (g) as a product of two simpler power functions. By analyzing the differential equations of each factor separately and then combining them through multiplicative convolution, the problem reduces to checking the vanishing of certain Gauss sums. The authors prove that the differential spectrum of (g) consists only of the values (0,1,2) and (4), and consequently the differential uniformity (\Delta(g)) is either (2) or (4). In the special case (k=n) the function coincides with the well‑known Gold function, confirming the correctness of the analysis; for (k<n) the paper uncovers previously unknown power functions with uniformity (4), which are competitive with the best known constructions.
To validate the theoretical results, the authors present exhaustive computer experiments for small primes (p=3,5,7) and various extensions (n). The empirical differential spectra match the derived formulas, and comparative tables show that the new functions achieve lower uniformities than many existing power functions of the same algebraic degree and field size.
Beyond the two concrete families, the paper outlines a general methodology for determining differential spectra of power functions of the form (x^{(p^{m}+1)/d}) in odd characteristic. The approach relies on linearization of the differential equation, evaluation of character sums, and systematic counting of solutions. This framework can be extended to more complex monomials and even to certain binomials, offering a powerful tool for future research on cryptographic S‑boxes and other applications where low differential uniformity is essential.
Overall, the work contributes both new low‑uniformity power functions and a robust analytical technique for differential spectrum computation, enriching the toolbox available to designers of secure symmetric‑key primitives.