New Ternary Power Mapping with Differential Uniformity Δ_f≤3 and Related Optimal Cyclic Codes

Abstract

In this letter, the differential uniformity of power function f(x)=x^e over GF(3^m) is studied, where m≥3 is an odd integer and $e=\frac{3^m-3}{4}$. It is shown that Δ_f≤3 and the power function is not CCZ-equivalent to the known ones. Moreover, we consider a family of ternary cyclic code C_(1,e), which is generated by m_ω(x)m_ω^e(x). Herein, ω is a primitive element of GF(3^m), m_ω(x) and m_ω^e(x) are minimal polynomials of ω and ω^e, respectively. The parameters of this family of cyclic codes are determined. It turns out that C_(1,e) is optimal with respect to the Sphere Packing bound.