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In this letter, the differential uniformity of power function f (x) = xe over GF(3m ) is studied, where m ≥ 3 is an odd integer and e = 3 m−3 4 . It is shown that ∆ 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 mω (x)mωe (x). Herein, ω is a primitive element of GF(3m ), mω (x) and mωe (x) are minimal polynomials of ω and ωe , 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. key words: power mapping, differential uniformity, cyclic code

ieice transactions on fundamentals of electronics, communications and computer sciences/ieice transactions on fundamentals of electronics, communications and computer science

New Ternary Power Mapping with Differential Uniformity Δ<i><sub>f</sub></i>≤3 and Related Optimal Cyclic Codes