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

Haode YAN  Dongchun HAN  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E102-A   No.6   pp.849-853
Publication Date: 2019/06/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E102.A.849
Type of Manuscript: LETTER
Category: Cryptography and Information Security
Keyword: 
power mapping,  differential uniformity,  cyclic code,  

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Summary: 
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 $e= rac{3^m-3}{4}$. It is shown that Δf≤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.