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New Ternary Power Mapping with Differential Uniformity Δ_{f}≤3 and Related Optimal Cyclic Codes
Haode YAN Dongchun HAN
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E102A
No.6
pp.849853 Publication Date: 2019/06/01
Online ISSN: 17451337
DOI: 10.1587/transfun.E102.A.849
Type of Manuscript: LETTER Category: Cryptography and Information Security Keyword: power mapping, differential uniformity, cyclic code,
Full Text: FreePDF(285.7KB)
Summary:
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 $e=rac{3^m3}{4}$. It is shown that Δ_{f}≤3 and the power function is not CCZequivalent to the known ones. Moreover, we consider a family of ternary cyclic code C_{(1,e)}, which is generated by m_{ω}(x)m_{ωe}(x). Herein, ω is a primitive element of GF(3^{m}), m_{ω}(x) and m_{ωe}(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.

