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

Haode YAN; Dongchun HAN

doi:10.1587/transfun.E102.A.849

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

Haode YAN, Dongchun HAN

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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 $e=rac{3^m-3}{4}$. 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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E102-A No.6 pp.849-853

Publication Date: 2019/06/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.E102.A.849

Type of Manuscript: LETTER

Category: Cryptography and Information Security

Authors

Haode YAN
Southwest Jiaotong University
Dongchun HAN
Southwest Jiaotong University

Keyword

power mapping, differential uniformity, cyclic code

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Haode YAN, Dongchun HAN, "New Ternary Power Mapping with Differential Uniformity Δf≤3 and Related Optimal Cyclic Codes" in IEICE TRANSACTIONS on Fundamentals, vol. E102-A, no. 6, pp. 849-853, June 2019, doi: 10.1587/transfun.E102.A.849.
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 $e=rac{3^m-3}{4}$. 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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.849/_p

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@ARTICLE{e102-a_6_849,
author={Haode YAN, Dongchun HAN, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={New Ternary Power Mapping with Differential Uniformity Δf≤3 and Related Optimal Cyclic Codes},
year={2019},
volume={E102-A},
number={6},
pages={849-853},
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 $e=rac{3^m-3}{4}$. 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.},
keywords={},
doi={10.1587/transfun.E102.A.849},
ISSN={1745-1337},
month={June},}

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TY - JOUR
TI - New Ternary Power Mapping with Differential Uniformity Δf≤3 and Related Optimal Cyclic Codes
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 849
EP - 853
AU - Haode YAN
AU - Dongchun HAN
PY - 2019
DO - 10.1587/transfun.E102.A.849
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E102-A
IS - 6
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - June 2019
AB - 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 $e=rac{3^m-3}{4}$. 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.
ER -