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52. A Note on Weight Distributions of Irreducible Cyclic Codes

Author

Tang, Chunming, Qi, Yanfeng, Xu, Maozhi, Wang, Baocheng, and Yang, Yixian

Subjects
Computer Science - Information Theory
Abstract

Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case when an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and sufficient condition. In this case, we also obtain a divisible property for the weight of a codeword. Further, we present a necessary and sufficient condition for an irreducible cyclic code with only one nonzero weight. Finally, we determine the weight distribution of an irreducible cyclic code for some cases.

Published
2012

53. A new class of hyper-bent Boolean functions in binomial forms

Author

Tang, Chunming, Qi, Yanfeng, Xu, Maozhi, Wang, Baocheng, and Yang, Yixian

Subjects
Computer Science - Information Theory
Abstract

Bent functions, which are maximally nonlinear Boolean functions with even numbers of variables and whose Hamming distance to the set of all affine functions equals $2^{n-1}\pm 2^{\frac{n}{2}-1}$, were introduced by Rothaus in 1976 when he considered problems in combinatorics. Bent functions have been extensively studied due to their applications in cryptography, such as S-box, block cipher and stream cipher. Further, they have been applied to coding theory, spread spectrum and combinatorial design. Hyper-bent functions, as a special class of bent functions, were introduced by Youssef and Gong in 2001, which have stronger properties and rarer elements. Many research focus on the construction of bent and hyper-bent functions. In this paper, we consider functions defined over $\mathbb{F}_{2^n}$ by $f_{a,b}:=\mathrm{Tr}_{1}^{n}(ax^{(2^m-1)})+\mathrm{Tr}_{1}^{4}(bx^{\frac{2^n-1}{5}})$, where $n=2m$, $m\equiv 2\pmod 4$, $a\in \mathbb{F}_{2^m}$ and $b\in\mathbb{F}_{16}$. When $a\in \mathbb{F}_{2^m}$ and $(b+1)(b^4+b+1)=0$, with the help of Kloosterman sums and the factorization of $x^5+x+a^{-1}$, we present a characterization of hyper-bentness of $f_{a,b}$. Further, we use generalized Ramanujan-Nagell equations to characterize hyper-bent functions of $f_{a,b}$ in the case $a\in\mathbb{F}_{2^{\frac{m}{2}}}$.

Published
2011

54. The Weight Distributions of Cyclic Codes and Elliptic Curves

Author

Wang, Baocheng, Tang, Chunming, Qi, Yanfeng, Yang, Yixian, and Xu, Maozhi

Subjects
Computer Science - Information Theory
Abstract

Cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes, have been studied for many years. However, the weight distributions of cyclic codes are difficult to determine. From elliptic curves, this paper determines the weight distributions of dual codes of cyclic codes with two zeros for a few more cases.

Published
2011

58. Secure Multi-party Comparison Protocol and Application

Author

Zhang, Jing, Luo, Shoushan, Yang, Yixian, Diniz Junqueira Barbosa, Simone, Series editor, Chen, Phoebe, Series editor, Filipe, Joaquim, Series editor, Kotenko, Igor, Series editor, Sivalingam, Krishna M., Series editor, Washio, Takashi, Series editor, Yuan, Junsong, Series editor, Zhou, Lizhu, Series editor, Zou, Beiji, editor, Li, Min, editor, Wang, Hongzhi, editor, Song, Xianhua, editor, Xie, Wei, editor, and Lu, Zeguang, editor

Published
2017
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