The Nonlinearity Optimization Algorithm of S-box Based on TD-ERCS Sequence

doi:10.3969/j.issn.1671-1122.2021.01.002

Abstract

Abstract:

Aiming at the problems that the S-boxes generated by chaotic systems have lower nonlinearity, in this paper, aims at the method of generating S-boxes based on a mapping system of tangent-delay ellipse reflecting cavity(TD-ERCS), proves the S-boxes have bijection firstly. On this basis, an improved hill-climbing algorithm is designed. By dynamically reducing the selection ranges of Walsh-Hadamard transform(WHT) of Boolean functions, and inverting six Boolean values satisfying the conditions, the nonlinearities of bijective S-boxes are improved. Theoretical and experimental simulation analysis shows that, the performance of S-boxes generated by optimization algorithm is improved effectively, and has better performance in algorithm efficiency, nonlinearity, strict avalanche criterion and differential approximation probability.

Key words: TD-ERCS, bijection, nonlinearity

CLC Number:

TP309

ZHANG Xuefeng, WEI Kaili, JIANG Wen. The Nonlinearity Optimization Algorithm of S-box Based on TD-ERCS Sequence[J]. Netinfo Security, 2021, 21(1): 10-18.

Figures/Tables 14

混沌值取值范围	${{\chi }_{{g}/{{{2}^{3}}}\;}}\left( X \right)$ 取值	d₃(X )取值
$0\le \left\| X \right\|<{1}/{8}\;$	0 0 0 0 0 0 0	0
${1}/{8}\;\le \left\| X \right\|<{2}/{8}\;$	1 0 0 0 0 0 0	1
${2}/{8}\;\le \left\| X \right\|<{3}/{8}\;$	1 1 0 0 0 0 0	0
${3}/{8}\;\le \left\| X \right\|<{4}/{8}\;$	1 1 1 0 0 0 0	1
${4}/{8}\;\le \left\| X \right\|<{5}/{8}\;$	1 1 1 1 0 0 0	0
${5}/{8}\;\le \left\| X \right\|<{6}/{8}\;$	1 1 1 1 1 0 0	1
${6}/{8}\;\le \left\| X \right\|<{7}/{8}\;$	1 1 1 1 1 1 0	0
${7}/{8}\;\le \left\| X \right\|\le 1$	1 1 1 1 1 1 1	1

算法	集合复杂度	非线性度提升上限
三点爬山算法^[24]	16581120 $\left( A_{256}^{3} \right)$	3
四点爬山算法^[25,26]	4195023360 $\left( A_{256}^{4} \right)$	4
雷鹏提出的算法^[27]	262144 $\left( 8C_{128}^{1}\left( C_{128}^{1}+C_{128}^{1} \right) \right)$	2
优化后的S盒算法	2731008 $\left( 8C_{128}^{3} \right)$	6

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	0	1	2	3	4	5	6	7
0	0.4688	0.4688	0.5781	0.5156	0.4688	0.4531	0.4844	0.5313
1	0.4219	0.5156	0.5781	0.5000	0.5156	0.5469	0.5000	0.4531
2	0.5156	0.4531	0.5000	0.5000	0.5469	0.5000	0.5156	0.5000
3	0.4375	0.4844	0.5156	0.5000	0.4844	0.4219	0.5469	0.6094
4	0.4531	0.4844	0.3906	0.4531	0.5625	0.5313	0.5313	0.4844
5	0.4688	0.5156	0.4219	0.4844	0.5625	0.5156	0.5313	0.4844
6	0.4823	0.4748	0.4593	0.4593	0.4748	0.4966	0.4672	0.4966
7	0.5000	0.5000	0.4219	0.5781	0.5000	0.5156	0.4688	0.5156

S盒	依赖矩阵
S盒	最大值	最小值	平均值
初始S盒^[16]	0.5938	0.3906	0.4914
本文优化S盒	0.6094	0.3906	0.4956
文献[15]	0.5781	0.4063	0.5050
文献[7]	0.5937	0.4218	0.5048
文献[8]	0.625	0.4219	0.5034
文献[9]	0.5625	0.4375	0.5000
文献[10]	0.6094	0.3909	0.4941

	0	1	2	3	4	5	6	7
0	0.0000	-0.0361	-0.0070	-0.0038	-0.0157	-0.0150	0.0361	0.0212
1	-0.0361	0.0000	-0.0180	0.0938	-0.0052	0.0159	0.0238	-0.0385
2	-0.0070	-0.0180	0.0000	0.0098	-0.0734	-0.0251	0.0102	-0.0520
3	-0.0038	0.0938	0.0098	0.0000	-0.0059	-0.0039	-0.0235	-0.0274
4	-0.0157	-0.0052	-0.0734	-0.0059	0.0000	0.0092	-0.0105	-0.0306
5	-0.0150	0.0159	-0.0251	-0.0039	0.0092	0.0000	0.0271	0.0035
6	0.0361	0.0238	0.0102	-0.0235	-0.0105	0.0271	0.0000	0.0307
7	0.0212	-0.0385	-0.0520	-0.0274	-0.0306	0.0035	0.0307	0.0000

S盒	BIC中非线性度均值
初始S盒^[16]	105.86
本文优化S盒	106.93
文献[15]	104.29
文献[7]	103.71
文献[8]	103.79
文献[9]	103.78
文献[10]	103.50

S盒	差分逼近概率最大值/%
初始S盒^[16]	3.9063
本文优化S盒	3.9063
文献[15]	4.6875
文献[7]	3.9063
文献[8]	3.9063
文献[9]	4.6875
文献[10]	3.9063