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

References 34

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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

	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
0	195	230	252	255	222	193	2	141	198	26	132	117	61	199	6	10
1	112	184	82	244	170	5	145	73	236	237	93	163	54	121	97	12
2	20	78	226	247	188	233	67	126	164	214	116	25	196	246	18	43
3	95	229	177	189	28	192	32	46	88	106	90	45	21	31	62	72
4	225	115	100	42	108	186	178	234	59	220	200	148	134	65	161	130
5	86	11	165	238	150	50	210	219	51	187	48	181	75	1	190	99
6	153	172	251	80	92	84	212	30	13	221	142	201	152	91	89	71
7	232	47	211	179	135	197	175	182	156	125	129	9	29	41	64	63
8	228	103	127	107	120	171	34	44	105	102	96	57	242	58	85	94
9	70	224	215	208	217	7	14	131	119	144	60	138	37	19	174	40
A	239	69	33	203	206	194	216	39	167	22	111	24	169	76	35	56
B	253	241	87	250	77	159	139	147	151	223	66	101	204	68	23	231
C	136	173	122	243	207	162	0	16	160	191	185	118	79	183	15	3
D	245	218	254	227	104	249	157	166	36	149	158	205	154	53	49	113
E	168	133	137	83	202	4	114	110	140	124	176	248	27	81	55	180
F	143	209	38	240	213	17	123	155	235	52	146	128	98	8	109	74

	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