7元旋转对称2-弹性函数的构造

doi:10.11959/j.issn.1000-436x.2024007

通信学报 ›› 2024, Vol. 45 ›› Issue (1): 194-200.doi: 10.11959/j.issn.1000-436x.2024007

• 学术通信 • 上一篇

7元旋转对称2-弹性函数的构造

杜蛟¹^,², 李琳¹^,², 赵紫薇¹^,², 李月月¹^,², 王天银¹^,³

¹ 河南师范大学数学与信息科学学院，河南新乡 453007
² 北京邮电大学网络与交换技术国家重点实验室，北京 100876
³ 洛阳师范学院数学科学学院，河南洛阳 471934

修回日期:2023-11-28 出版日期:2024-01-01 发布日期:2024-01-01
作者简介:杜蛟（1978- ），男，湖北英山人，博士，河南师范大学副教授、硕士生导师，主要研究方向为现代密码学中的布尔函数与序列设计、编码密码的数学理论
李琳（1998- ），女，河南信阳人，河南师范大学硕士生，主要研究方向为编码密码的数学理论
赵紫薇（1999- ），女，河南周口人，河南师范大学硕士生，主要研究方向为编码密码的数学理论
李月月（1990- ），女，河南安阳人，博士，河南师范大学讲师、硕士生导师，主要研究方向为仿射代数几何、编码密码的数学理论
王天银（1979- ），男，河南南阳人，博士，洛阳师范学院教授、硕士生导师，主要研究方向为密码学、隐私保护
基金资助:
国家自然科学基金资助项目(62372157);国家自然科学基金资助项目(62272208);国家自然科学基金资助项目(62172196);国家自然科学基金资助项目(12001173);国家自然科学基金资助项目(11971004);北京邮电大学网络与交换技术国家重点实验室开放基金资助项目(SKLNST-2022-1-01)

Concrete constructions of 2-resilient rotation symmetric Boolean functions with 7 variables

Jiao DU¹^,², Lin LI¹^,², Ziwei ZHAO¹^,², Yueyue LI¹^,², Tianyin WANG¹^,³

¹ College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
² State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China
³ Faculty of Mathematical Sciences, Luoyang Normal University, Luoyang 471934, China

Revised:2023-11-28 Online:2024-01-01 Published:2024-01-01
Supported by:
The National Natural Science Foundation of China(62372157);The National Natural Science Foundation of China(62272208);The National Natural Science Foundation of China(62172196);The National Natural Science Foundation of China(12001173);The National Natural Science Foundation of China(11971004);The Open Foundation of State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications(SKLNST-2022-1-01)

摘要/Abstract

摘要：

基于旋转对称轨道的数对分布矩阵的性质，给出了所有7元旋转对称2-弹性函数的具体构造。结果表明，在 $F_{2}^{7}$ 上有且仅有280个非线性旋转对称2-弹性函数。进一步地，对于任意的奇数k，在 $F_{2}^{7 k}$ 上至少有280个非线性旋转对称2-弹性函数。

关键词: 密码学, 旋转对称函数, 弹性函数, 支撑矩阵, 数对分布矩阵

Abstract:

Based on the properties of the 2-tuples distribution matrix of the rotation symmetric orbits, all 7-variable 2-resilient rotation symmetric Boolean functions were constructed concretely.The results show that there are only 280 nonlinear 2-resilient rotation symmetric Boolean functions over $F_{2}^{7}$ .Furthermore, there are at least 280 nonlinear 2-resilient rotation symmetric Boolean functions over $F_{2}^{7 k}$ for any odd number k.

Key words: cryptography, rotation symmetric function, resilient function, support matrix, 2-tuples distribution matrix

中图分类号:

TN918.1

杜蛟, 李琳, 赵紫薇, 李月月, 王天银. 7元旋转对称2-弹性函数的构造[J]. 通信学报, 2024, 45(1): 194-200.

Jiao DU, Lin LI, Ziwei ZHAO, Yueyue LI, Tianyin WANG. Concrete constructions of 2-resilient rotation symmetric Boolean functions with 7 variables[J]. Journal on Communications, 2024, 45(1): 194-200.

图/表 3

表1

F 2 7 上轨道代表元及其数对分布矩阵"

序号	旋转对称轨道代表元	数对分布矩阵	序号	旋转对称轨道代表元	数对分布矩阵
1	(0000000)	$(\begin{matrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix})$	11	(1111000)	$(\begin{matrix} 2 & 1 & 3 \\ 1 & 2 & 2 \\ 0 & 3 & 1 \end{matrix})$
2	(1000000)	$(\begin{matrix} 5 & 1 & 0 \\ 5 & 1 & 0 \\ 5 & 1 & 0 \end{matrix})$	12	(1101100)	$(\begin{matrix} 1 & 2 & 2 \\ 0 & 3 & 1 \\ 2 & 1 & 3 \end{matrix})$
3	(1100000)	$(\begin{matrix} 4 & 1 & 1 \\ 3 & 2 & 0 \\ 3 & 2 & 0 \end{matrix})$	13	(1101010)	$(\begin{matrix} 0 & 3 & 1 \\ 2 & 1 & 3 \\ 1 & 2 & 2 \end{matrix})$
4	(1001000)	$(\begin{matrix} 3 & 2 & 0 \\ 3 & 2 & 0 \\ 4 & 1 & 1 \end{matrix})$	14	(1110100)	$(\begin{matrix} 1 & 2 & 2 \\ 1 & 2 & 2 \\ 1 & 2 & 2 \end{matrix})$
5	(1010000)	$(\begin{matrix} 3 & 2 & 0 \\ 4 & 1 & 1 \\ 3 & 2 & 0 \end{matrix})$	15	(1110010)	$(\begin{matrix} 1 & 2 & 2 \\ 1 & 2 & 2 \\ 1 & 2 & 2 \end{matrix})$
6	(1110000)	$(\begin{matrix} 3 & 1 & 2 \\ 2 & 2 & 1 \\ 1 & 3 & 0 \end{matrix})$	16	(1111100)	$(\begin{matrix} 1 & 1 & 4 \\ 0 & 2 & 3 \\ 0 & 2 & 3 \end{matrix})$
7	(1100100)	$(\begin{matrix} 2 & 2 & 1 \\ 1 & 3 & 0 \\ 3 & 1 & 2 \end{matrix})$	17	(1110110)	$(\begin{matrix} 0 & 2 & 3 \\ 0 & 2 & 3 \\ 1 & 1 & 4 \end{matrix})$
8	(1010100)	$(\begin{matrix} 1 & 3 & 0 \\ 3 & 1 & 2 \\ 2 & 2 & 1 \end{matrix})$	18	(1111010)	$(\begin{matrix} 0 & 2 & 3 \\ 1 & 1 & 4 \\ 0 & 2 & 3 \end{matrix})$
9	(1101000)	$(\begin{matrix} 2 & 2 & 1 \\ 2 & 2 & 1 \\ 2 & 2 & 1 \end{matrix})$	19	(1111110)	$(\begin{matrix} 0 & 1 & 5 \\ 0 & 1 & 5 \\ 0 & 1 & 5 \end{matrix})$
10	(1011000)	$(\begin{matrix} 2 & 2 & 1 \\ 2 & 2 & 1 \\ 2 & 2 & 1 \end{matrix})$	20	(1111111)	$(\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{matrix})$

表1

表2

表3

|T|=56时T1和T2的选择方案"

序号	T₁	T₂	序号	T₁	T₂	序号	T₁	T₂
1	(6)(16)(17)(18)	(11)(14)(15)(198)	46	(7)(8)(9)(16)	(5)(12)(14)(15)	91	(7)(8)(9)(17)	(4)(12)(13)(14)
2	(7)(6)17)(18)	(12)(14)(15)(19)	47	(7)(8)(10)(16)	(3)(12)(13)(14)	92	(7)(8)(9)(17)	(4)(12)(13)(15)
3	(8)(16)(17)(18)	(13)(14)(15)19)	48	(7)(8)(10)(16)	(3)(12)(13)(15)	93	(7)(8)(10)(17)	(4)(12)(13)(14)
4	(9)(16)(17)(18)	(11)(12)(13)(19)	49	(7)(8)(10)(16)	(4)(11)(12)(13)	94	(7)(8)(10)(17)	(4)(12)(13)(15)
5	(10)(16)(17)(18)	(11)(12)(13)(19)	50	(7)(8)(10)(16)	(5)(12)(14)(15)	95	(7)(9)(10)(17)	(4)(12)(14)(15)
6	(2)(6)(6)(I7)	(3)(4)(11)(9)	51	(7)(9)(10)(16)	(3)(12)(14)(15)	96	(8)(9)(10)(17)	(4)(13)(14)(15)
7	(2)(7)(16)(17)	(3)(4)(12)(19)	52	(7)(9)(10)(16)	(4)(11)(12)(14)	97	(8)(9)(10)(17)	(5)(12)(13)(14)
8	(2)(8)(16)(17)	(3)(4)(13)(19)	53	(7)(9)(10)(16)	(4)(11)(12)(15)	98	(8)(9)(10)(17)	(5)(12)(13)(15)
9	(2)(8)(16)(17)	(4)(5)(14)(19)	54	(8)(9)(10)(16)	(3)(13)(14)(15)	99	(6)(7)(8)(18)	(3)(13)(14)(15)
10	(2)(8)(16)(17)	(4)(5)(15)(19)	55	(8)(9)(10)(16)	(4)(11)(13)(14)	100	(6)(7)(8)(18)	(4)(11)(13)(14)
11	(2)(9)(16)(17)	(3)(4)(14)(19)	56	(8)(9)(10)(16)	(4)(11)(13)(15)	101	(6)(7)(8)(18)	(4)(11)(13)(15)
12	(2)(9)(16)(17)	(3)(4)(15)19)	57	(8)(9)(10)(16)	(5)(11)(12)(13)	102	(6)(7)(8)(18)	(5)(11)(12)(13)
13	(2)(9)(16)(17)	(3)(5)(12)(19)	58	(2)(6)(17)(18)	(3)(5)(14)(19)	103	(6)(7)(9)(18)	(3)(11)(12)(13)
14	(2)(10)(16)(17)	(3)(4)(14)(19)	59	(2)(6)(17)(18)	(3)(5)(15)(19)	104	(6)(7)(9)(18)	(4)(11)(14)(15)
15	(2)(10)(16)(17)	(3)(4)(15)19)	60	(2)(6)(17)(18)	(4)(5)(11)(19)	105	(6)(7)(9)(18)	(5)(11)(12)(14)
16	(2)(10)(16)(17)	(3)(5)(12)(19)	61	(2)(7)(17)(18)	(4)(5)(12)(19)	106	(6)(7)(9)(18)	(5)(11)(12)(15)
17	(2)(6)(16)(18)	(3)(5)(11)(19)	62	(2)(8)(17)(18)	(4)(5)(13)(19)	107	(6)(7)(10)(18)	(3)(11)(12)(13)
18	(2)(7)(16)(18)	(3)(4)(14)19)	63	(2)(9)(17)(18)	(3)(4)(13)(19)	108	(6)(7)(10)(18)	(4)(11)(14)(15)
19	(2)(7)(16)(18)	(3)(4)(15)(19)	64	(2)(9)(17)(18)	(4)(5)(14)(19)	109	(6)(7)(10)(18)	(5)(11)(12)(14)
20	(2)(7)(16)(18)	(3)(5)(12)(19)	65	(2)(9)(17)(18)	(4)(5)(15)(19)	110	(6)(7)(10)(18)	(5)(11)(12)(15)
21	(2)(8)(16)(18)	(3)(5)(13)(19)	66	(2)10)(17)(18)	(3)(4)(13)(19)	111	(6)(8)(9)(18)	(5)(11)(13)(14)
22	(2)(9)(16)(18)	(3)(5)(14)(19)	67	(2)(10)(17)(18)	(4)(5)(14)(19)	112	(6)(8)(9)(18)	(5)(11)(13)(15)
23	(2)(9)(16)(18)	(3)(5)(15)(19)	68	(2)(10)(17)(18)	(4)(5)(15)(19)	113	(6)(8)(10)(18)	(5)(11)(13)(14)
24	(2)(9)(16)(18)	(4)(5)(11)(19)	69	(6)(7)(8)(17)	(3)(12)(13)(14)	114	(6)(8)(10)(18)	(5)(11)(13)(15)
25	(2)(10)(16)(18)	(3)(5)(14)(19)	70	(6)(7)(8)(17)	(3)(12)(13)(15)	115	(6)(8)(10)(18)	(3)(11)(13)(14)
26	(2)(10)(16)(18)	(3)(5)(15)(19)	71	(6)(7)(8)(17)	(4)(11)(12)(13)	116	(6)(9)(10)(18)	(3)(11)(13)(15)
27	(2)(10)(16)(18)	(4)(5)(11)(19)	72	(6)(7)(8)(17)	(5)(12)(14)(15)	117	(6)(9)(10)(18)	(5)(11)(14)(15)
28	(6)(7)(8)(16)	(3)(11)(12)(13)	73	(6)(7)(9)(17)	(3)(12)(14)(15)	118	(7)(8)(9)(18)	(4)(13)(14)(15)
29	(6)(7)(8)(16)	(4)(11)(14)(15)	74	(6)(7)(9)(17)	(4)(11)(12)(14)	119	(7)(8)(9)(18)	(5)(12)(13)(14)
30	(6)(7)(8)(6)	(5)(11)(12)(14)	75	(6)(7)(9)(17)	(4)(11)(12)(15)	120	(7)(8)(9)(18)	(5)(12)(13)(15)
31	(6)(7)(8)(16)	(5)(11)(12)(15)	76	(6)(7)(10)(17)	(3)(12)(14)(15)	121	(7)(8)(10)(18)	(4)(13)(14)(15)
32	(6)(7)(9)(16)	(3)(11)(12)(14)	77	(6)(7)(10)(17)	(4)(11)(12)(14)	122	(7)(8)(10)(18)	(5)(12)(13)(14)
33	(6)(7)(9)(6)	(3)(11)(12)(15)	78	(6)(7)(10)(17)	(4)(11)(12)(15)	123	(7)(8)(10)(18)	(5)(12)(13)(15)
34	(6)(7)(10)(16)	(3)(11)(12)(14)	79	(6)(8)(9)(17)	(3)(13)(14)(15)	124	(7)(8)(10)(18)	(3)(12)(13)(14)
35	(6)(7)(10)(16)	(3)(11)(12)(15)	80	(6)(8)(9)(17)	(4)(11)(13)(14)	125	(7)(9)(10)(18)	(3)(12)(13)(15)
36	(6)(8)(9)(16)	(3)(11)(13)(14)	81	(6)(8)(9)(17)	(4)(11)(13)(15)	126	(7)(9)(10)(18)	(4)(11)(12)(13)
37	(6)(8)(9)(16)	(3)(11)(13)(15)	82	(6)(8)(9)(17)	(5)(11)(12)(13)	127	(7)(9)(10)(18)	(5)(12)(14)(15)
38	(6)(8)(9)(16)	(5)(11)(14)(15)	83	(6)(8)(10)(17)	(3)(13)(14)(15)	128	(8)(9)(10)(18)	(5)(13)(14)(15)
39	(6)(8)(10)(6)	(3)(11)(13)(14)	84	(6)(8)(10)(17)	(4)(11)(13)(14)	129	(2)(6)(7)(8)	(3)(4)(5)(14)
40	(6)(8)(10)(16)	(3)(11)(13)(15)	85	(6)(8)(10)(17)	(4)(11)(13)(15)	130	(2)(6)(7)(8)	(3)(4)(5)(15)
41	(6)(8)10)(16)	(5)(11)(14)(15)	86	(6)(8)(10)(17)	(5)(11)(12)(13)	131	(2)(6)(9)(10)	(3)(4)(5)(11)
42	(6)(9)(10)(6)	(3)(11)(14)(15)	87	(6)(9)(10)(17)	(3)(11)(12)(13)	132	(2)(7)(9)(10)	(3)(4)(5)(12)
43	(7)(8)(9)(16)	(3)(12)(13)(l4)	88	(6)(8)(10)(17)	(4)(11)(14)(15)	133	(2)(8)(9)(10)	(3)(4)(5)(13)
44	(7)(8)(9)(16)	(3)(12)(13)(15)	89	(6)(9)(10)(17)	(5)(11)(12)(14)
45	(7)(8)(9)(16)	(4)(11)(12)(13)	90	(6)(9)(10)(17)	(5)(11)(12)(15)

表3

参考文献 25

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序号	T₁	T₂
1	(6)(20)	(1)(11)
2	(8)(20)	(1)(13)
3	(7)(20)	(1)(12)
4	(9)(20)	(1)(14)
5	(9)(20)	(1)(15)
6	(10)(20)	(1)(14)
7	(10)(20)	(1)(15)

7元旋转对称2-弹性函数的构造

Concrete constructions of 2-resilient rotation symmetric Boolean functions with 7 variables

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参考文献 25

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