7元旋转对称2-弹性函数的构造

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

Abstract

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

CLC Number:

TN918.1

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.

Figures/Tables 3

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)	$(\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{matrix})$

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)

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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)

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

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