The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups

Andrey V. Menyachikhin

doi:10.1515/dma-2021-0030

Article

The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups

Andrey V. Menyachikhin

Published/Copyright: October 13, 2021

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From the journal Discrete Mathematics and Applications Volume 31 Issue 5

Abstract

The limited deficit method is described, which allows constructing new orthomorphisms (almost orthomorphisms) of groups with the use of those already known. A class of transformations is described under which the set of all orthomorphisms (almost orthomorphisms) remains invariant. It is conjectured that the set of all orthomorphisms (almost orthomorphisms) is generated by transformations implemented by the limited deficit method. This conjecture is verified for all Abelian groups of order at most 12. The spectral-linear method and the spectral-differential method of design of permutations over the additive group of the field 𝔽_2^m (m = 4, …, 8) are used to construct orthomorphisms with sufficiently high values of the most important cryptographic parameters.

Keywords: orthomorphism; almost orthomorphism; permutation deficit; orthogonal Latin squares; permutation; s-box; spectral-linear method; spectral-differential method

Originally published in Diskretnaya Matematika (2019) 31,№3, 58–77 (in Russian).

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7 Appendix

Table 3

g ∈ Orth( 𝔽_2⁴)																δ_g	\|L(g, δ_g)\|	p_g	\|D(g, p_g)\|	λ_g
3	8	4	9	f	a	7	2	0	e	6	5	1	d	c	b	4/8	30	4/16	15	3

e	4	0	c	3	d	f	7	5	2	9	a	6	1	8	b	4/8	36	4/16	24	2

Table 4

g ∈ Orth( 𝔽_2⁵)																δ_g	\|L(g, δ_g)\|	p_g	\|D(g, p_g)\|	λ_g
17	1	11	4	1a	15	8	3	2	10	1c	0	19	16	d	9	6/16	68	4/32	57	3
12	1e	1b	f	e	18	13	1f	14	b	5	6	1d	c	a	7	6/16	68	4/32	57	3

1f	11	1b	17	3	7	10	6	16	d	1d	b	1	2	4	15	6/16	72	4/32	63	3
8	14	f	1a	1c	1e	a	5	9	c	19	0	12	e	18	13	6/16	72	4/32	63	3

8	6	1	1e	11	12	1f	1d	c	1b	f	7	10	19	3	9	6/16	74	4/32	66	3
b	0	a	1a	16	5	17	18	e	13	4	15	1c	2	d	14	6/16	74	4/32	66	3

7	17	9	1b	11	14	8	2	4	1e	b	6	16	1d	c	13	6/16	76	4/32	69	3
f	5	1a	1	a	1f	15	e	3	19	1c	12	18	0	d	10	6/16	76	4/32	69	3

9	1e	4	14	17	8	1f	6	0	1d	16	1b	e	d	a	12	6/16	78	4/32	72	3
13	1a	18	b	f	7	3	1	2	1c	15	5	10	c	19	11	6/16	78	4/32	72	3

1	3	17	18	15	f	12	2	0	13	4	6	8	1f	11	19	6/16	80	4/32	75	3
7	d	b	e	1b	1e	10	9	14	a	1a	1c	c	5	1d	16	6/16	80	4/32	75	3

Table 5

g ∈ Orth(𝔽_2⁶)																δ_g	\|L(g, δ_g)\|	p_g	\|D(g, p_g)\|	λ_g
14	2f	15	a	17	3b	25	1a	1e	9	8	3e	1	28	f	b	10/32	72	6/64	27	4
1c	31	2a	24	30	29	39	11	4	26	10	36	e	2e	2c	38
19	0	12	20	3f	3d	1b	16	13	33	2	35	23	6	7	5
21	34	22	1f	2d	3	c	3c	27	d	18	1d	32	3a	2b	37

14	1	c	5	21	f	1d	1e	29	3d	25	2f	32	1f	39	7	10/32	80	6/64	21	5
28	9	16	2a	22	3b	3	1a	2	31	b	2e	17	2c	3e	36
30	11	18	4	6	26	3a	20	2d	a	37	35	13	0	15	12
23	27	e	24	2b	19	3f	1c	34	38	10	8	33	1b	3c	d

29	18	1c	3b	e	3c	25	f	13	35	3d	36	0	3	19	c	10/32	83	6/64	27	5
24	37	a	2d	38	3e	1d	8	32	1f	1e	1b	34	28	33	1a
7	26	10	2e	12	39	4	3a	17	6	2b	11	d	9	15	1
22	20	2	16	14	23	5	27	31	2c	b	21	2f	3f	2a	30

2	18	35	36	3b	c	1a	2c	7	d	3a	15	2b	3f	1c	5	10/32	83	6/64	25	5
28	10	31	16	b	33	27	3e	26	17	11	37	1f	a	2d	22
1b	3c	1e	9	4	25	e	1d	20	3d	39	6	8	21	34	0
12	24	13	23	2f	32	2e	3	1	f	14	2a	19	30	38	29

1a	2e	16	27	15	e	36	21	22	30	4	1c	f	3a	1d	d	10/32	83	6/64	27	4
28	3d	3c	26	8	18	b	2c	0	39	5	1e	2f	3	35	2b
2a	29	1	2	32	17	3f	19	14	20	1b	2d	3e	a	c	10
6	24	1f	34	9	25	37	12	23	11	13	3b	38	7	31	33

13	28	19	f	3c	24	9	1f	2d	2b	20	d	31	6	14	39	10/32	85	6/64	30	4
10	22	18	c	21	25	b	3f	1a	27	3	38	3b	32	2	e
35	15	36	34	a	1	17	1c	26	2a	1d	2c	16	29	5	3d
2e	11	1e	23	12	30	1b	3a	7	2f	8	33	0	4	37	3e

f	0	30	2f	13	2a	24	1a	6	39	3d	2b	4	12	d	a	10/32	86	6/64	19	5
e	1b	2c	22	3c	21	25	c	31	33	9	16	38	36	19	7
3	18	1f	28	2d	3f	34	1c	3e	15	2e	32	1	8	14	17
11	20	27	5	b	37	26	2	1e	35	1d	3b	3a	29	10	23

39	22	7	1e	30	5	0	28	10	34	3a	3d	3b	1d	1b	2b	10/32	88	6/64	27	5
1c	19	2a	35	38	11	2	37	a	3e	6	31	b	4	25	1
15	26	21	2d	2e	2c	e	36	27	3f	3	18	16	c	2f	13
23	14	29	d	1f	17	9	1a	33	8	20	24	12	f	3c	32

Table 6

g ∈ Orth(𝔽_2⁷)																δ_g	\|L(g, δ_g)\|	p_g	\|D(g, p_g)\|	λ_g
6c	34	60	19	7d	5d	43	2d	d	24	5b	18	29	10	26	50	18/64	16	6/128	147	5
b	47	33	1a	2c	1d	20	15	72	32	6a	4b	63	1	52	59
13	4a	16	4c	31	c	58	7a	7f	22	71	65	5e	57	64	9
46	55	79	7e	38	56	2e	14	62	41	4	2	28	6f	f	8
67	3	6d	36	40	5	7c	1b	3f	4f	6e	a	12	5a	4d	21
6b	37	44	7	69	54	3b	2b	1c	75	48	3e	2f	3a	23	51
11	70	25	5c	3d	30	e	6	68	76	73	49	17	5f	4e	27
39	61	78	3c	0	45	35	1e	66	2a	77	74	7b	53	42	1f

3e	23	39	4f	3b	46	20	f	7a	78	69	6e	4c	15	2a	13	18/64	17	6/128	143	5
70	66	79	6b	3d	5c	17	56	37	3c	d	40	2d	35	7c	11
68	49	12	61	31	36	76	7d	7b	10	2c	65	58	1b	5e	7e
a	6d	1	75	3f	4a	48	73	26	42	16	c	52	1a	14	32
60	38	1c	2f	6f	22	1f	6a	21	5f	45	5b	19	47	28	2
41	4b	51	4e	5d	34	44	24	2e	b	62	1e	59	9	6	8
4	1d	29	67	64	50	30	5a	27	72	7	57	2b	74	33	0
3a	43	18	7f	55	77	71	54	6c	4d	25	e	63	53	3	5

6d	f	5f	4b	1b	15	2b	6c	71	3	5b	4a	5e	70	49	44	18/64	19	6/128	190	5
3a	c	58	40	4c	4f	2f	29	14	8	46	23	7d	52	4	3e
34	41	5d	13	3f	18	45	21	2c	7f	1c	22	16	35	9	63
2e	17	e	20	39	7c	42	4d	5a	6e	56	78	68	54	36	38
0	43	60	77	7b	2d	31	3c	30	7	1a	10	47	7e	59	53
73	64	79	25	27	67	33	57	1e	6	61	6a	74	32	69	7a
12	48	3b	1f	76	1	65	62	2a	37	4e	26	75	28	6f	5
66	24	2	5c	a	55	11	19	d	3d	b	1d	50	72	6b	51

7d	2c	1e	51	27	68	19	12	58	66	3b	18	74	47	34	7c	18/64	20	6/128	165	5
36	45	29	70	e	5	56	4d	3f	1d	1a	57	16	78	1b	6f
49	b	43	2f	79	4	5f	4f	63	3d	65	3	2	6b	48	26
4c	11	35	38	6	2b	37	59	5a	5d	7e	5b	10	7a	6d	28
52	c	4a	1c	72	32	17	7f	4b	a	61	53	15	5c	40	71
60	6e	24	55	42	3e	62	64	2a	2d	d	2e	73	6a	77	7
21	6c	39	5e	46	f	23	31	13	20	4e	76	0	54	1f	33
69	44	8	14	3a	50	9	22	30	7b	75	25	67	41	3c	1

35	58	70	a	21	1	56	3b	e	3a	43	50	37	72	28	f	18/64	23	6/128	156	5
18	5	39	4e	24	2	6d	7e	36	47	66	22	79	1c	26	9
4c	46	65	5b	19	4a	3e	52	51	44	1d	49	5f	20	5a	55
6f	13	67	5d	7c	2a	8	30	23	7d	60	78	3f	73	1f	16
64	c	74	33	59	7f	69	1b	2b	1a	1e	68	7	29	42	25
61	31	48	2e	3	15	0	12	2f	57	7a	14	d	3c	4f	54
32	27	4	76	77	3d	4b	2d	40	63	4d	41	53	2c	5c	11
75	45	5e	71	7b	6c	6a	6	34	6b	38	10	62	b	6e	17

38	70	29	8	69	24	f	7b	5f	46	e	34	78	10	1c	32	18/64	24	6/128	178	5
6d	3d	31	12	6e	5	2	5a	79	37	51	63	2c	44	60	35
4e	3b	7e	40	57	7f	39	48	2f	23	1b	6c	72	2e	49	3c
c	28	45	9	5e	71	14	1f	15	75	74	73	5c	1d	58	4a
77	2d	19	6a	4	4d	43	18	50	4f	33	6f	27	62	55	6
16	1	a	53	36	59	0	3f	7d	54	68	3e	42	17	67	1e
5b	1a	56	30	66	20	41	52	2b	d	76	5d	7a	4b	11	61
25	3	65	7c	26	64	22	7	3a	47	13	2a	21	b	6b	4c

56	65	a	6d	24	67	73	16	20	34	5	6a	3d	74	3a	e	18/64	26	6/128	161	5
1d	2	61	55	2f	42	0	19	43	d	70	52	35	30	4a	79
75	4	10	7b	50	4e	18	77	6f	2b	1	44	4b	6c	53	36
3c	32	7c	76	4f	13	49	2d	8	3e	46	3f	39	2a	72	5c
64	1f	f	45	68	7d	5e	4c	5a	1b	22	27	59	28	c	7
1a	6e	6b	71	48	11	3b	37	29	47	3	6	2c	12	69	78
23	54	38	40	1e	7a	14	4d	5b	17	1c	5d	33	51	2e	66
21	5f	62	b	7e	26	57	58	63	25	31	41	15	60	9	7f

13	7b	37	48	4d	d	4a	7	5e	10	c	5a	50	58	2b	69	18/64	28	6/128	191	5
14	21	3e	31	5c	23	28	77	6b	3d	6e	7f	1e	5	25	3f
2c	64	4b	22	36	65	1f	74	43	67	75	18	6a	39	7e	72
73	2a	2e	1a	79	30	61	2d	4f	78	6	51	62	16	53	19
7d	66	34	9	38	54	2	32	57	11	35	27	41	5d	6f	70
a	52	3a	45	33	5b	7c	42	46	68	15	0	56	3c	71	24
3	76	20	4e	63	6c	6d	55	40	1b	49	e	1d	59	29	3b
12	5f	b	1c	26	7a	2f	f	8	17	4	4c	1	60	44	47

Table 7

Linear orthomorpism g₀																Orthomorphism g obtained from g₀ with the use of the new method

δ_g₀ = 1, p_g₀ = 1, λ_g₀ = 1																δ_g = 28/128 , p_g = 8/256 , λ_g = 7
rg0=1,rg01=8,rg02=100,rg03=604																rg=3,rg1=0,rg2=0,rg3=441
0	66	81	e7	90	f6	11	77	c1	a7	40	26	51	37	d0	b6	67	c8	2e	5	9e	70	8c	79	56	d7	63	9c	bd	e3	3a	d6
17	71	96	f0	87	e1	6	60	d6	b0	57	31	46	20	c7	a1	21	57	9f	58	36	e1	6	9b	a2	7	e2	42	7d	20	d0	e8
a9	cf	28	4e	39	5f	b8	de	68	e	e9	8f	f8	9e	79	1f	72	e7	8d	fb	11	60	a5	92	68	61	2b	83	6e	8a	38	93
be	d8	3f	59	2e	48	af	c9	7f	19	fe	98	ef	89	6e	8	6d	6b	d8	ad	d2	a3	3d	f0	19	98	b4	c4	53	fe	c	e0
42	24	c3	a5	d2	b4	53	35	83	e5	2	64	13	75	92	f4	2d	fa	24	db	8f	3f	ce	f4	a7	5b	52	89	45	b7	aa	cf
55	33	d4	b2	c5	a3	44	22	94	f2	15	73	4	62	85	e3	41	5c	5d	51	c5	69	ee	39	ef	26	de	88	b0	c2	13	64
eb	8d	6a	c	7b	1d	fa	9c	2a	4c	ab	cd	ba	dc	3b	5d	17	90	4c	f1	34	71	73	85	c6	fc	81	95	f5	d	ca	3
fc	9a	7d	1b	6c	a	ed	8b	3d	5b	bc	da	ad	cb	2c	4a	d9	76	54	96	fd	df	31	6c	9	c9	1e	2f	59	28	7e	1c
f3	95	72	14	63	5	e2	84	32	54	b3	d5	a2	c4	23	45	f3	4d	ab	94	dc	4f	1b	7a	b2	30	3c	10	af	5a	be	4a
e4	82	65	3	74	12	f5	93	25	43	a4	c2	b5	d3	34	52	e4	82	f8	12	bc	37	9a	cc	7f	a	a9	bb	47	80	32	a0
5a	3c	db	bd	ca	ac	4b	2d	9b	fd	1a	7c	b	6d	8a	ec	99	8b	a1	c1	eb	75	91	44	87	14	6a	d3	c7	d1	b1	33
4d	2b	cc	aa	dd	bb	5c	3a	8c	ea	d	6b	1c	7a	9d	fb	62	3e	49	43	e5	ac	f2	f9	4b	ea	1f	cb	a6	b3	f7	cd
b1	d7	30	56	21	47	a0	c6	70	16	f1	97	e0	86	61	7	ed	a4	b9	23	15	77	27	84	b5	8	9d	97	ba	0	3b	1a
a6	c0	27	41	36	50	b7	d1	67	1	e6	80	f7	91	76	10	18	b	50	f	2	2c	2a	dd	5e	4	ec	65	35	d5	e6	7c
18	7e	99	ff	88	ee	9	6f	d9	bf	58	3e	49	2f	c8	ae	74	25	a8	ff	c3	4e	29	b8	48	bf	55	46	e9	66	86	ae
f	69	8e	e8	9f	f9	1e	78	ce	a8	4f	29	5e	38	df	b9	d4	da	f6	1	40	1d	22	8e	e	5f	b6	16	7b	78	c0	6f

Permutations g₀ and g have 13 values in common (the corresponding positions are marked in bold face)

Table 8

Piecewise linear orthomorphism g₀																orthomorphism g obtained from g₀ with the use of the new method

δ_g₀ = 22/128 , p_g₀ = 8/256 , λ_g₀ = 7																δ_g = 24/128 , p_g = 8/256 , λ_g = 7
rg0=2,rg01=0,rg02=2,rg03=441																rg=3,rg1=0,rg2=0,rg3=441
0	1f	2f	62	d9	5a	74	be	33	da	f2	a4	a9	87	80	a5	0	1f	2f	62	d9	aa	74	be	33	da	f2	5a	a9	87	80	a5
c2	4	6d	ab	5d	46	52	9f	32	d	2b	b5	e5	ba	a8	bf	c2	4	6d	ab	5d	46	52	9f	32	d	2b	b5	e5	ba	a8	bf
7a	20	d6	15	bb	7	85	72	44	1	9a	cd	43	84	9d	58	7a	20	d6	15	bb	7	85	72	44	1	9a	cd	43	84	9d	58
19	36	88	71	f1	b2	3f	39	d0	9b	31	f3	b3	e2	c0	fb	19	36	88	71	f1	b2	3f	39	d0	9b	31	f3	b3	e2	c0	fb
fc	ff	70	29	59	a7	d7	c6	35	cf	d4	fe	6b	23	af	dd	fc	ff	70	29	59	a7	d7	c6	35	cf	d4	fe	6b	23	af	dd
48	93	d2	ae	82	1d	4f	34	50	99	d3	6e	7c	6a	90	cc	48	93	d2	ae	82	1d	4f	34	50	99	d3	6e	7c	6a	90	cc
54	3d	75	89	28	96	27	f0	38	12	8d	2	49	ee	b6	17	54	3d	75	89	28	96	27	f0	38	12	8d	2	49	ee	b6	17
53	d5	97	1e	7e	c4	57	a2	1c	a	4a	e3	98	67	40	64	53	d5	97	1e	7e	c4	57	a2	1c	a	4a	e3	98	67	40	64
e8	f8	c1	d8	bd	db	4c	4b	13	b	f	df	81	25	63	c8	e8	f8	c1	d8	bd	db	4c	4b	13	b	f	df	81	25	63	c8
61	c9	b4	18	7f	83	94	dc	ce	ed	2e	45	10	66	f5	3e	61	c9	b4	18	7f	83	94	dc	ce	ed	2e	45	10	66	f5	3e
79	37	c7	73	69	8b	b9	68	6f	1b	56	e1	ec	1a	5b	22	79	37	c7	73	69	8b	b9	68	6f	1b	56	e1	ec	1a	5b	22
e7	5e	8f	78	fa	55	ef	a6	2c	f6	91	b8	a0	7b	4e	26	e7	5e	8f	78	fa	55	ef	a6	2c	f6	91	b8	a0	7b	4e	26
86	b7	bc	c5	5	e9	ca	30	3a	65	16	ac	5c	3b	2d	92	86	b7	bc	c5	5	e9	ca	30	3a	65	16	ac	5c	3b	2d	92
f4	4d	9	b1	3	d1	b0	42	eb	8	77	60	41	76	e4	e0	f4	4d	9	b1	3	d1	b0	42	eb	8	77	60	41	76	e4	e0
ad	47	5f	e6	a1	f7	de	7d	6c	f9	6	9e	c3	24	51	95	ad	47	5f	e6	a1	f7	de	7d	6c	f9	6	9e	c3	24	51	95
2a	a3	fd	c	e	21	8a	cb	11	ea	14	aa	9c	8c	8e	3c	2a	a3	fd	c	e	21	8a	cb	11	ea	14	a4	9c	8c	8e	3c

Permutations g₀ and g differ in 3 values (the corresponding positions are marked in bold face)

Received: 2019-05-26

Published Online: 2021-10-13

Published in Print: 2021-10-26

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Articles in the same Issue

https://doi.org/10.1515/dma-2021-0030

Keywords for this article

orthomorphism; almost orthomorphism; permutation deficit; orthogonal Latin squares; permutation; s-box; spectral-linear method; spectral-differential method