# Finite field special functions

I am looking for special class of involutive functions (f) with exactly one (zero) fixed point over finite field $$F_q$$ with properties:

1) $$f(f(x))=x$$ for any $$x \in F_q$$ , $$f(x)=x$$ iff $$x=0$$

2) For any fixed $$a \in F_q$$, $$a \ne 0$$ the following holds:

$$f(x+a)-f(x)=g_a(x)$$

where $$g_a(x_1) \ne g_a(x_2)$$ for any $$x_1 \ne x_2$$, $$x_1 \ne -a$$ and $$x_1 \ne 0$$, $$x_2 \ne -a$$ and $$x_2 \ne 0$$

and

for any $$x \ne -a$$ and $$x \ne 0$$

$$g_a(x) \ne a$$

EDIT(25 Aug 2019) I'd like to add 3'rd constraint:

3) For any fixed $$a \in F_q$$, $$a \ne 0$$ the following holds:

$$f(x+a)-x=g_a(x)$$

where $$g_a(x_1) \ne g_a(x_2)$$ for any $$x_1 \ne x_2$$, $$x_1 \ne -a$$ and $$x_1 \ne 0$$, $$x_2 \ne -a$$ and $$x_2 \ne 0$$

• Actually 3'rd constraint directly follows from 1 and 2. – user144684 Sep 15 '19 at 9:55

## 5 Answers

An involution with exactly one fixed point must be over an odd-sized set, so characteristic 2 is ruled out.

By my computation, there are only three such functions for odd prime powers between 3 and 29 inclusive:

• $$f_1: GF(3) \to GF(3)$$ given in cycle form as $$(0)(12)$$
• $$f_2: GF(7) \to GF(7)$$ is $$(0)(13)(26)(45)$$
• $$f_3: GF(7) \to GF(7)$$ is $$(0)(15)(23)(46)$$ (so $$f_3(x) = -f_2(-x)$$ and there's fundamentally only one solution for $$GF(7)$$).

However, $$GF(31)$$ has three pairs:

0, 12, 24, 8, 17, 28, 16, 9, 3, 7, 25, 30, 1, 27, 18, 21, 6, 4, 14, 20, 19, 15, 29, 26, 2, 10, 23, 13, 5, 22, 11
0, 13, 26, 23, 21, 7, 15, 5, 11, 25, 14, 8, 30, 1, 10, 6, 22, 27, 19, 18, 28, 4, 16, 3, 29, 9, 2, 17, 20, 24, 12
0, 14, 28, 5, 25, 3, 10, 16, 19, 24, 6, 23, 20, 30, 1, 22, 7, 18, 17, 8, 12, 27, 15, 11, 9, 4, 29, 21, 2, 26, 13
0, 18, 5, 29, 10, 2, 27, 22, 20, 16, 4, 19, 23, 14, 13, 24, 9, 30, 1, 11, 8, 25, 7, 12, 15, 21, 28, 6, 26, 3, 17
0, 19, 7, 11, 14, 29, 22, 2, 28, 15, 27, 3, 13, 12, 4, 9, 25, 21, 30, 1, 23, 17, 6, 20, 26, 16, 24, 10, 8, 5, 18
0, 20, 9, 26, 18, 8, 21, 29, 5, 2, 16, 12, 11, 17, 27, 25, 10, 13, 4, 30, 1, 6, 24, 28, 22, 15, 3, 14, 23, 7, 19


My search strategy is to enumerate involutions by maintaining a set of unpaired elements, from which I take the smallest and one other, check for failure of injection in any $$g_a$$, and then recurse.

I note that nothing in the definition (functions which are involutions and perfect nonlinear) relies on the multiplicative structure of $$GF(q)$$, and the additive structure of $$GF(p)$$ (for $$p$$ prime) is just the additive structure of $$\mathbb{Z} / p\mathbb{Z}$$.

I note that the three fields for which I have found solutions have orders which are Mersenne primes. I therefore extended my search to $$\mathbb{Z} / n\mathbb{Z}$$ for the Mersenne number $$n=15$$, and found one pair of solutions:

0, 4, 8, 14, 1, 10, 13, 9, 2, 7, 5, 12, 11, 6, 3
0, 12, 9, 4, 3, 10, 8, 13, 6, 2, 5, 14, 1, 7, 11


I further observe that all of the functions I've found so far are exponential, in the sense that they satisfy $$f(2x) = 2f(x)$$. By restricting the search to involutions which satisfy this criterion, I am able to extend it to higher Mersenne numbers, finding for $$n=63$$ the functions

0, 6, 12, 32, 24, 62, 1, 26, 48, 45, 61, 25, 2, 35, 52, 23, 33, 47, 27, 56, 59, 42, 50, 15, 4, 11, 7, 18, 41, 60, 46, 34, 3, 16, 31, 13, 54, 44, 49, 43, 55, 28, 21, 39, 37, 9, 30, 17, 8, 38, 22, 53, 14, 51, 36, 40, 19, 58, 57, 20, 29, 10, 5
0, 8, 16, 53, 32, 38, 43, 62, 1, 45, 13, 51, 23, 10, 61, 44, 2, 41, 27, 34, 26, 42, 39, 12, 46, 30, 20, 18, 59, 48, 25, 35, 4, 58, 19, 31, 54, 57, 5, 22, 52, 17, 21, 6, 15, 9, 24, 49, 29, 47, 60, 11, 40, 3, 36, 56, 55, 37, 33, 28, 50, 14, 7
0, 25, 50, 55, 37, 40, 47, 53, 11, 27, 17, 8, 31, 15, 43, 13, 22, 10, 54, 51, 34, 42, 16, 28, 62, 1, 30, 9, 23, 49, 26, 12, 44, 59, 20, 58, 45, 4, 39, 38, 5, 57, 21, 14, 32, 36, 56, 6, 61, 29, 2, 19, 60, 7, 18, 3, 46, 41, 35, 33, 52, 48, 24
0, 39, 15, 11, 30, 28, 22, 17, 60, 45, 56, 3, 44, 61, 34, 2, 57, 7, 27, 31, 49, 42, 6, 58, 25, 24, 59, 18, 5, 43, 4, 19, 51, 37, 14, 40, 54, 33, 62, 1, 35, 47, 21, 29, 12, 9, 53, 41, 50, 20, 48, 32, 55, 46, 36, 52, 10, 16, 23, 26, 8, 13, 38
0, 56, 49, 13, 35, 30, 26, 8, 7, 27, 60, 23, 52, 3, 16, 34, 14, 39, 54, 48, 57, 42, 46, 11, 41, 58, 6, 9, 32, 44, 5, 59, 28, 38, 15, 4, 45, 43, 33, 17, 51, 24, 21, 37, 29, 36, 22, 61, 19, 2, 53, 40, 12, 50, 18, 62, 1, 20, 25, 31, 10, 47, 55
0, 58, 53, 34, 43, 6, 5, 44, 23, 27, 12, 49, 10, 41, 25, 55, 46, 33, 54, 26, 24, 42, 35, 8, 20, 14, 19, 9, 50, 32, 47, 60, 29, 17, 3, 22, 45, 56, 52, 59, 48, 13, 21, 4, 7, 36, 16, 30, 40, 11, 28, 61, 38, 2, 18, 15, 37, 62, 1, 39, 31, 51, 57


and for $$n=127$$

0, 7, 14, 63, 28, 54, 126, 1, 56, 90, 108, 87, 125, 55, 2, 31, 112, 43, 53, 29, 89, 57, 47, 82, 123, 105, 110, 66, 4, 19, 62, 15, 97, 77, 86, 109, 106, 46, 58, 100, 51, 75, 114, 17, 94, 68, 37, 22, 119, 122, 83, 40, 93, 18, 5, 13, 8, 21, 38, 104, 124, 88, 30, 3, 67, 95, 27, 64, 45, 107, 91, 79, 85, 78, 92, 41, 116, 33, 73, 71, 102, 118, 23, 50, 101, 72, 34, 11, 61, 20, 9, 70, 74, 52, 44, 65, 111, 32, 117, 103, 39, 84, 80, 99, 59, 25, 36, 69, 10, 35, 26, 96, 16, 115, 42, 113, 76, 98, 81, 48, 121, 120, 49, 24, 60, 12, 6
0, 10, 20, 92, 40, 64, 57, 85, 80, 126, 1, 82, 114, 25, 43, 105, 33, 111, 125, 81, 2, 88, 37, 96, 101, 13, 50, 79, 86, 113, 83, 54, 66, 16, 95, 38, 123, 22, 35, 60, 4, 69, 49, 14, 74, 56, 65, 55, 75, 42, 26, 121, 100, 97, 31, 47, 45, 6, 99, 122, 39, 93, 108, 68, 5, 46, 32, 106, 63, 41, 76, 116, 119, 104, 44, 48, 70, 103, 120, 27, 8, 19, 11, 30, 98, 7, 28, 91, 21, 124, 112, 87, 3, 61, 110, 34, 23, 53, 84, 58, 52, 24, 115, 77, 73, 15, 67, 109, 62, 107, 94, 17, 90, 29, 12, 102, 71, 118, 117, 72, 78, 51, 59, 36, 89, 18, 9
0, 14, 28, 37, 56, 105, 74, 64, 112, 115, 83, 77, 21, 126, 1, 23, 97, 62, 103, 107, 39, 12, 27, 15, 42, 94, 125, 22, 2, 80, 46, 72, 67, 58, 124, 95, 79, 3, 87, 20, 78, 96, 24, 91, 54, 110, 30, 76, 84, 89, 61, 98, 123, 59, 44, 86, 4, 120, 33, 53, 92, 50, 17, 70, 7, 82, 116, 32, 121, 102, 63, 75, 31, 117, 6, 71, 47, 11, 40, 36, 29, 111, 65, 10, 48, 109, 55, 38, 108, 49, 93, 43, 60, 90, 25, 35, 41, 16, 51, 101, 122, 99, 69, 18, 119, 5, 118, 19, 88, 85, 45, 81, 8, 114, 113, 9, 66, 73, 106, 104, 57, 68, 100, 52, 34, 26, 13
0, 19, 38, 13, 76, 62, 26, 44, 25, 63, 124, 97, 52, 3, 88, 84, 50, 119, 126, 1, 121, 99, 67, 47, 104, 8, 6, 59, 49, 85, 41, 66, 100, 79, 111, 96, 125, 120, 2, 53, 115, 30, 71, 58, 7, 105, 94, 23, 81, 28, 16, 55, 12, 39, 118, 51, 98, 122, 43, 27, 82, 92, 5, 9, 73, 70, 31, 22, 95, 112, 65, 42, 123, 64, 113, 87, 4, 93, 106, 33, 103, 48, 60, 90, 15, 29, 116, 75, 14, 91, 83, 89, 61, 77, 46, 68, 35, 11, 56, 21, 32, 107, 110, 80, 24, 45, 78, 101, 109, 108, 102, 34, 69, 74, 117, 40, 86, 114, 54, 17, 37, 20, 57, 72, 10, 36, 18
0, 21, 42, 30, 84, 95, 60, 19, 41, 22, 63, 68, 120, 118, 38, 65, 82, 98, 44, 7, 126, 1, 9, 78, 113, 56, 109, 124, 76, 46, 3, 102, 37, 119, 69, 93, 88, 32, 14, 75, 125, 8, 2, 47, 18, 61, 29, 43, 99, 72, 112, 79, 91, 117, 121, 104, 25, 116, 92, 70, 6, 45, 77, 10, 74, 15, 111, 73, 11, 34, 59, 96, 49, 67, 64, 39, 28, 62, 23, 51, 123, 110, 16, 101, 4, 87, 94, 85, 36, 103, 122, 52, 58, 35, 86, 5, 71, 100, 17, 48, 97, 83, 31, 89, 55, 114, 107, 106, 115, 26, 81, 66, 50, 24, 105, 108, 57, 53, 13, 33, 12, 54, 90, 80, 27, 40, 20
0, 31, 62, 7, 124, 82, 14, 3, 121, 44, 37, 34, 28, 91, 6, 63, 115, 69, 88, 106, 74, 40, 68, 116, 56, 86, 55, 117, 12, 101, 126, 1, 103, 84, 11, 118, 49, 10, 85, 57, 21, 66, 80, 76, 9, 50, 105, 89, 112, 36, 45, 94, 110, 73, 107, 26, 24, 39, 75, 81, 125, 102, 2, 15, 79, 67, 41, 65, 22, 17, 109, 95, 98, 53, 20, 58, 43, 122, 114, 64, 42, 59, 5, 92, 33, 38, 25, 108, 18, 47, 100, 13, 83, 104, 51, 71, 97, 96, 72, 111, 90, 29, 61, 32, 93, 46, 19, 54, 87, 70, 52, 99, 48, 119, 78, 16, 23, 27, 35, 113, 123, 8, 77, 120, 4, 60, 30
0, 39, 78, 109, 29, 104, 91, 101, 58, 74, 81, 27, 55, 80, 75, 107, 116, 41, 21, 63, 35, 18, 54, 28, 110, 123, 33, 11, 23, 4, 87, 90, 105, 26, 82, 20, 42, 68, 126, 1, 70, 17, 36, 115, 108, 79, 56, 60, 93, 51, 119, 49, 66, 62, 22, 12, 46, 97, 8, 96, 47, 113, 53, 19, 83, 118, 52, 114, 37, 77, 40, 117, 84, 95, 9, 14, 125, 69, 2, 45, 13, 10, 34, 64, 72, 121, 103, 30, 89, 88, 31, 6, 112, 48, 120, 73, 59, 57, 102, 122, 111, 7, 98, 86, 5, 32, 124, 15, 44, 3, 24, 100, 92, 61, 67, 43, 16, 71, 65, 50, 94, 85, 99, 25, 106, 76, 38
0, 41, 82, 73, 37, 111, 19, 36, 74, 97, 95, 110, 38, 121, 72, 83, 21, 28, 67, 6, 63, 16, 93, 102, 76, 48, 115, 92, 17, 120, 39, 81, 42, 123, 56, 62, 7, 4, 12, 30, 126, 1, 32, 107, 59, 47, 77, 45, 25, 109, 96, 75, 103, 61, 57, 69, 34, 54, 113, 44, 78, 53, 35, 20, 84, 100, 119, 18, 112, 55, 124, 105, 14, 3, 8, 51, 24, 46, 60, 104, 125, 31, 2, 15, 64, 117, 87, 86, 118, 101, 94, 98, 27, 22, 90, 10, 50, 9, 91, 116, 65, 89, 23, 52, 79, 71, 122, 43, 114, 49, 11, 5, 68, 58, 108, 26, 99, 85, 88, 66, 29, 13, 106, 33, 70, 80, 40
0, 55, 110, 88, 93, 97, 49, 20, 59, 25, 67, 48, 98, 120, 40, 81, 118, 19, 50, 17, 7, 83, 96, 94, 69, 9, 113, 63, 80, 41, 35, 53, 109, 56, 38, 30, 100, 116, 34, 42, 14, 29, 39, 121, 65, 103, 61, 75, 11, 6, 18, 119, 99, 31, 126, 1, 33, 117, 82, 8, 70, 46, 106, 27, 91, 44, 112, 10, 76, 24, 60, 104, 73, 72, 105, 47, 68, 95, 84, 90, 28, 15, 58, 21, 78, 124, 115, 101, 3, 123, 79, 64, 122, 4, 23, 77, 22, 5, 12, 52, 36, 87, 111, 45, 71, 74, 62, 114, 125, 32, 2, 102, 66, 26, 107, 86, 37, 57, 16, 51, 13, 43, 92, 89, 85, 108, 54
0, 73, 19, 42, 38, 35, 84, 114, 76, 111, 70, 90, 41, 20, 101, 61, 25, 125, 95, 2, 13, 65, 53, 56, 82, 16, 40, 91, 75, 115, 122, 105, 50, 104, 123, 5, 63, 48, 4, 124, 26, 12, 3, 49, 106, 69, 112, 99, 37, 43, 32, 59, 80, 22, 55, 54, 23, 67, 103, 51, 117, 15, 83, 36, 100, 21, 81, 57, 119, 45, 10, 94, 126, 1, 96, 28, 8, 109, 121, 116, 52, 66, 24, 62, 6, 88, 98, 113, 85, 93, 11, 27, 97, 89, 71, 18, 74, 92, 86, 47, 64, 14, 118, 58, 33, 31, 44, 120, 110, 77, 108, 9, 46, 87, 7, 29, 79, 60, 102, 68, 107, 78, 30, 34, 39, 17, 72
0, 87, 47, 57, 94, 21, 114, 98, 61, 39, 42, 28, 101, 19, 69, 59, 122, 116, 78, 13, 84, 5, 56, 48, 75, 104, 38, 62, 11, 36, 118, 77, 117, 37, 105, 100, 29, 33, 26, 9, 41, 40, 10, 63, 112, 125, 96, 2, 23, 67, 81, 103, 76, 119, 124, 113, 22, 3, 72, 15, 109, 8, 27, 43, 107, 92, 74, 49, 83, 14, 73, 93, 58, 70, 66, 24, 52, 31, 18, 102, 82, 50, 80, 68, 20, 95, 126, 1, 97, 115, 123, 120, 65, 71, 4, 85, 46, 88, 7, 110, 35, 12, 79, 51, 25, 34, 111, 64, 121, 60, 99, 106, 44, 55, 6, 89, 17, 32, 30, 53, 91, 108, 16, 90, 54, 45, 86
0, 89, 51, 21, 102, 28, 42, 33, 77, 62, 56, 111, 84, 60, 66, 35, 27, 103, 124, 83, 112, 3, 95, 122, 41, 29, 120, 16, 5, 25, 70, 68, 54, 7, 79, 15, 121, 96, 39, 38, 97, 24, 6, 55, 63, 93, 117, 114, 82, 125, 58, 2, 113, 118, 32, 43, 10, 87, 50, 90, 13, 75, 9, 44, 108, 74, 14, 80, 31, 119, 30, 81, 115, 105, 65, 61, 78, 8, 76, 34, 67, 71, 48, 19, 12, 91, 110, 57, 126, 1, 59, 85, 107, 45, 101, 22, 37, 40, 123, 104, 116, 94, 4, 17, 99, 73, 109, 92, 64, 106, 86, 11, 20, 52, 47, 72, 100, 46, 53, 69, 26, 36, 23, 98, 18, 49, 88
0, 97, 67, 123, 7, 50, 119, 4, 14, 92, 100, 104, 111, 49, 8, 79, 28, 75, 57, 40, 73, 108, 81, 34, 95, 66, 98, 37, 16, 55, 31, 30, 56, 76, 23, 44, 114, 27, 80, 109, 19, 102, 89, 94, 35, 122, 68, 85, 63, 13, 5, 84, 69, 107, 74, 29, 32, 18, 110, 105, 62, 86, 60, 48, 112, 125, 25, 2, 46, 52, 88, 103, 101, 20, 54, 17, 33, 82, 91, 15, 38, 22, 77, 118, 51, 47, 61, 106, 70, 42, 117, 78, 9, 116, 43, 24, 126, 1, 26, 115, 10, 72, 41, 71, 11, 59, 87, 53, 21, 39, 58, 12, 64, 121, 36, 99, 93, 90, 83, 6, 124, 113, 45, 3, 120, 65, 96
0, 107, 87, 100, 47, 37, 73, 115, 94, 114, 74, 70, 19, 22, 103, 77, 61, 46, 101, 12, 21, 20, 13, 72, 38, 96, 44, 30, 79, 110, 27, 56, 122, 41, 92, 69, 75, 5, 24, 91, 42, 33, 40, 123, 26, 111, 17, 4, 76, 104, 65, 99, 88, 63, 60, 78, 31, 68, 93, 116, 54, 16, 112, 53, 117, 50, 82, 121, 57, 35, 11, 102, 23, 6, 10, 36, 48, 15, 55, 28, 84, 98, 66, 109, 80, 125, 119, 2, 52, 113, 95, 39, 34, 58, 8, 90, 25, 124, 81, 51, 3, 18, 71, 14, 49, 118, 126, 1, 120, 83, 29, 45, 62, 89, 9, 7, 59, 64, 105, 86, 108, 67, 32, 43, 97, 85, 106
0, 109, 91, 117, 55, 70, 107, 90, 110, 73, 13, 41, 87, 10, 53, 58, 93, 25, 19, 18, 26, 49, 82, 103, 47, 17, 20, 95, 106, 71, 116, 92, 59, 81, 50, 66, 38, 44, 36, 113, 52, 11, 98, 112, 37, 67, 79, 24, 94, 21, 34, 123, 40, 14, 63, 4, 85, 62, 15, 32, 105, 96, 57, 54, 118, 122, 35, 45, 100, 84, 5, 29, 76, 9, 88, 115, 72, 111, 99, 46, 104, 33, 22, 120, 69, 56, 97, 12, 74, 125, 7, 2, 31, 16, 48, 27, 61, 86, 42, 78, 68, 121, 119, 23, 80, 60, 28, 6, 126, 1, 8, 77, 43, 39, 124, 75, 30, 3, 64, 102, 83, 101, 65, 51, 114, 89, 108
0, 114, 101, 93, 75, 27, 59, 70, 23, 21, 54, 61, 118, 14, 13, 119, 46, 82, 42, 39, 108, 9, 122, 8, 109, 58, 28, 5, 26, 76, 111, 86, 92, 102, 37, 67, 84, 34, 78, 19, 89, 72, 18, 79, 117, 62, 16, 98, 91, 87, 116, 80, 56, 121, 10, 96, 52, 64, 25, 6, 95, 11, 45, 120, 57, 110, 77, 35, 74, 94, 7, 123, 41, 83, 68, 4, 29, 66, 38, 43, 51, 97, 17, 73, 36, 103, 31, 49, 107, 40, 124, 48, 32, 3, 69, 60, 55, 81, 47, 125, 105, 2, 33, 85, 112, 100, 115, 88, 20, 24, 65, 30, 104, 126, 1, 106, 50, 44, 12, 15, 63, 53, 22, 71, 90, 99, 113
0, 118, 109, 38, 91, 68, 76, 49, 55, 10, 9, 56, 25, 115, 98, 37, 110, 33, 20, 65, 18, 60, 112, 54, 50, 12, 103, 75, 69, 43, 74, 104, 93, 17, 66, 124, 40, 15, 3, 106, 36, 99, 120, 29, 97, 116, 108, 119, 100, 7, 24, 57, 79, 83, 23, 8, 11, 51, 86, 64, 21, 95, 81, 122, 59, 19, 34, 88, 5, 28, 121, 82, 80, 96, 30, 27, 6, 101, 85, 52, 72, 62, 71, 53, 113, 78, 58, 123, 67, 92, 105, 4, 89, 32, 111, 61, 73, 44, 14, 41, 48, 77, 114, 26, 31, 90, 39, 125, 46, 2, 16, 94, 22, 84, 102, 13, 45, 126, 1, 47, 42, 70, 63, 87, 35, 107, 117
0, 121, 115, 67, 103, 78, 7, 6, 79, 46, 29, 51, 14, 85, 12, 111, 31, 101, 92, 117, 58, 91, 102, 68, 28, 47, 43, 88, 24, 10, 95, 16, 62, 83, 75, 53, 57, 118, 107, 66, 116, 93, 55, 26, 77, 104, 9, 25, 56, 54, 94, 11, 86, 35, 49, 42, 48, 36, 20, 82, 63, 100, 32, 60, 124, 97, 39, 3, 23, 89, 106, 119, 114, 122, 109, 34, 87, 44, 5, 8, 105, 90, 59, 33, 110, 13, 52, 76, 27, 69, 81, 21, 18, 41, 50, 30, 112, 65, 108, 123, 61, 17, 22, 4, 45, 80, 70, 38, 98, 74, 84, 15, 96, 125, 72, 2, 40, 19, 37, 71, 126, 1, 73, 99, 64, 113, 120


I think there is starting to be sufficient evidence to conjecture that all Mersenne numbers $$n$$ will have involutive perfect nonlinear exponential functions over $$\mathbb{Z}/n\mathbb{Z}$$, and for Mersenne primes these will be involutive perfect nonlinear functions over $$GF(n)$$.

• Good idea to use algorithmic approach, great results !!! I tried to find using mathematical abstractions. – user144684 Aug 22 '19 at 9:54
• Incidentally, since there's no use of multiplication in the definition of these functions, it just occurred to me to try searching for suitable functions over $Z_{15}$, and there is one pair, with representative $0(1,4)(2,8)(3,14)(5,10)(6,13)(7,9)(11,12)$. – Peter Taylor Aug 22 '19 at 10:15
• Hmm. These examples are all exponential (specifically, $f(2x) = 2f(x)$). I should be able to enumerate exponential functions much more efficiently, so I can check whether this class extends to $GF(127)$... – Peter Taylor Aug 22 '19 at 10:29
• Little note: all of such functions produces starter. I'd give the name for such starter. – user144684 Aug 22 '19 at 10:33
• Amazing result !! – user144684 Aug 22 '19 at 16:32

What you are looking for are called perfect nonlinear or differentially $$1-$$uniform functions.

They don't exist over even characteristic since if $$x_0$$ satisfies $$f(x+a)-f(x)=b,$$ so does $$x_0+a.$$

For a long time only some power functions or functions equivalent to them were known. A recent paper lists the following known examples among others.

$$x^2~~ in ~~GF(p^n),$$

$$x^{p^k+1} ~~in ~~GF(p^n),\quad k \leq n/2~~and ~~n/(k,n)~~odd$$

$$x^{10} + x^6 − x^2 ~~in~~ GF(3^n), ~~n \geq 5 ~~odd$$

See New families of perfect nonlinear polynomial functions by Zhengbang Zha, Xueli Wang, Journal of Algebra (322):3912-3918.

for more. One class of such polynomials are called Dembowski-Ostrom polynomials. All of the above are, but a recent example $$x^{(3^k+1 )/2}$$ isn't.

Edit: I apologise, I shouldn't post before my morning coffee. This is now essentially a long comment.

The functions displayed are not involutions. Recent work on involutions is here. I think your question is quite difficult.

• You mentioned x^2 \in GF(p^n) . is it involution for any field of odd characteristic ? – user144684 Aug 21 '19 at 5:55
• I need special involutions among perfect nonlinear or differentially 1−uniform functions – user144684 Aug 21 '19 at 8:26
• I tried to find these functions last 3 years :) I was aware about described papers and types of functions. I know that this question is very difficult :) – user144684 Aug 21 '19 at 12:33

Taking into account Peter's Tailor analysis I can say the following. For example we have finite field $$GF(p)$$ for $$p$$ prime;

We can take element $$2$$ and obtain subgroup generated by $$2$$: $$<2>$$. This subgroup generates set of cosets:

$$<2>$$, $$C_1<2>$$ .... $$C_{K-1}<2>$$

where $$k=(|GF^*(p)|/(|<2>|))$$.

So exponential functions means that

f($$C_i<2>$$) = $$C_j<2>$$

and we have involution of cosets (taking into account that their number is even)

$$(i_1,j_1),...,(i_r,j_r)$$.

Plus we have the following condition:

for any two cosets $$i,j$$ if $$(a,b)\in C_i<2>$$, $$(f(a),f(b))\in C_j<2>$$

we have: $$a/b$$=$$f(a)/f(b)$$.

So we can iterate over all involutions on the set of such Cosets and iterate over corresponding mapping for each pair of cosets.

• The biggest questions is: how to describe cosets mapping to get such functions. We dive to the Cyclotomy of Finite Field. – user144684 Aug 28 '19 at 9:48
• This is a place where I stuck and can't find solution. – user144684 Aug 28 '19 at 9:48
• 3'rd constraint works for 0, 12, 24, 8, 17, 28, 16, 9, 3, 7, 25, 30, 1, 27, 18, 21, 6, 4, 14, 20, 19, 15, 29, 26, 2, 10, 23, 13, 5, 22, 11 – user144684 Sep 14 '19 at 3:48

$$GF(31)$$:

all these involutions satisfy all 3 constraints:

0, 12, 24, 8, 17, 28, 16, 9, 3, 7, 25, 30, 1, 27, 18, 21, 6, 4, 14, 20, 19, 15, 29, 26, 2, 10, 23, 13, 5, 22, 11

0, 13, 26, 23, 21, 7, 15, 5, 11, 25, 14, 8, 30, 1, 10, 6, 22, 27, 19, 18, 28, 4, 16, 3, 29, 9, 2, 17, 20, 24, 12

0, 14, 28, 5, 25, 3, 10, 16, 19, 24, 6, 23, 20, 30, 1, 22, 7, 18, 17, 8, 12, 27, 15, 11, 9, 4, 29, 21, 2, 26, 13

0, 18, 5, 29, 10, 2, 27, 22, 20, 16, 4, 19, 23, 14, 13, 24, 9, 30, 1, 11, 8, 25, 7, 12, 15, 21, 28, 6, 26, 3, 17

0, 19, 7, 11, 14, 29, 22, 2, 28, 15, 27, 3, 13, 12, 4, 9, 25, 21, 30, 1, 23, 17, 6, 20, 26, 16, 24, 10, 8, 5, 18

0, 20, 9, 26, 18, 8, 21, 29, 5, 2, 16, 12, 11, 17, 27, 25, 10, 13, 4, 30, 1, 6, 24, 28, 22, 15, 3, 14, 23, 7, 19

$$Z/255Z$$:

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$$0, 99, 198, 205, 141, 87, 155, 158, 27, 144, 174, 67, 55, 44, 61, 237, 54, 68, 33, 247, 93, 181, 134, 165, 110, 154, 88, 8, 122, 209, 219, 242, 108, 18, 136, 83, 66, 175, 239, 103, 186, 125, 107, 243, 13, 184, 75, 241, 220, 127, 53, 238, 176, 50, 16, 12, 244, 59, 163, 57, 183, 14, 229, 178, 216, 213, 36, 11, 17, 109, 166, 116, 132, 235, 95, 46, 223, 140, 206, 131, 117, 91, 250, 35, 214, 170, 231, 5, 26, 159, 113, 81, 150, 20, 227, 74, 185, 104, 254, 1, 106, 126, 221, 39, 97, 197, 100, 42, 32, 69, 24, 120, 233, 90, 118, 192, 71, 80, 114, 153, 111, 143, 28, 195, 203, 41, 101, 49, 177, 230, 171, 79, 72, 161, 22, 246, 34, 251, 218, 210, 77, 4, 232, 121, 9, 169, 215, 179, 190, 249, 92, 248, 191, 119, 25, 6, 157, 156, 7, 89, 234, 133, 182, 58, 245, 23, 70, 193, 173, 145, 85, 130, 207, 168, 10, 37, 52, 128, 63, 147, 226, 21, 162, 60, 45, 96, 40, 204, 199, 225, 148, 152, 115, 167, 208, 123, 253, 105, 2, 188, 212, 217, 252, 124, 187, 3, 78, 172, 194, 29, 139, 224, 200, 65, 84, 146, 64, 201, 138, 30, 48, 102, 240, 76, 211, 189, 180, 94, 236, 62, 129, 86, 142, 112, 160, 73, 228, 15, 51, 38, 222, 47, 31, 43, 56, 164, 135, 19, 151, 149, 82, 137, 202, 196, 98$$
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• Since $255$ is not a prime power (so that $\mathbb Z / (255)$ is not a field), it is not clear what this post has to do with the question. – Alex M. Oct 14 '19 at 15:21
• It mean that people found such involutions not only for fields but also for Rings (which are not Fields) – user144684 Oct 14 '19 at 15:30