Finite field special functions

Question

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 $x_2 \ne 0$

Answer 1

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 $GF(n)$.

Answer 2

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.

Answer 3

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.

Answer 4

$GF(31)$:

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

Answer 5

$Z/255Z$:

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

Answer 6

The perfect nonlinear function $f$ can be represented as a special sequence ($a_{1},...,a_{n}$).

The main property: $f(a_{i}) = a_{-i}$.

Other properties:

${a_{2^k}}$ = $2^k$;

There exist such element $\alpha$ that

${a_{-i} = a_{i} + i*(\alpha-1)}$

For $i \ne 0, i+1 \ne 0$ we have:

$ f(a_{i}) - f(a_{i}-a_{i+1}) = 1$

For any 2 not equal $i,k \ne 0, i+k \ne 0 $:

$ f(a_{i}) - f(a_{i}-a_{i+k}) = a_{k}$

I know yet 5 of the same nature sequences for the $GF(31)$.

I know a sequence of the same nature for the $GF(2^{k}-1)$ and prime $2^{k}-1$.

Looks like the following property holds as well:

$ a_{j} - a_{k} = a_{j+ \frac{a_{k}-a_{k-j}}{\alpha-1}} $

$ a_{i} = -f(i*(\alpha-1)) $

Example ($\alpha=18$):

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