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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 $x \ne 0$

$g_a(x) \ne a$

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 $x_2 \ne 0$

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 $x \ne 0$

$g_a(x) \ne a$

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 $x \ne 0$

$g_a(x) \ne a$

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 $a \in F_q$ the following holds:

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

where $g(x_1) \ne g(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 $x \ne 0$

$g(x) \ne a$

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 $x \ne 0$

$g_a(x) \ne a$