$K$ is a finite field which not equal to its base field $F_2$.
Let $f: V \rightarrow K$ be a function and $B(x,y)=f(x+y)+f(x)+f(y)$ such that $B(x+y,z)=B(x,z)+B(y,z)$ and $B(z,x+y)=B(z,x)+B(z,y)$ for every $x,y,z \in V$ (vector space over $K$) but $B(ax,y)$ is not necessary to equal to $aB(x,y)$ for every $a \in K$, same for $B(x,ay)$. Actually $B$ is linear over base field of $K$, but not linear over $K$.
Is there any such research area?
Example: $f(x)=Tr(x^9+x^5)$ where $Tr: F_{256} \rightarrow F_4$ is trace function.