Let $G = {0, cdots , p-1 }$ be a field. Let $K = G^{m times n}$ and $F:K times G^n to G^m$ be a family of functions.

For $A in G^{m times n}$ and $x in G^n$ we have $F(A,x) = Ax$.

I need to check if $F$ is a secure pseudo random function.

We say that PRF $H: K times X to Y$ is $(T, epsilon)$-secure if for every algorithm $B: X to {0,1}$ of size $T$ it follows:

$$|P(B^{H_k()} = 1) – (B^{R()} = 1)| le epsilon$$

where $H_k(x) = H(k,x)$, $R:X to Y$ is a random function, and $B^{S()}$ means $B$ has oracle access to the function $S$.

Now, back to the question. Since the matrix $A in G^{m times n}$, we can take $m times n$ base elements of $G^{m times n}$, and check that for elements in $G^m$ if they are in $Ax$.

I tried to describe the following adversary B:

on input $x in G^m$ and access to oracle $Z()$, A will querry $Z$ on $x$, then it will somehow check if $x$ is in the image. But I am not sure how to do this.

Help would be very appreciated!

There is also the same question but with $K = G^{m times n} times G^m$ and $F:K times G^n to G^m$, $F((A,b), x) = Ax+b$.