Proof that $(k+m)^2 \mod{n}$ is not injective

Question

$M=K=C=\mathbb{Z}_n$

Lets say $n=10$

$E(k,m) = (k+m)^2 \mod{n}$

Proof that E is not injective (over $\mathbb{Z}_n$) for any key $k \in K$.

So what this means, is that any $n$ consecutive squares (for $k=0$: $0^2,1^2,...,9^2$ etc.) are never injective $\mod{n}$.

How can i prove that?

Answer 1

Hint:

$(-1)^2=1^2$

................................