assume two random uniform independent variables X,Y~Unif([N]) prove that the modulo of the sum: (( X + Y ) modulo N) follows a uniform distribution on the set {0,1,…,N−1}
i need to general proof for any N that belongs to Z
i tried: defining another variable Z = (x+y)ModuleN and then saying P(Z=z)=P((x+y)ModuleN = z)
and then i thought of using the fact that they are independent
maybe something like for every z P(Z=z)= sum for every z P(x=i)*P(y=n-i)moduleN
Consider an arbitrary $k \in \left\{0, 1, ..., N-1\right\}$. For any $X$ value in $\left\{0, 1, ..., N-1\right\}$, there is exactly one $Y$ value in $\left\{0, 1, ..., N-1\right\}$ such that $X+Y \equiv k \pmod{N}$.
Specifically, this value is: $\,\,k - X \pmod{N}$.
Therefore the following cardinality is constant and is equal to $N$:
$$\left|\left\{\left(X,Y\right) \in \left\{0, 1, ..., N-1\right\}^2 : X+Y \equiv k \pmod{N}\right\}\right| \, = \, N$$
Because our choice of $k$ was arbitrary, this implies that the distribution you describe is uniform.