Independence of events in spaces of equally probable ciphervalues

Within the framework of the probabilistic cipher model, the problem of decomposition in some orthogonal coordinate system of the discrete space $\Omega$ of elementary events into pairs of families of incompatible events independent of any event of another family is considered. It is shown that the binary event independence relation is related to the number-theoretic nature of the number $N$ — the power of the discrete space $\Omega$ of elementary events. It is proved that for a composite number $N$ there are pairs of independent subspaces of the space $\Omega$, and for prime $N$ there are no independent subspaces. Examples illustrating the obtained theoretical statements are constructed.