# Is correlation in vector distributions “dangerous”?

Consider the two vector distributions $\xi,\chi$ described below, each one outputting integer vectors of length $n$ with coefficients in $\{0,\dots,n\}$.

1. Distribution $\xi$ samples each coefficient $v_i$ following a distribution $\alpha_i\sim Binomial(n,p)$ and outputs $\mathbf{v}=(v_1,\dots,v_{n})$. The $\alpha_i$'s are pairwise independent.

2. Distribution $\chi$ samples each coefficient $w_i$ following a distribution $\beta_i\sim Binomial(n,p)$ and outputs $\mathbf{w}=(w_1,\dots,w_{n})$. The $\beta_j$'s are dependent: for each $a\neq b$, we have $$\operatorname{Corr}(\beta_a,\beta_b)=r>0,$$

where $\operatorname{Corr}=\operatorname{Cov}(\beta_a,\beta_b)/\sigma_a\sigma_b$ is the Pearson correlation coefficient.

I know both distributions are distinguishable, but how many samples do we need to actually distinguish them?

Let's say that the use of distribution $\xi$ is secure in a protocol. Is the use of $\chi$ less secure ? I.e., does correlation helps the attacker? The answer depends maybe on the attack, but is there a general result or fact that will help me understand to what extent is correlation dangerous?

• This question, while interesting, is not fully specified, I believe. What exactly do the black boxes output? $P(X)$ for some random $X$? but where does $X$ live? Where does $P(X)$ live? Is $P(X)$ an integer in $\{0,1,\ldots,1023\}$ in $\mathbb{N}$? The field $\mathbb{F}_{1024}$? Or do they simply output the random vector $(p_0,\ldots,p_{1023})$? Some of these choices are equivalent, some not. I noticed you say sample polynomials not sample vectors. – kodlu Apr 26 '16 at 23:58
• All is stated above, the boxes output polynomials on the symbolic variable $X$, i.e. they output the coefficient vectors. I'll edit this question though, using vectors as you suggest. – Tal-Botvinnik Apr 27 '16 at 12:07
• "how many samples do we need to actually distinguish them?": With what level of certainty? – Aleph Apr 28 '16 at 16:39
• Well if we make a guess, we are 50% sure we distinguished them correctly. So better than that – Tal-Botvinnik Apr 28 '16 at 16:49