Statistical Distance and Learning with Rounding

Question

Given an integer $b$ modulo a prime $q$, one can define a `rounding’ function $\lfloor b\rceil_p$ for a prime $p$, $p<q$, as follows: $$\lfloor b\rceil_p = \lfloor \frac{p}{q}\cdot b\rceil\bmod p.$$ In the paper Pseudorandom Functions and Lattices, it is claimed that the statistical distance between the uniform distribution over $\mathbb{Z}_p$, $U(\mathbb{Z}_p)$, and the distribution obtained by sampling from the uniform distribution over $\mathbb{Z}_q$ and then rounding, $\lfloor U(\mathbb{Z}_q)\rceil_p$, is at most $\frac{p}{q}$. Why is this the maximum statistical distance?

Answer 1

The statistical distance between two probability distributions $\mathbf{A}$ and $\mathbf{B}$ with support $S$ is defined as $$ \mathbf{SD}(\mathbf{A}, \mathbf{B}) = \frac{1}{2} \sum_{x \in S} |\mathbf{Pr}[\mathbf{A} = x] - \mathbf{Pr}[\mathbf{B} = x]|\,. $$

Applying this to the rounding function, with positive integers $q\ge p\ge 2$ as assumed in the paper, let $d$ and $r$ be the smallest integers such that $q = d\cdot p + r$ (i.e., $d = \lfloor q/p \rfloor$ and $r = q \bmod p$). Then we have $r$ elements of $\mathbb{Z}_p$ occurring with probability $\frac{d+1}{q}$ and $p-r$ elements with probability $\frac{d}{q}$.

Plugging this in the above formula we get

\begin{align*} \mathbf{SD}(\mathbf{U}(\mathbb{Z}_p), {\lfloor \mathbf{U}(\mathbb{Z}_q) \rceil}_p) &= \frac{1}{2}\left( \left|r\left(\frac{1}{p} - \frac{d+1}{q}\right)\right| + \left|(p-r)\left(\frac{1}{p} - \frac{d}{q}\right)\right| \right) \\ &= \frac{1}{2}\left( \left|\frac{r}{p} - \frac{r(d+1)}{q}\right| + \left|\frac{p-r}{p} - \frac{d(p-r)}{q}\right| \right) \\ &= \frac{1}{2}\left( \left|\frac{r}{p} - \frac{r(d+1)}{q}\right| + \left|\left( 1- \frac{r}{p}\right) - \left(1 - \frac{r(d+1)}{q}\right)\right| \right) \\ &= r\left(\frac{d+1}{q} - \frac{1}{p}\right)\,. \end{align*}

Since $r = q \bmod p$ we can upper bound $r$ by $p$, and thus we can simplify the above to $\frac{p(q/p+1)}{q} - \frac{p}{p} = \frac{q+p}{q} - 1 = p/q$.

This same reasoning also applies to the ${\lfloor \mathbf{U}(\mathbb{Z}_q) \rfloor}_p, {\lceil \mathbf{U}(\mathbb{Z}_q) \rceil}_p$, and $\mathbf{U}(\mathbb{Z}_q) \bmod p$ functions.