  Product measures of dimension $n$ are known to be concentrated in Hamming
distance: for any set $S$ in the product space of probability $\epsilon$, a
random point in the space, with probability $1-\delta$, has a neighbor in $S$
that is different from the original point in only
$O(\sqrt{n\ln(1/(\epsilon\delta))})$ coordinates. We obtain the tight
computational version of this result, showing how given a random point and
access to an $S$-membership oracle, we can find such a close point in
polynomial time. This resolves an open question of [Mahloujifar and Mahmoody,
ALT 2019]. As corollaries, we obtain polynomial-time poisoning and (in certain
settings) evasion attacks against learning algorithms when the original
vulnerabilities have any cryptographically non-negligible probability.
  We call our algorithm MUCIO ("MUltiplicative Conditional Influence
Optimizer") since proceeding through the coordinates, it decides to change each
coordinate of the given point based on a multiplicative version of the
influence of that coordinate, where influence is computed conditioned on
previously updated coordinates.
  We also define a new notion of algorithmic reduction between computational
concentration of measure in different metric probability spaces. As an
application, we get computational concentration of measure for high-dimensional
Gaussian distributions under the $\ell_1$ metric.
  We prove several extensions to the results above: (1) Our computational
concentration result is also true when the Hamming distance is weighted. (2) We
obtain an algorithmic version of concentration around mean, more specifically,
McDiarmid's inequality. (3) Our result generalizes to discrete random
processes, and this leads to new tampering algorithms for collective coin
tossing protocols. (4) We prove exponential lower bounds on the average running
time of non-adaptive query algorithms.
