  We continue the study of statistical/computational tradeoffs in learning
robust classifiers, following the recent work of Bubeck, Lee, Price and
Razenshteyn who showed examples of classification tasks where (a) an efficient
robust classifier exists, in the small-perturbation regime; (b) a non-robust
classifier can be learned efficiently; but (c) it is computationally hard to
learn a robust classifier, assuming the hardness of factoring large numbers.
The question of whether a robust classifier for their task exists in the large
perturbation regime seems related to important open questions in computational
number theory. In this work, we extend their work in three directions.
  First, we demonstrate classification tasks where computationally efficient
robust classification is impossible, even when computationally unbounded robust
classifiers exist. For this, we rely on the existence of average-case hard
functions.
  Second, we show hard-to-robustly-learn classification tasks in the
large-perturbation regime. Namely, we show that even though an efficient
classifier that is robust to large perturbations exists, it is computationally
hard to learn any non-trivial robust classifier. Our first construction relies
on the existence of one-way functions, and the second on the hardness of the
learning parity with noise problem. In the latter setting, not only does a
non-robust classifier exist, but also an efficient algorithm that generates
fresh new labeled samples given access to polynomially many training examples
(termed as generation by Kearns et. al. (1994)).
  Third, we show that any such counterexample implies the existence of
cryptographic primitives such as one-way functions. This leads us to a win-win
scenario: either we can learn an efficient robust classifier, or we can
construct new instances of cryptographic primitives.
