  We show that if NC$^1 \neq$ L, then for every element $\alpha$ of the
alternating group $A_t$, circuits of depth $O(\log t)$ cannot distinguish
between a uniform vector over $(A_t)^t$ with product $= \alpha$ and one with
product $=$ identity. Combined with a recent construction by the author and
Viola in the setting of leakage-resilient cryptography [STOC '13], this gives a
compiler that produces circuits withstanding leakage from NC$^1$ (assuming
NC$^1 \neq$ L). For context, leakage from NC$^1$ breaks nearly all previous
constructions, and security against leakage from P is impossible. %In the
multi-query setting, circuits produced by this compiler use a simple secure
hardware component.
  We build on work by Cook and McKenzie [J.\ Algorithms '87] establishing the
relationship between L $=$ logarithmic space and the symmetric group $S_t$. Our
techniques include a novel algorithmic use of commutators to manipulate the
cycle structure of permutations in $A_t$.
