We show that if NC^1 \neq L, then for every element g of the alternating group A_t, circuits of depth O(log t) cannot distinguish between a uniform vector over (A_t)^t with product = g 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.
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.
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