International Association
for Cryptologic Research

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.