Some Limitations of the Sum of Small-Bias Distributions

We present two approaches to constructing ε-biased distributions D on n bits and functions f : {0,1}n→ {0,1} such that the XOR of two independent copies (D+D) does not fool f . Using them, we give constructions for any of the following choices: 1. ε = 2−Ω(n) and f is in P/poly; 2. ε = 2−Ω(n/ logn) and f is in NC2; 3. ε = n−c and f is a one-way space O(c logn) algorithm, for any c; 4. ε = n−Ω(1) and f is a mod 3 linear function. All the results give one-sided distinguishers, and extend to the XOR of more copies for suitable ε . We also give conditional results for AC0 and DNF formulas. Meka and Zuckerman (RANDOM 2009) prove 4 with ε = O(1). Bogdanov, Dvir, Verbin, and Yehudayoff (Theory of Computing 2013) prove 2 with ε = 2−O( √ n). Chen and Zuckerman (personal communication) give an alternative proof of 3. ACM Classification: F.1.3, G.3, F.2.3 AMS Classification: 68Q17