Gradual Small-Bias Sample Spaces

A (k, ε)-biased sample space is a distribution over {0, 1}n that ε-fools every nonempty linear test of size at most k. Since they were introduced by Naor and Naor (SIAM J. Computing, 1993), these sample spaces have become a central notion in theoretical computer science with a variety of applications. When constructing such spaces, one usually attempts to minimize the seed length as a function of n, k and ε. Given such a construction, if we reverse the roles and consider a fixed seed length, then the smaller we pick k, the better the bound on the bias ε becomes. However, once the space is constructed we have a single bound on the bias of all tests of size at most k. In this work we initiate the study of a new pseudorandom object, which we call a gradual (k, ε)-biased sample space. Roughly speaking, this is a sample space that ε-fools linear tests of size exactly k and moreover, the bound on the bias for linear tests of size i ≤ k decays as i gets smaller. We show how to construct gradual (k, ε)-biased sample spaces of size comparable to the (non-gradual) spaces constructed by Alon et al. (Random Structures and Algorithms, 1992), and prove a lower bound on their size. Our construction is based on the lossless expanders of Guruswami et al. (J. ACM, 2009), combined with the Quadratic Character Construction of Alon et al. (Random Structures and Algorithms, 1992) ∗Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. Email: [email protected]. †Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. Email: [email protected]. Research supported by Israel Science Foundation (ISF) grant. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 3 of Report No. 50 (2012)