Efficiently Testing Sparse GF(2) Polynomials
  We give the first algorithm that is both query-efficient and time-efficient for testing whether an unknown function $f: \{0,1\}^n \to \{0,1\}$ is an $s$-sparse GF(2) polynomial versus $\eps$-far from every such polynomial. Our algorithm makes $\poly(s,1/\eps)$ black-box queries to $f$ and runs in time $n \cdot \poly(s,1/\eps)$. The only previous algorithm for this testing problem \cite{DLM+:07} used poly$(s,1/\eps)$ queries, but had running time exponential in $s$ and super-polynomial in $1/\eps$.   Our approach significantly extends the ``testing by implicit learning'' methodology of \cite{DLM+:07}. The learning component of that earlier work was a brute-force exhaustive search over a concept class to find a hypothesis consistent with a sample of random examples. In this work, the learning component is a sophisticated exact learning algorithm for sparse GF(2) polynomials due to Schapire and Sellie \cite{SchapireSellie:96}. A crucial element of this work, which enables us to simulate the membership queries required by \cite{SchapireSellie:96}, is an analysis establishing new properties of how sparse GF(2) polynomials simplify under certain restrictions of ``low-influence'' sets of variables. 
Ilias Diakonikolas, Homin K. Lee, Kevin Matulef, Rocco A. Servedio,   Andrew Wan