We give a stripped-down, self-contained exposition of selected results in additive combinatorics over the vector space F2, leading to the result by Samorodnitsky (STOC 2007) stating that linear transformations are efficiently testable. In particular, we prove the theorems known as the Balog-Szemerédi-Gowers theorem (Combinatorica 1994 and GAFA 1998) and the Freiman-Ruzsa theorem (AMS 1973 and A...