Lifting theorems are theorems that relate the query complexity of a function f : {0, 1}^n → {0, 1} to the communication complexity of the composed function f ◦ g^n, for some “gadget” g : {0, 1}^b × {0, 1}^b →{0, 1}. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget g. We prove a new lifting theorem that works for all gadgets g that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we increase the range of gadgets for which such lifting theorems hold considerably. Our result has two main motivations: First, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications the theorem. Second, our result can be seen a strong generalization of a direct-sum theorem for functions with low discrepancy. Joint work with Arkadev Chattopadhyay, Yuval Filmus, Or Meir, Toniann Pitassi