On the road towards higher memory density and computer performance, a significant improvement in energy efficiency constitutes the dominant goal in future information technology. Passive crossbar arrays of memristive elements were suggested a decade ago as non-volatile random access memories (RAM) and can also be used for reconfigurable logic circuits(1-5). As such they represent an interesting alternative to the conventional von Neumann based computer chip architectures. Crossbar architectures hold the promise of a significant reduction in energy consumption because of their ultimate scaling potential and because they allow for a local fusion of logic and memory, thus avoiding energy consumption by data transfer on the chip(6-8). However, the expected paradigm change has not yet taken place because the general problem of selecting a designated cell within a passive crossbar array without interference from sneak-path currents through neighbouring cells has not yet been solved satisfactorily. Here we introduce a complementary resistive switch. It consists of two antiserial memristive elements and allows for the construction of large passive crossbar arrays by solving the sneak path problem in combination with a drastic reduction of the power consumption.