We present Constant Power Reconfigurable Computing, a general and device-independent framework based on a closed-loop control system used to keep the power consumption constant for any reconfigurable computing design targeting FPGA implementation. We develop an on-chip power consumer, an on-chip power monitor and a proportional-integral-derivative controller with circuit primitives available in most commercial FPGAs. We demonstrate the effectiveness of the proposed methodology on a square-and-multiply exponentiation circuit implemented on a Spartan-6 LX45 FPGA board. By reducing the peak autocorrelation values by a factor of 2.7 on average, the proposed Constant Power Reconfigurable Computing approach decreases the information leaked by the power consumption of this system with only 26% area overhead and 28% power overhead.