Cloud service providers are interested in deploying multi-tenant Field-Programmable Gate Arrays (FPGAs) for virtualized computation platforms. A primary concern towards such shared FPGA platforms is ensuring the security of critical applications (such as encryption cores) against hardware-based attacks, such as remote power Side-Channel Attacks (SCAs) intended to steal secret assets like encryption keys. To address this issue, we propose ShapeShifter, a novel defense methodology based on design diversity that generates isofunctional variants of the target application at the design stage using different synthesis, placement, and routing procedures. ShapeShifter leverages the dynamic partial reconfiguration feature of modern FPGAs to exchange the variants at run-time, causing dynamic variations in the power trace (vertical obfuscation) and introducing misalignment in the time domain (horizontal obfuscation) to thwart SCAs. ShapeShifter successfully thwarts the Correlation Power Analysis (CPA) attack on an Advanced Encryption Standard (AES) implementation, ensuring unsuccessful key byte recovery for up to 10× more traces. It also decreases the CPA value by 0.69×, reducing the attacker's confidence in key recovery.