We introduce a new technique for doing the key recovery part of an integral or higher order differential attack. This technique speeds up the key recovery phase significantly and can be applied to any block cipher with small S-boxes. We show several properties of this technique, then apply it to PRINCE and report on the improvements in complexity from earlier integral and higher order differential attacks on this cipher. Our attacks on 4 and 6 rounds were the fastest and the winner of PRINCE Challenge's last round in the category of chosen plaintext attack.