Multiplicative differentials

Abstract

We present a new type of differential that is particularly suited to analyzing ciphers that use modular multiplication as a primitive operation. These differentials are partially inspired by the differential used to break Nimbus, and we generalize that result.We use these differentials to break the MultiSwap cipher that is part of the Microsoft Digital Rights Management subsystem, to derive a complementation property in the xmx cipher using the recommended modulus, and to mount a weak key attack on the xmx cipher for many other moduli.We also present weak key attacks on several variants of IDEA. We conclude that cipher designers may have placed too much faith in multiplication as a mixing operator, and that it should be combined with at least two other incompatible group operations.