Exploiting Linearity of Modular Multiplication

Abstract

The XOR ⊕ and the addition operations have been widely used as building blocks for many cryptographic primitives. These operations and the multiplication operation are successively used in the design of IDEA and the MESH block ciphers. This work presents several interesting algebraic properties of the multiplication operation. By fixing one operand, we obtain vector valued function g Z on Z n 2 , associated with . In this paper we show that the nonlinearity of g Z remains the same under some transformations of Z and moreover we give an upper bound for the nonlinearity of g Z when Z is a power of 2. Under weak-key assumptions, we furthermore present a list of new linear relations for 1round IDEA cipher, some of directly derived and others algorithmically generated using these relations and known ones. We extend the largest linear weak key class for IDEA cipher with size 2 23 to derive such a class with sizes 2 24 . Under the independent key subblocks (subkeys) and weak-key assumptions we derive many linear relations for IDEA cipher using linear relations for 1-round IDEA cipher.