Exploiting linearity of modular multiplication

Abstract

The XOR Open image in new window 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 ggZggZ on Zn2Z2n, associated with ⊙⊙. In this paper we show that the nonlinearity of ggZggZ remains the same under some transformations of Z and moreover we give an upper bound for the nonlinearity of ggZggZ when Z is a power of 2. Under weak-key assumptions, we furthermore present a list of new linear relations for 1-round 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 223223 to derive such a class with sizes 224224. 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.