Abstract: Modern cipher components ususlly combine mixed operations of different groups. In cryptanalysis,one alwalys uses the XOR operation to linearly approximate modular addition,which requires calculation of the distribution of noise.In this paper, we propose a reduced linear representation of such distribution based on the formal power series on free monoid which can be easily used to calculate the pobability diatribution.We also extend the result to the addition modulo p^n.This provides a solid basis for linear cryptanalysis.

Key words: cryptanalysis, reduced linear representation, probability distribution, mixed operations