International Association
for Cryptologic Research

In this paper, we investigate some statistical properties of multiplication mod $2^n$ for cryptographic use.

For this purpose, we introduce a family of T-functions similar to modular multiplication, which we call

M-functions as vectorial Boolean functions. At first, we determine the joint probability distribution of

arbitrary number of the output of an M-function component bits. Then, we obtain the probability distribution

of the component Boolean functions of combination of a linear transformation with an M-function. After that,

using a new measure for computing the imbalance of maps, we show that the restriction of the output of an

M-function to its upper bits is asymptotically balanced.