Statistical properties of modular multiplication modulo a power of two

In this paper, we investigate statistical properties of modular multiplication modulo a power of two. In fact, we obtain the distribution of every single bit of modular multiplication. Moreover, we determine the distribution of modular multiplication as a vectorial Boolean function. Then, the joint distribution of modular multiplication component bits is discussed: an explicit formula in the case of two component bits is determined and for the joint distribution of any number of component bits we prove a theorem. As a result of this theorem, we calculate the joint distribution of any number of component bits of modular multiplication modulo a power of two with reasonable computational complexity.