Cryptographic properties of modular multiplication modulo a power of 2

Modular multiplication modulo a power of two is one of the operators that is used in cryptography specially symmetric cryptography. In this paper we investigate the statistical and algebraic properties of this operator from the cryptographic viewpoint. At first we compute the distribution of the output of modular multiplication modulo a power of two and the distribution of its component Boolean functions. Then after presenting a criterion for measuring the imbalance of maps, we obtain the imbalance of this operator and its component Boolean functions. At last we investigate the algebraic degree of the component Boolean functions of this operator and present a lower bound for the degree of these functions.