Statistical properties of square and square root maps

The square map is one of the functions that is used in cryptography. For instance, the square map is used in Rabin encryption scheme, block cipher RC6 and stream cipher Rabbit, in different forms. In this paper, we study statistical properties of the output of the square map as a vectorial Boolean function. We obtain the joint probability distribution of arbitrary number of the upper and the lower bits of the output of square map along with asymptotic probability distribution of the upper bits of its output. Based upon a measure for evaluating the imbalance of maps, we study the imbalance of limit distribution of the restriction of square map to its upper bits. At last, we introduce the square root map and we examine this map as a vectorial Boolean function; we compute probability distribution of the component Boolean functions of the proposed map and also we obtain the imbalance of the square root map.