Statistical properties of the square map

Document Type: Regular Paper


1 Faculty of Mathematical and Computer Sciences, Kharazmi University,Tehran, Islamic Republic of Iran

2 Faculty of Sciences, Shahid Rajaee Teacher Training University, Tehran, Islamic Republic of Iran

3 Faculty of Mathematics and Computer Science, Shahid Bahonar University, Kerman, Islamic Republic of Iran


The square map is one of the functions 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
the 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.
Last, we introduce the square root map and examine this map as a vectorial Boolean function; we compute
probability distribution of the component Boolean functions of this new map and also obtain the imbalance of the
square root map.