**Statistical Properties of Multiplication mod $2^n$**

*A. Mahmoodi Rishakani and S. M. Dehnavi and M. R. Mirzaee Shamsabad and Hamidreza Maimani and Einollah Pasha*

**Abstract: **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.

**Category / Keywords: **foundations / Modular multiplication, Boolean function, Joint probability distribution, T-function, Symmetric cryptography

**Date: **received 4 Mar 2015, last revised 5 Mar 2015

**Contact author: **std_dehnavism at khu ac ir

**Available format(s): **PDF | BibTeX Citation

**Version: **20150306:132127 (All versions of this report)

**Short URL: **ia.cr/2015/201

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