## Cryptology ePrint Archive: Report 2016/181

Cryptographic Properties of Addition Modulo $2^n$

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

Abstract: The operation of modular addition modulo a power of two is one of the most applied operations in symmetric cryptography. For example, modular addition is used in RC6, MARS and Twofish block ciphers and RC4, Bluetooth and Rabbit stream ciphers. In this paper, we study statistical and algebraic properties of modular addition modulo a power of two. We obtain probability distribution of modular addition carry bits along with conditional probability distribution of these carry bits. Using these probability distributions and Markovity of modular addition carry bits, we compute the joint probability distribution of arbitrary number of modular addition carry bits. Then, we examine algebraic properties of modular addition with a constant and obtain the number of terms as well as algebraic degrees of component Boolean functions of modular addition with a constant. Finally, we present another formula for the ANF of the component Boolean functions of modular addition modulo a power of two. This formula contains more information than representations which are presented in cryptographic literature, up to now.

Category / Keywords: Modular addition and Boolean function and Component Boolean function and Carry bit and Algebraic degree

Date: received 12 Feb 2016, last revised 23 Feb 2016

Contact author: std_dehnavism at khu ac ir

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2016/181

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