Cryptographic Properties of Addition Modulo $2^n$

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

Paper 2016/181

Cryptographic Properties of Addition Modulo $2^n$

S. M. Dehnavi, A. Mahmoodi Rishakani, M. R. Mirzaee Shamsabad, 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.

Metadata

Available format(s): PDF
Publication info: Preprint. MINOR revision.
Contact author(s): std_dehnavism @ khu ac ir
History: 2016-02-23: revised; 2016-02-23: received; See all versions
Short URL: https://ia.cr/2016/181
License: CC BY

BibTeX

@misc{cryptoeprint:2016/181,
      author = {S.  M.  Dehnavi and A.  Mahmoodi Rishakani and M.  R.  Mirzaee Shamsabad and Hamidreza Maimani and Einollah Pasha},
      title = {Cryptographic Properties of Addition Modulo $2^n$},
      howpublished = {Cryptology {ePrint} Archive, Paper 2016/181},
      year = {2016},
      url = {https://eprint.iacr.org/2016/181}
}