Linear approximations of addition modulo 2n-1

Authors:
Chunfang Zhou;Xiutao Feng;Chuankun Wu
Affiliations:
State Key Laboratory of Information Security, Institute of Software, Chinese Academy of Sciences, Beijing, China and Graduate University of the Chinese Academy of Science, Beijing, China;State Key Laboratory of Information Security, Institute of Software, Chinese Academy of Sciences, Beijing, China;State Key Laboratory of Information Security, Institute of Software, Chinese Academy of Sciences, Beijing, China
Venue:
FSE'11 Proceedings of the 18th international conference on Fast software encryption
Year:
2011

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Abstract

Addition modulo 231 - 1 is a basic arithmetic operation in the stream cipher ZUC. For evaluating ZUC's resistance against linear cryptanalysis, it is necessary to study properties of linear approximations of the addition modulo 231 - 1. In this paper we discuss linear approximations of the addition of k inputs modulo 2n -1 for n ≥ 2. As a result, an explicit expression of the correlations of linear approximations of the addition modulo 2n -1 is given when k = 2, and an iterative expression when k 2. For a class of special linear approximations with all masks being equal to 1, we further discuss the limit of their correlations when n goes to infinity. It is shown that when k is even, the limit is equal to zero, and when k is odd, the limit is bounded by a constant depending on k.