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**2010/521**

**Title:**

Linear Approximations of Addition Modulo 2^n-1

**Authors:**
Xiutao Feng, Chunfang Zhou, Chuankun Wu

**Abstract:**

Addition modulo 2^{31}-1 is a basic arithmetic operation in the stream cipher ZUC. For evaluating ZUC in resistance to linear cryptanalysis, it is necessary to study properties of linear approximations of the addition modulo 2^{31}-1. In this paper we discuss linear approximations of the addition modulo 2^n-1 for integer n\ge2. As results, an exact formula on the correlations of linear approximations of the addition modulo 2^n-1 is given for the case when two inputs are involved, and an iterative formula for the case when more than two inputs are involved. 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. Let k be the number of inputs of the addition modulo 2^n-1. Itâ€™s shows 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.

**ePrint:**
https://eprint.iacr.org/2010/521

**Slides: **http://www.iacr.org/cryptodb/archive/2011/FSE/presentation/23562.pdf

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