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Linear approximations of addition modulo $$2^n-1$$. (English) Zbl 1307.94115
Joux, Antoine (ed.), Fast software encryption. 18th international workshop, FSE 2011, Lyngby, Denmark, February 13–16, 2011. Revised selected papers. Berlin: Springer (ISBN 978-3-642-21701-2/pbk). Lecture Notes in Computer Science 6733, 359-377 (2011).
Summary: Addition modulo $$2^{31} - 1$$ 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 $$2^{31} - 1$$. In this paper we discuss linear approximations of the addition of $$k$$ inputs modulo $$2^{n } - 1$$ for $$n \geq 2$$. As a result, an explicit expression of the correlations of linear approximations of the addition modulo $$2^{n } - 1$$ 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 $$k$$.
For the entire collection see [Zbl 1217.68011].

##### MSC:
 94A60 Cryptography 68M07 Mathematical problems of computer architecture
SNOW
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