I'm studying a block cipher based on ARX(Addition, Rotation, XOR).

In a block cipher, nonlinearity comes from S-box or modular addition($Z=X\boxplus Y$ mod $2^n$).

As far as I know, in the case of S-Box, its security can be explained in the algebraic aspect.

How to explain the modular addition($\boxplus$ mod $2^n$) in the algebraic aspect?


1 Answer 1


So you have addition with carry bitwise if the sum is smaller than $2^n-1$ as well as the subtraction of $2^n$ if the sum is greater than $2^n-1.$

Helger Lipmaa has analysis that is relevant in "Efficient Algorithms for Computing Differential Properties of Addition" available at


You can use that work as a starting point.

  • $\begingroup$ kodlu, thank you for your answer. The paper you told me was about statisitical analysis method, but it became good starting point to search the answer. In this weekend, I found a paper for the aspect of modular addtion, and you can also check it at nicolascourtois.com/papers/usnow.pdf Thank you again kodlu! $\endgroup$
    – Tylor Yoo
    Sep 5, 2016 at 7:01

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