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
    Commented Sep 5, 2016 at 7:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.