# Modular Addition in RC5 is linear or not?

So, far my understanding was Modular addition is non linear function which is mainly used in ARX based ciphers.

While I was glancing through RC5 paper (https://link.springer.com/content/pdf/10.1007/3-540-60590-8_7.pdf) the author has mentioned that the only non-linear operation is "data dependent left-rotation", even though it has modulo 2^n addition.

In the given paper, the author, has not mentioned about finite field details, so it is assumed that the addition operation is happening on n-bit register.

I am attaching the image for the reference and also note the last 3 lines.

Consider $$f: x\mapsto f(x)=\underbrace{x+x\ldots x}_{a\text{ times}}+b\bmod 2^{32}$$ in the the ring $$(\mathbb Z_{2^{32}},+,\times)$$, where $$a$$ and $$b$$ are constants.
Per one meaning, $$f$$ is linear, that is of the form $$x\mapsto a\times x+b$$, for constants $$a$$ and $$b$$. Per another meaning, $$f$$ is not (in general) a linear boolean function. $$f$$ is also not linear in the field $$(\mathbb F_{2^{32}},\oplus,\otimes)$$, where $$\otimes$$ is polynomial multiplication modulo an irreducible polynomial of degree $$32$$.