Why Addition Mod 32?

Question

I was looking at the algorithm for Twofish, and I noticed that in some places a XOR is used, but in others, they use "addition modulo-32." What makes modulo-32 special? Why not always use XOR? Why not always addition mod 32?

Answer 1

The combination between addition modulo $2^{32}$ (not modulo $32 = 2^5$) - indicated by $\boxplus$ in the diagram - and XOR (i.e. bitwise addition modulo $2$) - indicated by $\oplus$ - makes the algorithm more non-linear.

Each of them for itself is a linear operation, but over different groups (addition in $GF(2^{32})$ vs. addition in $Z/2^{32})$, and the combination is slightly non-linear over both groups.

Why modulo $2^{32}$? This operation is already implemented in many processors (Java's int addition is this, for example), and this makes implementation easy and efficient.

The later Threefish cipher used in Skein takes this to a new level, its nonlinearity depending exclusively on this simple combination, in a really large number of rounds.