Swap elements using integer addition and multiplication gates
Let's say we have inputs $x, y$ and $c$, where $c$ is either 0 or 1, 0 = no swap, 1 = swap. We can make a conditional swap function like this:

notc = 1 + (-1)*c
x' = notc*x + c*y
y' = c*x + notc*y


Even if you have no negative integers, as long as your integers are modulo $2^n$, you can use $-1 \equiv 2^n - 1 \mod 2^n$ and everything works out.