Modulo operation in monotone complexity

Question

Given $x\in\Bbb N$, I would like to find $x\bmod N$, where $N$ is composite. For example $N=35$, $x=53$ and $x\bmod N=18$. Is this operation considered monotone in circuit/algebraic complexity language? I also want to have consider $x_1+x_2\bmod N = x_1\bmod N+x_2\bmod N$ and $x_1x_2\bmod N = (x_1\bmod N)(x_2\bmod N)$.

Answer 1

In circuit complexity, a monotone operation is a function $f$ which is also monotone: if $x \leq y$ then $f(x) \leq f(y)$, where $x \leq y$ is a shorthand for $x_i \leq y_i$ for all $i$. There are many ways to define modulo $N$, but most of them are probably either uninteresting or not monotone. For example, consider the function which is true if the input is divisible by $N$. Then $f(0) = 1$ while $f(1) = 0$, and $0 \leq 1$ no matter what the actual length of the input is.