# Binary Number Theory-like problem

1. Oct 21, 2009

### Kreizhn

1. The problem statement, all variables and given/known data
I have a function $f: \mathbb Z_p \times \mathbb Z_p \to \mathbb Z_p$ for some prime p. I am given $(r_1, r_2) \in \mathbb Z_p \times \mathbb Z_p\setminus_{\left\{0,0\right\} }$, and told that

$f(a_1, a_2) = f(b_1,b_2) \Leftrightarrow (a_1,a_2)-(b_1,b_2) = m (r_1,r_2)$
for some integer m. That is, their difference is an integer multiple of $(r_1, r_2)$. I need to find some way of converting this statement into binary, specifically something using XOR.

3. The attempt at a solution
I've been trying different things, such as $f(a_1,a_2) = f(b_1,b_2) \Leftrightarrow (a_1,a_2) \oplus (b_1,b_2) = (r_1,r_2)$ but I'm not even sure how to check if this is correct. Any ideas?