Binary Number Theory-like problem

[Kreizhn]

Homework Statement

I have a function [itex] f: \mathbb Z_p \times \mathbb Z_p \to \mathbb Z_p [/itex] for some prime p. I am given [itex] (r_1, r_2) \in \mathbb Z_p \times \mathbb Z_p\setminus_{\left\{0,0\right\} } [/itex], and told that

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

The Attempt at a Solution

I've been trying different things, such as [itex] f(a_1,a_2) = f(b_1,b_2) \Leftrightarrow (a_1,a_2) \oplus (b_1,b_2) = (r_1,r_2) [/itex] but I'm not even sure how to check if this is correct. Any ideas?

 

[mighty2000]

Thank you for your question. It seems like you are trying to convert a mathematical statement into a binary form using XOR. XOR (exclusive OR) is a logical operation that outputs true only when the inputs differ. In order to convert a mathematical statement into a binary form using XOR, you will need to first understand the properties of XOR and how it can be used to represent mathematical operations.

In your case, you are given a function f: \mathbb Z_p \times \mathbb Z_p \to \mathbb Z_p for some prime p, and you are trying to find a way to represent the statement f(a_1, a_2) = f(b_1,b_2) \Leftrightarrow (a_1,a_2)-(b_1,b_2) = m (r_1,r_2) using XOR. One approach you can take is to use the properties of XOR to represent the operations involved in the statement.

For example, you can represent the operation of subtracting (b_1,b_2) from (a_1,a_2) using XOR by first converting the numbers into their binary representations and then applying XOR to each corresponding bit. This will result in a binary number that represents the difference between the two numbers.

Similarly, you can use XOR to represent the operation of multiplying (r_1,r_2) by an integer m. This can be done by using the properties of XOR to perform bit-wise multiplication and addition, as XOR is equivalent to addition in modulo 2.

Once you have represented each operation involved in the statement using XOR, you can combine them to form a binary representation of the statement. However, please note that this may not be the most efficient or practical way of representing the statement, as it may involve a large number of XOR operations.

I hope this helps. Good luck with your research!