I am wondering whether the following form is a one way function with collision resistance: $$ax^2+by^2 \mod p$$ where $a$ and $b$ are given, and $p$ is a prime number.
Since the QR (quadratic residue) $x^2 \mod p$ is one way, $ax^2 \mod p$ is also one way. The same goes for $by^2$.
So, my guess is that the form $ax^2+by^2 \mod p$ is one way. Is that correct?
A few more follow-up questions:
- How to formalize a proof that a numeric equation is one way?
- Does anyone know an archive collecting all of known one way functions?
- If $ax^2+by^2 \mod p$ is not one way, how about $a^2x+b^2y \mod p$?
- What is instead of a prime $p$ we have a composite number (possibly with certain assumptions on its factorization)?