constructing QR-like one way function 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)?
From your comment, I replace $p$ with a composite $n$.
The answer is no.

The problem is determining whether $f(x,y) = a x^2 + b y^2 \bmod{n}$ is one-way or not.
We want to find $(x',y')$ such that $f(x',y') = z$, given $a, b, n$ and $z = a x^2 + b y^2 \bmod{n}$.

Let us consdier $f'(x,y) = x^2 + h y^2 \bmod{n}$, where $h = b a^{-1} \bmod{n}$.
The problem is now finding $(x',y')$ satisfying $f'(x',y') = m$, given $m = z a^{-1} \bmod{n}$, $h = b a^{-1} \bmod{n}$, and $n$.

The problem is a basis of the Ong-Schnorr-Shamir signature (STOC 1984, CRYPTO 1984).
We already know that Pollad and Schnorr (1987) cryptanalyzed this scheme without factoring $n$.
The algorithm and example is available in a short note written by Jeffrey Shallit.