Post-Quantum Public Key Cryptography with EC math properties Is there any quantum resistant public key cryptography with similar properties of elliptic curves?

Assuming lowercase for scalars and uppercase for points. The properties I'm interested are:


Reusing the same public key.
Given $k = a + b$ then $k \times G = (a + b) \times G$
Distributive Scalars: $(a + b) \times G = a \times G + b \times G$
Distributive Points: $k \times (A + B) = k \times A + k \times B$
Commutative Sum: $(a + b) \times G = (b + a) \times G$
Commutative Mul: $a \cdot b \times G = b \cdot a \times G$


How existing solutions map to the same properties?
I'm interested in the direct applications not how it fundamentally works.
Is there any quantum resistant public key cryptography with similar properties of elliptic curves?


Not at the level you're asking about.

The issue is that Shor's algorithm can generically solve the problem "given $A, x \times A$, recover $x$" for any finite group (and going to an infinite group looks problematic for large $x$).  Hence, we'd need to base the algorithm on a different hard problem, and I don't know of any suitable one that fits in the framework you're looking for.

In any case, basing things on a different hard problem would imply that the crypto would look quite different (even if it happens to satisfy the identities you're looking for)


  How existing solutions map to the same properties?


They don't satisfy those identities; instead, they use different properties to provide public key encryption and signatures (and more advanced primitives, such as IBE and PAKE).