# 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:

1. Reusing the same public key.
2. Given $$k = a + b$$ then $$k \times G = (a + b) \times G$$
3. Distributive Scalars: $$(a + b) \times G = a \times G + b \times G$$
4. Distributive Points: $$k \times (A + B) = k \times A + k \times B$$
5. Commutative Sum: $$(a + b) \times G = (b + a) \times G$$
6. 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.

• What were you hoping to do with such a beast, if it were to exist? – Squeamish Ossifrage Oct 23 at 15:27

Is there any quantum resistant public key cryptography with similar properties of elliptic curves?

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).

• I was hoping for a way to migrate custom schemes to new post-quantum schemes. But these rely heavily on those math properties. – shumy Oct 23 at 15:56
• I was hoping that random walks in Supersingular Isogeny could map to $x \times A$, but I dont understand its properties enough. Does it have any subset of the requirements? – shumy Oct 23 at 16:01
• @shumy: well, I don't know of good way to add either 'two random walks' or 'two elliptic curves' (in a way that preserves the properties), so relations 2-6 are out. Safe key reuse is possible, but tricky. That leaves 7 (which SIDH depends on). – poncho Oct 24 at 2:35
• Sophism time! "The issue is that Shor's algorithm can generically solve the problem "given A,x×A, recover x" for any finite group". If you look at CSIDH, you realize this statement is not accurate: it depends on the way you "represent" elements of the group. But the representation in CSIDH takes away the possibility of adding elements (basically what you are saying about "adding random walks"), and actually it even takes away the possibility of saying who the identity element is. – Luca De Feo Oct 24 at 9:56
• @LucaDeFeo: I was using the meaning $x \times A = A + A + A + … + A$ ($x$ times). I believe (and you would be the expert, so feel free to correct me) that CSIDH uses a different meaning of $\times$, and so my statement doesn't apply. For that matter, if you don't have an 'identity element', then you don't have a 'group'... – poncho Oct 24 at 13:30