> For the complete documentation index, see [llms.txt](https://esper.gitbook.io/esperchain-docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://esper.gitbook.io/esperchain-docs/architecture/publish-your-docs/types-of-post-quantum-cryptography.md).

# Types of Post-Quantum Cryptography

### **Isogeny-Based Cryptography Recap**

Isogeny-based cryptography uses the hardness of finding isogenies between elliptic curves (morphisms preserving structure) as its core security assumption. Key examples include:

| Scheme    | Based on                                       | Status          |
| --------- | ---------------------------------------------- | --------------- |
| SIDH/SIKE | Supersingular isogenies (Diffie-Hellman-style) | Broken          |
| CSIDH     | Class group action on ordinary curves          | Active research |
| SQISign   | Quaternion algebra + isogeny path compression  | Promising       |
| OSIDH     | Oriented isogenies in isogeny graphs           | Experimental    |

They all use **complex multiplication**, **class groups**, or **quaternion algebras** tied to elliptic curves (genus 1).

***

### **2. Are There Other Algebraic Frameworks Like Isogenies?**

#### A. **Higher Genus Curves (Hyperelliptic, etc.)**

* Use Jacobians of curves of genus g>1g > 1g>1 as the underlying group.
* Can define DLP in Jacobians or use isogenies between Jacobians.
* Some schemes (e.g., isogeny between genus-2 Jacobians) are explored.
* Still suffer from quantum attacks if they rely on DLP.
* But **isogeny-based hard problems** in higher genus are not fully understood and **still active research**.

#### B. **Quaternion Algebras** (used in SQISign)

* Already used to compactly encode isogenies.
* Act on the endomorphism rings of supersingular elliptic curves.
* Very effective and cryptographically useful.

***

### **3. Octonion-Based Cryptography?**

#### What are Octonions?

* Octonions O\mathbb{O}O are an 8-dimensional non-associative extension of quaternions.
* Constructed using the **Cayley-Dickson process**.
* Non-commutative and non-associative (even worse than quaternions in that sense).
* Form a **normed division algebra**, but lose many desirable algebraic properties.

#### Why They’re *Not* Currently Used in Cryptography

| Issue                        | Explanation                                                                          |
| ---------------------------- | ------------------------------------------------------------------------------------ |
| **Non-associativity**        | Makes algebraic operations very complex; no consistent multiplication for groups     |
| **No clear group action**    | Cryptography relies on well-behaved groups or rings for security assumptions         |
| **No known hard problem**    | There’s no widely accepted "octonion discrete log problem" or analogous hard problem |
| **Poor algebraic structure** | Makes efficient key generation, signing, or verification difficult to define         |

#### Could They Be Used in the Future?

* **Speculative**: Some theoretical research explores using **non-associative algebras** (like octonions, Jordan algebras) in cryptography, often for exotic constructions.
* No practical or provably secure scheme exists **yet**.
* Might have theoretical applications in **obfuscation**, **nonlinear algebraic cryptosystems**, or **lattice-style constructs**, but nothing mature.

***

### **4. Other Non-Isogeny, Exotic Ideas (Some Involving Algebraic Geometry)**

| Direction                          | Description                                                          | Quantum-safe?   |
| ---------------------------------- | -------------------------------------------------------------------- | --------------- |
| **Lattice-based**                  | Uses hard lattice problems like LWE, SIS                             | Yes             |
| **Code-based**                     | Based on error-correcting codes (e.g., McEliece)                     | Yes             |
| **Multivariate**                   | Solving systems of nonlinear equations                               | Somewhat        |
| **Hash-based**                     | Stateless (SPHINCS+) or Merkle-tree-based                            | Yes             |
| **Geometric group theory**         | Braid groups, non-commutative groups                                 | Mostly broken   |
| **Superspecial Abelian varieties** | Generalizations of isogenies to higher-dimensional abelian varieties | Active research |


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