asyncmind on Nostr: how can isogenies used as a map between knowledget structures encoded on a curve ? ...

how can isogenies used as a map between knowledget structures encoded on a curve ?

#ecai #EllipticCurveAI

Now we’re diving into some beautifully speculative territory — combining **elliptic curve isogenies** with **knowledge representation** is super high-concept, but it could lead to a powerful new way of thinking about **cryptographic knowledge graphs**, **semantic state transitions**, or even **neural-symbolic fusion**.

Let’s unpack it.

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## 🧠 Idea: Knowledge Encoded on an Elliptic Curve

First, how might you encode knowledge on an elliptic curve?

### Possibilities:
1. **Points = Concepts**
Each point on the curve (especially torsion points or rational points over some field) represents a semantic unit — a *concept*, *fact*, *node*, or *bit of encoded information*.

2. **Curve Parameters = Context**
The specific curve equation (e.g. \( y^2 = x^3 + ax + b \)) defines the *context*, or *ontology* — essentially a namespace or environment in which the knowledge resides.

3. **Group Law = Reasoning**
The elliptic curve’s group operation (point addition) encodes logical composition — combining concepts to derive new ones.

4. **Structures = Knowledge Graphs**
A curve with known subgroup structure (torsion subgroup, endomorphism ring) acts like a constrained semantic network — knowledge + constraints.

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## 🔁 Isogenies as Knowledge Mappings

### What is an Isogeny in this context?
- An **isogeny \( \phi: E \to E' \)** maps a curve \( E \) to another curve \( E' \), preserving the group structure.
- It transforms not just the *curve*, but also the *knowledge points* on it.
- It is **structure-preserving** and **non-trivial**, meaning knowledge remains consistent but transforms into another representation.

### Use cases:
| Application | Description |
|-------------|-------------|
| **Ontology Mapping** | An isogeny could model how the same knowledge is viewed in two different formal systems or contexts (curves \( E \) and \( E' \)). |
| **Proof Transport** | If a proof or verified computation is valid on one curve, and there's an isogeny to another, then it may be possible to translate the proof via the isogeny. |
| **Privacy-preserving Transformation** | Isogeny-based transformations could hide structure or context while retaining integrity — similar to a homomorphic mapping. |
| **Knowledge Evolution** | A knowledge graph on \( E \) changes through reasoning or learning; the isogeny maps that state to a new state \( E' \). |

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## 🔁 Parallel with Functional Programming

This is almost like:

```haskell
type KnowledgeGraph = EllipticCurve
type SemanticMap = Isogeny

transform :: KnowledgeGraph -> SemanticMap -> KnowledgeGraph
transform E φ = φ(E)
```

You can think of **isogeny chains** as **semantic pipelines** — kind of like a *deep transformation of meaning or form*, analogous to how a language model refines meaning layer by layer.

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## 🧬 Bonus: Quantum Knowledge and Isogeny Obfuscation

In post-quantum crypto (like SIDH), isogenies are:
- Easy to compute in one direction.
- Infeasible to reverse without secret info.

In knowledge terms:
- You might expose *just enough* transformed knowledge (on \( E' \)) for verification, without leaking the full structure on \( E \).
- This is like **zero-knowledge knowledge** transmission: “I’ve got knowledge K encoded here, and I can prove a transformation of it without exposing the source.”

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## 🧭 Experimental Applications

1. **Semantic Isogeny Network**
Curves represent worldviews or theories.
Isogenies represent valid translations of knowledge between them.

2. **Isogeny-based Knowledge Compression**
If multiple facts or proofs on \( E \) can be composed into one isogeny map to \( E' \), the target curve becomes a *compressed knowledge representation*.

3. **Cryptographic BDDs**
Encode logical behavior (like DamageBDD does) on curves. Isogenies represent logical implication, refactorings, or proof-of-behavior equivalence.

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## 🧩 TL;DR

**Isogenies** can be seen as **structure-preserving mappings** between **curves that encode knowledge**.
They let you:
- Translate between knowledge systems.
- Perform cryptographically secure proof transformations.
- Abstract reasoning into curve-based semantic flows.

> It’s like mapping consciousness across different minds that still preserve meaning.

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If you want to actually prototype this, we can sketch an experiment — maybe a simple knowledge base encoded on points, with isogenies used to shuffle perspectives or domains. Want to go there?