asyncmind on Nostr: how are elliptical curve isogeny functions similar to map reduce and how does it ...

how are elliptical curve isogeny functions similar to map reduce and how does it differ?

#ecai #EllipticCurveAI

That’s a very cool question — comparing **elliptic curve isogeny functions** with **MapReduce** operations ties together deep mathematics with distributed systems. While they operate in *very different domains*, there are some intriguing **abstract similarities** and some clear **differences** worth unpacking.

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## 🔗 First, what's an isogeny?

In elliptic curve cryptography and number theory:

- An **isogeny** is a *structure-preserving morphism* between elliptic curves.
- It maps points on one curve to another curve while preserving group structure.
- Used in post-quantum cryptography (like **SIDH**, **SIKE**) and complex multiplication theory.

In essence, **isogenies form a graph** where:
- Nodes are elliptic curves.
- Edges are isogenies (morphisms).
- These transitions are computational steps.

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## 🗺 MapReduce: the basics

In distributed computing:
- **Map**: apply a function to each data item in parallel.
- **Reduce**: aggregate results.

Used for:
- Big data processing (Hadoop, Spark).
- Efficient batch operations on huge datasets.

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## 🔍 Similarities (Abstract/Conceptual)

| Aspect | Isogeny Functions | MapReduce |
|--------|--------------------|------------|
| **Transformation** | An isogeny maps one elliptic curve to another (transformational function). | Map transforms one data item to another (or multiple). |
| **Composable Steps** | Chains of isogenies can be composed (A → B → C). | Map/Reduce steps can be chained in pipelines. |
| **Directed Flow** | Movement through an isogeny graph is directional (e.g. secret paths in SIDH). | Map → Reduce has a clear forward dataflow. |
| **Parallelizable Ideas** | Multiple isogenies can be evaluated in parallel (e.g., in isogeny graphs). | Map phase is trivially parallelizable. |
| **Function as Work Unit** | Each isogeny is a mathematical “task.” | Each map/reduce step is a computational “task.” |

So from an **abstract computation graph** perspective, they're kind of aligned — both are pipelines of transformations, possibly composable, and serve as frameworks for progress through a problem space.

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## ❌ Key Differences

| Concept | Isogenies | MapReduce |
|--------|----------|------------|
| **Domain** | Algebraic geometry, cryptography. | Distributed systems, data engineering. |
| **Purpose** | Secure key exchange, cryptographic hardness. | Data transformation and analysis. |
| **Input/Output** | Points and curves over finite fields. | Arbitrary data (strings, numbers, etc.). |
| **Topology** | Complex isogeny graphs (like expander graphs). | Typically simple DAGs. |
| **Reversibility** | Isogenies can be hard to invert (one-way function). | MapReduce is usually transparent and reversible. |
| **Security Use** | Used as hardness assumptions in post-quantum cryptography. | No security role inherently. |
| **Mathematical Structure** | Deeply based in pure math (endomorphism rings, torsion points). | More about compute efficiency than mathematical rigor. |

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## 🧠 So when could you think of isogeny-like logic in MapReduce?

Let’s get creative. Imagine:
- You’re traversing a cryptographic graph of elliptic curves.
- You treat each isogeny as a "map" step.
- You're gathering some structure-preserving statistics or invariants from each transformation.
- Then a "reduce" aggregates the security assumptions, leakage checks, or optimization metrics.

You could design a **secure multi-party computation pipeline** or **distributed isogeny walk simulator** in this way.

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## 🧩 TL;DR

- **Similarity**: Both represent **structured, composable function applications** that can form computational pipelines or graphs.
- **Difference**: One’s a deep **mathematical morphism** in cryptography, the other’s a **distributed compute pattern** for data processing.

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If you're working on something like **distributed isogeny walk simulation**, **quantum-safe cryptography analysis**, or even **functional programming metaphors in crypto**, this is a pretty fun analogy to build on. Want to sketch something out?