The Moon and the Maths: The Sea of Tranquility
Source: Dev.to
Chapter 1 – The Sea of Tranquility
This is an exploration of the Moon and the “Moon Maths” that underlies Zero‑Knowledge Proofs (ZKPs). Starting at the Apollo 11 landing site, our journey will take us around the Moon, each chapter introducing a new area of mathematics.
- The Moon Math Manual – [download link] – the guide to the maths we will discuss (with exercises we can try as we travel).
- Music – an appropriate soundtrack is Apollo Atmospheres and Soundtracks by Brian Eno.
Charts & Companions
NASA has made many charts available to help us navigate. Our companions are Peggy and Victor:
- Victor is a sceptic, always worried that Peggy might try to cheat him with an incorrect proof.
We are ready to set off, but where are we going?
Our journey takes us from the well‑understood landscape of the Sea of Tranquility to the mysterious dark side of the Moon, from basic mathematics to elliptic curves and the processes used in ZKP systems.
Itinerary
| Chapter | Destination | Topic |
|---|---|---|
| 1 | Sea of Tranquility | The basics |
| 2 | Bay of Rainbows | Polynomials |
| 3 | Ocean of Storms | Finite fields |
| 4 | Apennine Mountains | Abstract algebra |
| 5 | The Far Side | Elliptic curves |
| 6 | Sea of Moscow | R1CS |
| 7 | Crater Tycho | Trusted setup |
| 8 | The North Pole | Groth16 & true zero‑knowledge |
The Conflict: Verification vs. Privacy
Banks emphasise privacy: your details remain private, but transactions are not verifiable, so we must trust the bank.
Zero‑Knowledge Proofs resolve this conflict: we obtain both verifiability (mathematical proofs) and privacy (the verifier learns nothing new).
Analogies
- Imagine our journey to the Moon began as a secret search for an alien artifact (as in 2001: A Space Odyssey).
- Physically covering information is useless for digital assets; we need mathematical constructs that hide information while still allowing verification.
Our aim in this first chapter is to establish the background in a well‑known setting: the Sea of Tranquility (Mare Tranquillitatis).
Setting the Scene
- To the right: the Sun is rising in the East.
- Ground: grey, flat regolith – a layer of shattered rock and charcoal‑grey powder covering the bedrock, formed by meteorite impacts. The particles are sharp and can damage equipment.
- Horizon: deceptively close (a few kilometres away) and curves sharply because the Moon is small.
Just as the lunar horizon can be misleading, the realm of proofs can cause us to lose perspective. We need clear rules.
What Makes a ZKP?
A Zero‑Knowledge Proof system must satisfy three properties:
- Completeness – If Peggy is honest and the statement she proves is true, the math always works; Victor will always accept her proof.
- Soundness – If the statement is false, Peggy cannot cheat. The probability of fooling Victor is negligible.
- Zero‑Knowledge – If the statement is true, Victor learns nothing else; he cannot reverse‑engineer the proof to discover any secrets.
Terminology Primer
| Symbol | Meaning |
|---|---|
| $\mathbb{Z}$ | The set of all integers (whole numbers). |
| $\mathbb{Q}$ | The set of rational numbers (ratios of integers). |
We will focus on integers and modular arithmetic (often called clock maths).
- Congruence groups together integers that give the same remainder when divided by a modulus.
- Example: $13 \equiv 17 \equiv 21 \pmod{4}$ because each leaves remainder 1.
More arithmetic will be explored in Chapter 3 (Ocean of Storms) when we study fields.
Landscape Features
- Apennine Mountain Range – rising to our left as we leave the Sea of Tranquility.
- Crater Plinius – a sharp, 43 km‑wide crater with a central peak, standing sentinel nearby.
We are leaving the familiar behind and heading for the unknown, so more terminology and concepts are needed.
Key Concepts
- Witness – the secret data Peggy knows and wants to prove she knows without revealing it to Victor.
- Asymmetry – a fundamental cryptographic concept we will revisit many times.
The Two‑Step Analogy
- Finding the Alien Artifact (The Witness) – hard, requires vehicle, fuel, time, luck; computationally expensive.
- Checking the Photograph (The Statement) – easy, takes a split second.
If solving a problem were easy, we wouldn’t need a proof—we would just solve it ourselves.
Hence ZKP systems aim to be succinct: the proof should be small enough to handle, and verification should be reasonably fast.
Traversing the Terrain
We cannot drive over the Apennine Mountains; we must pass through the Hadley‑Apennine Gap (the landing site of Apollo 15). This narrow corridor where the mountains meet the marshy Palus Putridi (the “Marsh of Decay”) will be our next waypoint.
edinis (Marsh of Decay)
To our immediate left is a massive winding canyon called Hadley Rille; to the right, the towering Mount Hadley.
We won’t climb the Apennines yet—that challenge is saved for Chapter 4, when we tackle the heavy lifting of Abstract Algebra.
The Journey Continues
Once we clear the mountains, the world opens up and we enter the Sea of Rains. We navigate by three massive craters in the distance:
- Archimedes
- Autolycus
- Aristillus
Traveling northwest, we come to the terminator line, marking the boundary between day and night.
Such a sharp distinction between light and dark is what we want for our proofs: an exact, easily tested cutoff between a valid and an invalid proof.
The Mathematics Behind It
In the underlying math we achieve this through probability. While theoretically probabilistic, the chance of a false proof being accepted is so vanishingly small (like guessing a specific grain of sand in this desert) that, for all practical purposes, it is a sharp, solid wall.
The shadows are getting longer. The math is about to get steeper.
Sinus Iridum – where we will learn that every secret is just a point on a polynomial curve.
Further Resources
For more information about Zero‑Knowledge Proofs (ZKPs) and the mathematics behind them, visit our Academy.
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