I’ve spent the last couple of years circling one question from as many directions as I could find: why is it hard to split a number into its two prime factors, when multiplying them together is trivial? Everyone in cryptography treats this asymmetry as a given. RSA is built on it. But “it’s hard” is a description, not an explanation. I wanted to see the difficulty — literally, geometrically — and that’s what this book tries to do.
The Wall at Two is the result. It’s a short book, and it’s written for anyone who’s comfortable with a bit of high-school algebra and willing to sit with a picture for a while. No prerequisites in number theory. No proofs you have to take on faith.
The picture at the center
Here’s the frame the whole book hangs on. When you multiply two primes p and q together, the product n is public — anyone can see it. Think of n as a circle everyone can look at. The factorization is a specific angle hidden inside that circle. The circle tells you the angle exists. It tells you almost nothing about where it points.
Trial division, the naive attack, is just walking around the circle checking angles one at a time. And the book’s title comes from where that walk hits a wall: every composite number has a factor at or below its square root, so the search space is bounded — but for a balanced semiprime, both factors sit right up against that boundary, packed into the region where candidates are densest. The wall at two is the observation that you can always split the search in half, but you can never do better than that from structure alone. The number keeps its secret not by hiding it far away, but by hiding it among too many neighbors.
What’s actually in the book
The chapters build one geometric object at a time:
The public circle and the secret angle — the core picture, developed slowly enough that you can reconstruct it yourself on paper.
The shadow prime — a companion quantity I call q̆ that travels alongside the hidden factor. It turns out to be provably irrational, always, and that irrationality is doing real work: it’s part of why no simple ruler ever lines up with the answer.
Balance and the square root — why the hardest semiprimes are the most symmetric ones, and what that symmetry looks like when you plot it.
The hinge — a construction (via Rabin’s connection between square roots and factoring) that shows the secret isn’t just hidden, it’s load-bearing: knowing certain square roots mod n is the same as knowing the factorization. The difficulty isn’t decorative. It’s structural.
Who this is for
I wrote it for the reader I was before I started: curious about why RSA works, tired of hand-waving, and more convinced by a diagram than by a citation. If you’ve ever wondered what the “hard problem” under modern encryption actually looks like from the inside, this is my best attempt at showing you.
I should be clear about what I am and am not. I’m an independent researcher — no institutional affiliation, no journal publications on this topic. The proofs in the book stand or fall on their own, and I’ve tried to make every step checkable by the reader. Where something is an observation rather than a theorem, I say so.
Where to get it
The Wall at Two: Seeing Why a Number Keeps a Secret is available now on Kindle ($4.99), with the paperback edition arriving within days.
If you read it and something doesn’t hold up — or something does — I’d genuinely like to hear about it. This project has always moved forward through people pushing back on the pictures.