Imagine a world defined by numbers and shapes, where certain operations are easy and straightforward, while others are almost impossible to reverse. In this world, we have a special kind of shape, a curve, that isn’t like the curves you might see on a piece of paper. This curve exists in a number space, and it has unique properties that make it particularly interesting.

This curve, known as an elliptic curve, isn’t a physical object but rather a concept—a definition of how numbers relate to each other within a certain space. In this number space, you can multiply numbers together quite easily. But if you try to do the opposite—divide—you quickly realize it’s much, much harder and actually impossible. The curve is special because it defines the rules of multiplication and division in this space.

Now, let’s talk about multiplication in this space. Imagine you have a number that everyone knows—let’s call it the base point. This base point is a common factor, something everyone starts with. Then, you have your own special number, something you keep secret. When you multiply your secret number by the base point, you get a new number. This new number is unique to you because it’s the product of your secret number and the base point.

What’s fascinating about this space is that while multiplying your secret number by the base point is easy, trying to work backwards from the product to figure out the secret number is practically impossible. This isn’t just a case of challenging arithmetic—it’s a fundamental property of the curve. The rules of this number space make division so complex that it’s infeasible to do it, especially when the numbers involved are large.

In this world of numbers and curves, two key concepts emerge from this process of multiplication: one is something everyone can see, and the other is something only you know. The number that everyone can see, the product of your secret number and the base point, is like an identifier—let’s call it the public key. It’s what you share with the world, something that anyone can know and use.

But behind this public key is something more personal, something hidden: your secret number. This is your private key, and it’s the foundation of everything that happens in this number space. The reason it’s called the private key is simple: even though everyone knows the public key and the base point, they cannot figure out the private key because division, in this space, is practically impossible.

So, what is a wallet in this context? A wallet is like a trusted tool that handles these numbers for you. It performs the necessary multiplications, using your private key and the base point to create your public key, without ever revealing your private key. The wallet ensures that while you interact with the number space, your secret remains safe.

A wallet simply performs these multiplications while keeping your private key hidden. In the context of blockchains, this multiplication takes on a crucial role. When you want to make a transaction, that transaction is first converted into a number within the same number space defined by the elliptic curve. Your wallet then multiplies this transaction number by your private key, producing a new product—a unique signature for that transaction. This signature is something only you can compute because only you know your private key.

When you create a signature by multiplying your private key with a transaction (which has been converted into a number), you generate a unique result that only you could produce because only you have the private key. However, anyone who knows your public key can verify that this result is correct. Here’s how this works in simpler terms:

Think of your public key as a special tool that lets others check your work without knowing how you did it (your private key). When you multiply your private key with the transaction, the result is like a coded message. Now, anyone with your public key can perform a different operation on the transaction—let’s call it a simplified multiplication or a ”check” operation—that helps them compare it with your coded message.

The public key allows them to mimic the multiplication process in such a way that if the transaction was really created with your private key, the numbers will match up in a way that proves it’s legitimate. It’s like solving a puzzle: the public key is a piece of the puzzle that lets anyone see if the final picture fits, but without revealing the secret steps you took to solve it (your private key). So, while they can verify that the multiplication was done correctly using your public key and the transaction, they can’t reverse-engineer or ”divide” the result to discover your private key.