Montgomery reduction

Okay kiddo, so let me tell you about Montgomery reduction. It's like counting money, but for big numbers. Imagine you have a lot of pennies, like a hundred or even a thousand! If you want to add or subtract these pennies, it's going to take you a long time to count them all.

Now, imagine you have a special machine that can help you count these pennies faster. This machine lets you turn your pennies into special tokens or chips that represent a certain amount of pennies, but you don't have to count them all individually.

Montgomery reduction is like that special machine. It helps you do math faster with big numbers. It's used a lot in computer science and cryptography, which is where people try to keep information safe from bad guys who might want to steal it or hack into it.

Here's how it works: say you have a big number, let's call it A, and you want to subtract another big number, let's call it B, from it. Normally, you'd have to count all the digits in A and B and do the subtraction one digit at a time. But with Montgomery reduction, you can turn those big numbers into special tokens that are easier to work with.

First, you pick a special number, let's call it R. R has to be bigger than A and B, and it has to be an odd number. Then you multiply A and B by R, and divide the result by another big number, let's call it N.

This gives you two new numbers, let's call them A' and B'. A' and B' are special tokens that represent the original numbers A and B. But they're easier to work with because they're smaller and they don't have as many digits.

Now, you can subtract A' and B' like normal, and you'll get a new number, let's call it C'. But C' is also a special token, and it's not the final answer yet. To get the actual answer, you have to do one more step.

You multiply C' by another special number, let's call it R^{-1}. R^{-1} is a number that's equivalent to dividing 1 by R, but it's also a special token. When you multiply C' by R^{-1}, you get a final answer, let's call it C.

C is the answer you were looking for, the result of subtracting B from A. But you didn't have to count all the digits one by one, because Montgomery reduction helped you turn those big numbers into special tokens that were easier to work with.

So there you have it, kiddo. Montgomery reduction is like counting money with a special machine, only instead of pennies, you're counting big numbers. It helps mathematicians and computer scientists do math faster and keep information safe from bad guys.