> For the complete documentation index, see [llms.txt](https://ir0nstone.gitbook.io/notes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://ir0nstone.gitbook.io/notes/cryptography/overview/choice-of-primes/fermat-factorisation.md).

# Fermat Factorisation

If $$p$$ and $$q$$ are numerically close, we can use [Fermat Factorisation](https://en.wikipedia.org/wiki/Fermat's_factorization_method).

During Fermat Factorisation, we hope to find $$a$$ and $$b$$ such that

$$
a^2 - b^2 = N
$$

Because that then means we can factorise the left-hand expression into

$$
(a+b)(a-b)=N
$$

As thus we get the two factors of $$N$$ as $$(a+b)$$ and $$(a-b)$$.

The reason we use this when $$p$$ and $q$ are numerically close is because the closer they are to each other the closer they are to $$\sqrt{N}$$. If we say $$a=\sqrt{N}$$ rounded up to the nearest number, we can calculate $$b^2 = a^2-N$$ (as rearranged from before) until $$b$$ is a whole number, after which we've solved the equation.

An example of this attack can be found in [this writeup](https://www.notion.so/Factorize-b96056dc70f54cc7b42b32f8984cb7cf), which may make it a bit clearer.


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