# Reading a book on cryptology and it talks about Euclid's algorithm. I'm wondering if I should I read more about it? Other than finding the GCD, is it useful?

I'd say no, with the caveat that finding the GCD is *extremely* useful and that that application alone makes it worth understanding.

You can use it to find the GCD of numbers and do all sorts of fancy number theory, but you can also find the GCD of polynomials, and more generally Euclidean domains, which are basically "places where the Euclidean algorithm works". Proving that the Euclidean algorithm works is a common way (possibly the most common?) to prove that a ring is a unique factorization domain; this covers the rings of Gaussian integers and of Eisenstein integers, for example. I don't have time to give more details or background, but I guess these things are all on Wikipedia.

In an even more real-world application, you can crack badly generated RSA keys with it: https://sbseminar.wordpress.com/2012/02/16/the-recent-difficulties-with-rsa/

You can use it to find the GCD of numbers and do all sorts of fancy number theory, but you can also find the GCD of polynomials, and more generally Euclidean domains, which are basically "places where the Euclidean algorithm works". Proving that the Euclidean algorithm works is a common way (possibly the most common?) to prove that a ring is a unique factorization domain; this covers the rings of Gaussian integers and of Eisenstein integers, for example. I don't have time to give more details or background, but I guess these things are all on Wikipedia.

In an even more real-world application, you can crack badly generated RSA keys with it: https://sbseminar.wordpress.com/2012/02/16/the-recent-difficulties-with-rsa/