eli5: what do elliptical curves have to do with cryptography?
In: Mathematics
Many cryptographic schemes use arithmetic to conceal information. For instance, if you and a buddy share a secret number (say, 803) and you want to securely tell them your credit card pin (say, 3829) then you can just tell them 4632. They can subtract the shared number to retrieve the actual pin but no one else who sees the message really can.
Real cryptographic schemes are more sophisticated and work differently but the important thing is that arithmetic structure – like addition and multiplication – can be used to encode information.
The RSA cryptosystem is a widely used one and it relies on “modular multiplication” to encode things. This is an arithmetic operation which is uses a base reference number to switch things up a bit. This base number is the product of two prime numbers and the security of the RSA algorithm is due to the fact that it is hard to factor such numbers.
Elliptic curves are very important objects in math and number theory. This is because if you have two points on an elliptic curve then you can create a third point in a natural way. That is, elliptic curves have their own unique arithmetic on them. And this arithmetic is, in ways, fundamentally more complex than addition and multiplication so they see quite a bit of attention in math to unlock their secrets.
For cryptography, however, this gives more options for arithmetic to encode things and so we have things like Elliptic Curve RSA Cryptography. Now we might think that elliptic curves are just more secure than ordinary RSA, but what happens is that the sophistication of elliptic curve arithmetic is to encode things more effectively using less space. If you would need a 512-bit key in regular RSA cryptography, then you could get the same security with something like a 64-bit key in elliptic curve cryptography. This results in speed and space saving security measures.
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Elliptic curves have been used in other fundamentally different ways. There was a proposed post-quantum algorithm, a cryptosystem that would remain secure even under attacks from quantum computers, which used exotic features of large collections of very special elliptic curves laid out in a net to encode things as paths between different curves. It was a pretty cool algorithm but, unfortunately, a classical (non-quantum) algorithm was found that cracks it. But, ultimately, cryptographers and number theorists and modern geometers are all interested in elliptic curves because of their elevated arithmetic sophistication.
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