Cloudflare (a web security company) has a primer on elliptic curve cryptography and its uses for privacy and security online. Quick version: RSA cryptography relies on the fact that multiplying integers is easy but factoring them is hard. Elliptic curve cryptography relies on the fact that there’s a group law for elliptic curves over the integers mod n, and applying that group law repeatedly (exponentiation) is easy but determining how many times that law was applied (taking the discrete logarithm) is hard.
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