What the hell is an elliptic curve?

Summary

Have you ever been browsing the web and come across a term that made you go, "huh?" Well, if you're even slightly cryptographcally inclined, you might have stumbled upon the term elliptic curve and thought to yourself, "What the hell?" Don't worry, you're not alone in feeling a bit lost. An elliptic curve is a type of mathematical function that has some pretty cool properties. It's defined by a specific equation, usually in the form of y2=x3+ax+by^2 = x^3 + ax + by2=x3+ax+b, where aaa and bbb are constants. The graph of this equation looks like a smooth, looping curve, which is where the name "elliptic" comes from. These curves have a unique feature: they can be used to create a group of points that can be added together in a unique way to form new points. This is what makes them so useful in cryptography. (The graphs below are interactive, you can zoom, pan, etc.) SHORT WEIERSTRASS FORM:y² = x³ - x + 1f: R → R Elliptic curves are used in various cryptographic algorithms, including digital signatures, key exchange, and encryption. The most common use is in Elliptic Curve Cryptography, which is a form of public-key cryptography that uses the mathematics of elliptic curves to secure data. The fundamental algorithm behind ECC is based on the Elliptic Curve Discrete Logarithm Problem. This problem is hard to solve, which means that it's difficult for an attacker to break the encryption. In simple terms, if you have a point on the curve and you know how to add points together, it's easy to find the result. But if you only have the result and one of the points, it's really hard to figure out the other point. The ECDLP involves finding the integer kkk such that P=k⋅GP = k \cdot GP=k⋅G, where PPP is a point on the curve, GGG is a known point (the generator point), and kkk is the ephemeral key. The difficulty of this problem is what makes ECC secure. y² = x³ + 2x² + 3Public Key (Q)80, 10Generator Point (G)3, 6Decrypted<no data>Point (P), Elliptic curves are also more efficie...