# 2023 Mar 05

Stuff on elliptic curves - haven’t really looked into it yet, but it seems that multiplication/division with an elliptic curve generator point corresponds to exponents/logarithm with a modulo group generator.

Fun visualization on desmos for elliptic curve addition. Elliptic curve addition: the sum of two points is where their slope intersects the curve, then reflected over the x-axis (negate y).

Okay, I’m learning math. A set is a collection of elements, say, integers.

A group is a way to describe structure (of a set), i.e. a group is a set and an operation, such that they have the following properties

- the operation performed on two elements of the set results in a third element of the set
- there exists an identity element
- every element has an inverse which means the operation performed on an element and it’s inverse results in the identity element
- the operation is associative (e.g. (A+B)+C = A+(B+C))

- The most basic group is the set of integers and addition.
- An abelian group is one where the operation is commutative (points on elliptic curves and their “addition” is an abelian group).
- Bonus: a space can be characterized by their
*fundamental group*, which is the set of “loops” (like a path that starts and stops in the same point I think) from a single point.

A field is a group with a multiplicative identity and inverses for each element. Importantly a field’s multiplication operation should distribute over it’s addition, i.e. A*(B+C)=A

*B+B*C.- Technically what we call addition and multiplication here don’t always have to correspond to “actual” addition and multiplication, but generally have the same properties so they’re often called so.
- The most basic field is the set of rational numbers with basic addition and multiplication.

Was suggested to see this paper on how discrete log exists in elliptic curves.

If you check the security info in your browser on an HTTPS connection, it’ll show it uses something like “TLS_ECDHE_RSA_WITH_AES_GCM_128_SHA256.” What does this mean? The root of trust is that you trust federated certificate authorities to verify clients’ and servers’ public keys. These public keys are used initially to sign messages. Then they use Elliptic Curve Diffie Hellman Ephemeral to establish a shared secret key, which is used for symmetric encryption (more practical/faster than public/asymmetric), AES, in GCM, which is just counter mode (full name Galois/Counter Mode - Galois because of the associated message authentication system that uses Galois (finite) field multiplication smth?). Still kind of unsure where SHA256 is used, because authentication and data integrity seems to be already provided with RSA signatures and the encryption scheme. Maybe for the PRF somewhere?