on the elliptic curve

So elliptic curves over finite fields has points that form a (abelian) group and rational points that form an abelian group. Why?

Supersingular elliptic curves have unusually large endomorphism rings of rank 4. Elliptic curves are supersingular iff its endomorphism algebra is a quaternion algebra.
(Here algebra means a vector space and a ring.)

What is the relation of the point at infinity to the chord and tangent method?
Point of infinity serves as the identity element How does it achieve that? The field makes no difference. file:///.file/id=6571367.13918117

Is the point at infinity a part of the elliptic curve? - it’s both

Regarding surjectiveness: Mathematics expressing the relationship of a set to its image under a mapping when every element of the image set has an inverse image in the first set.

Algebraiclly closure are unique - every field has a larger field and that field algebraically closes that field. You can be algebraically closed by yourself, no finite field is algebraically closed. An algebraically closed field has no irreducible polynomials of degree larger than 1.

Why is it suitable for cryptography? - the key is that it is large enough group that DLP is hard to solve.

Maybe you can define another operations on

Try brute force attack for ECDLP:
Compute 1g, 2g, 3g, 4g… until y is found. Take the integer bits as the ones to store in computer memory, O(2^logN) where N is the order of G.

Nondeterministic: can take random bits on the choice

Use Euclidean algorithm to find solution for Chinese remainder theorem - page 257 of textbook.

What is a trace? - plus or minus a quantity that they are calling the trace.2

1. A map from one finite field to another finite field.

Np - polynomial time algorithms that is not deterministic
If it’s in exponential time, it’s not in P or in NP.

NP problem: giving a candidate solution you can verify in polynomial time
Np-complete: you can reduce a problem to another one so you reduce difficulty.