Introduction To Elliptic Curves

Core idea: ECC turns points on a curve into an arithmetic system. Repeating its point operation is easy; reversing a large repetition is hard.

Progression

$$\text{real curve} \longrightarrow \text{point addition} \longrightarrow \text{scalar multiplication} \longrightarrow \text{finite-field point group} \longrightarrow \text{cryptographic subgroup}$$

Remember

• Real curves provide geometric intuition.
• Finite fields make the coordinates finite, discrete, and exact.
• Cryptographic systems publicly select a base point and work with its subgroup.
• This collection builds the mathematics; later collections build keys, signatures, and protocols from it.