Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC requires smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security.[1]
Elliptic curves are applicable for key agreement, digital signatures, pseudo-random generators and other tasks. Indirectly, they can be used for encryption by combining the key agreement with a symmetric encryption scheme. They are also used in several integer factorization algorithms based on elliptic curves that have applications in cryptography, such as Lenstra elliptic-curve factorization.
Theory
For current cryptographic purposes, an elliptic curve is a plane curve over a finite field (rather than the real numbers) which consists of the points satisfying the equation
{\displaystyle y^{2}=x^{3}+ax+b,\,}
along with a distinguished point at infinity, denoted ∞. (The coordinates here are to be chosen from a fixed finite field of characteristic not equal to 2 or 3, or the curve equation will be somewhat more complicated.)
This set together with the group operation of elliptic curves is an abelian group, with the point at infinity as an identity element. The structure of the group is inherited from the divisor group of the underlying algebraic variety.
{\displaystyle \mathrm {Div} ^{0}(E)\to \mathrm {Pic} ^{0}(E)\simeq E,\,}
Cryptographic schemes
Several discrete logarithm-based protocols have been adapted to elliptic curves, replacing the group
{\displaystyle (\mathbb {Z} _{p})^{\times }}
with an elliptic curve:
At the RSA Conference 2005, the National Security Agency (NSA) announced Suite B which exclusively uses ECC for digital signature generation and key exchange. The suite is intended to protect both classified and unclassified national security systems and information.[8]
Recently, a large number of cryptographic primitives based on bilinear mappings on various elliptic curve groups, such as the Weil and Tate pairings, have been introduced. Schemes based on these primitives provide efficient identity-based encryption as well as pairing-based signatures, signcryption, key agreement, and proxy re-encryption.
Reference :
https://en.wikipedia.org/wiki/Elliptic-curve_cryptography
Reading :
An Introduction to the Theory of Elliptic Curves Joseph H. Silverman Brown University and NTRU Cryptosystems, Inc.
https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf
Craig Costello A gentle introduction to elliptic curve cryptography. Summer School on Real-World Crypto and Privacy
https://summerschool-croatia.cs.ru.nl/2017/slides/A%20gentle%20introduction%20to%20elliptic%20curve%20cryptography.pdf
https://github.com/bitcoinbook/bitcoinbook/blob/develop/ch04.asciidoc
Guide to Elliptic Curve Cryptography BOOK
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.394.3037&rep=rep1&type=pdf
You tube Reference :
https://www.youtube.com/results?search_query=Elliptic+Curve+Cryptography
Elliptic Curve Cryptography Overview
https://www.youtube.com/watch?v=dCvB-mhkT0w
Elliptic Curve Diffie Hellman
https://www.youtube.com/watch?v=F3zzNa42-tQ
Elliptic Curve Point Addition
https://www.youtube.com/watch?v=XmygBPb7DPM
Elliptic curves
Explore the history of counting points on elliptic curves, from ancient Greece to present day. Inaugural lecture of Professor Toby Gee.
https://www.youtube.com/watch?v=6eZQu120A80
Martijn Grooten – Elliptic Curve Cryptography for those who are afraid of maths
https://www.youtube.com/watch?v=yBr3Q6xiTw4&t=119s
Elliptic Curve Cryptography, A very brief and superficial introduction
https://www.youtube.com/watch?v=oPJrWYmqGRs
ECC intro Broer
Rizal Broer Bahaweres 