Elliptic Curves

Question 1

What are Elliptic Curves?

Answer 1

Elliptic Curves follow a general equation, defined as:
y^2 = x^3 + ax + b
In cryptography, the equation changes to be:
y^2 = x^3 + ax + b mod p
Which also includes a neutral element (weird O)

Question 2

How do you ensure an elliptic Curve implements a Discrete Logarithm Problem?

Answer 2

The elliptic curve needs a cyclic group, where it is made up of elements within the group and a group operation. For elliptic curves, these elements are points on the curve.

Question 3

How does Point Addition work in Elliptic Curves?

Answer 3

P + Q = R
In summary:
Any line through two points will intersect a third point on the line. If you reflect that intersection in the X-axis, then you get your third point.

Question 4

What is Point Doubling in Elliptic Curves?

Answer 4

P + P = 2P
In summary:
The tangent at a point will intersect another point on the curve. If you reflect that intersection, then you get your point doubled value

Question 5

How is a cyclic group formed in an elliptic curve?

Answer 5

The points on an elliptic curve, including the neutral element, form cyclic subgroups. Under certain conditions, all points form a cyclic group.