Elliptic curve — cubic curve on the projective plane.
Y2 + A1 * X * Y + A3 = X3 + A2 * X2 + A4 * X + A6 — complete curve formula
Y2 = X3 + 7 — Bitcoin elliptic curve
Below is an extremely simplified description of the computational scheme.
0 (Zero) — a convention referring to an infinitely distant point.
If a blue line intersects a red line (elliptic curve) at two points, there is always a third point of intersection.
Let's assume that A+B+C=0 point at infinity.
Let's move term A to the right side of the equation A+B=-C.
This formula shows how the addition operation maps a pair of points on an elliptic curve to a point on the same curve.
Note: A+B=B+A=-C addition of dots commutes, just like ordinary addition of numbers.
The next step is to multiply the point by an integer.
A + A + ... + A = n*A |..... n .....| A + A = 2*A
Adding identical dots is equivalent to doubling. Geometrically, doubling a point corresponds to constructing a tangent line to a point on an elliptic curve.
Using modular arithmetic on an elliptic curve, we can transition to a cyclic group and implement the concept of a discrete logarithm.
Comparison of Diffie-Hellman Scheme on Discrete Logarithm and Elliptic Curve.
Warning!
In the following forms the Y-coord's notation has been changed.
Instead of plus and minus signs, used the evenness and oddness.
For the Y-coordinate you can use any number - even or odd.
The parity of the entered number will be used to select the coordinate value.
If the Y-coord is not specified, the positive (smallest) value will be set.