Elliptic curve is the core mathematical function in elliptic curve cryptography (ECC). By utilizing its properties over finite field, namely the discrete logarithm problem, ECC works in a way different from traditional public key system where the basis is built upon large prime numbers and factorization.
The equation for elliptic curve is in this form
y2 = x3 + a • x + b
which is known as Weierstrass equation. The plot with a = -1 and b = 1 is shown in the below graph.
As can be seen from the plot, the curve is symmetric over the X-axis. Taking another example with a=1 and b=1, and a prime field of GF(5) for the equation
y2 = x3 + a • x + b, there are 9 points on the curve, namely
and therefore the order or cardinality is 9.
Nspire Graph application can be utilized to dynamically visualize and explore the properties of this curve. With the help of two sliders controlling variable a and b, the shape of the curve can be manipulated in real time.