Elliptic Curve Cryptography, or ECC, is public-key cryptography that uses the properties of an elliptic curve over a finite field for encryption. ECC requires smaller keys than non-ECC cryptography to provide equivalent security. For example, a 256-bit ECC public key provides comparable security to a 3072-bit RSA public key. Let’s try to understand how Elliptic Curve Cryptography works.

What is an Elliptic Curve?

An elliptic curve is a set of points described by the equation :

y² = x³ + ax + b

We can define a group G, where elements are points on the elliptic curve, and apply that to generate a public-private key pair for encryption.

How does Elliptic Curve Cryptography work?

If d is a random integer chosen from {1, 2, …, n}, where n is the order of a subgroup (number of elements in the subgroup) and G is the base point (beginning and ending point) of the subgroup, then we can always apply scalar multiplication and find H, which is another element of the subgroup, such that

H = dG

The random integer d can be used as a private key and H as a public key.

Does this look confusing? Let’s understand what the above statement actually means.

A group in Number Theory is a set with the following properties :

• If a and b are any two elements of the group and + is a binary operation, then (a + b) is also a group member.
• If a, b and c are any three elements of the group, then (a + b) + c = a + (b + c)
• For any element a of the group, a + 0 = a
  0 is called the identity element of the group.
• For any element a in the group, there will always be another element b in the group, such that
  a + b = 0

We can define such a group G, such that elements of the group are points on the elliptic curve.

If P, Q, and R are three points on the elliptic curve, then

P + Q + R = 0

This means if we join any two points P and Q on the curve with a straight line, the straight line …