A group is a set of elements and one binary operator that satisfies the following axioms:
Closure: for any, the element is in .
Associativity: for any , .
Identity: , for all .
Invertibility: , for all .
In addition to the above properties, if a group exhibits the commutative property of , it is called an abelian group.
A single-generator group contains an element , called the generator point, such that repeated additions of with itself can generate every element in .
Additionally, our group is cyclic, which means it has an order , such that .
Let us now represent the generator point in Python, used in Bitcoin.
# Generator point of the abelian group used in BitcoinG = Point(x=0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,y=0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8,curve=secp256k1)# Order of the group generated by G, such that nG = IN = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141