Group Theory

A group is a set of elements Z={a,b,c,...}Z = \{a, b, c, ...\} and one binary operator ++ that satisfies the following axioms:

  • Closure: for anya,bZa,b ∈ Z, the element a+ba + b is in ZZ .

  • Associativity: for any a,b,cZa,b,c ∈ Z, (a+b)+c=a+(b+c)(a + b) + c = a + (b + c).

  • Identity: a+I=aa + I = a , for all aZa ∈ Z.

  • Invertibility: a+(a)=Ia + (-a) = I, for all aZa ∈ Z.

In addition to the above properties, if a group exhibits the commutative property of a+b=b+aa + b = b + a, it is called an abelian group.

The elliptic curve used in Bitcoin is actually a mathematical group, that is finite, cyclic, abelian, and has a single-generator point, defined over the binary addition operator. These properties form the bedrock for an efficient signature and verification mechanism in Bitcoin.

A single-generator group contains an element GZG ∈ Z, called the generator point, such that repeated additions of GG with itself can generate every element in ZZ.

In fact, in prime order elliptic curves, every point is a generator point.

Z={G,2G,3G,4G,...}Z = \{G, 2G, 3G, 4G, ...\}

Additionally, our group is cyclic, which means it has an order nn, such that nG=InG = I.

Let us now represent the generator point GGin Python, used in Bitcoin.

# Generator point of the abelian group used in Bitcoin
G = Point(
x=0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,
y=0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8,
curve=secp256k1
)
# Order of the group generated by G, such that nG = I
N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141

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