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 any
    a,bโˆˆZa,b โˆˆ Z
    , the element
    a+ba + b
    is in
    ZZ
    .
  • Associativity: for any
    a,b,cโˆˆZa,b,c โˆˆ Z
    ,
    (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)
    .
  • Identity:
    a+I=aa + I = a
    , for all
    aโˆˆZa โˆˆ Z
    .
  • Invertibility:
    a+(โˆ’a)=Ia + (-a) = I
    , for all
    aโˆˆZa โˆˆ 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
GโˆˆZG โˆˆ 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.
โ€‹
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
GG
in Python, used in Bitcoin.
# Generator point of the abelian group used in Bitcoin
G = Point(
x=0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,
y=0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8,
curve=secp256k1
)
โ€‹
# Order of the group generated by G, such that nG = I
N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141

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