🕓Galois Fields

Galois Fields (aka Finite Fields)

• A Galois field is finite set of elements and two operations $+$ (addition) and $\times$ (multiplication), with the following properties:

  • Closure: if $a$ and $b$ are in the set, $a + b$ and $a \times b$ are also in the set.

  • Additive identity: $a + 0 = a$

  • Multiplicative identity: $a \times 1 = a$

  • Additive inverse: $a + (-a) = 0$

  • Multiplicative inverse: $a \times a^-¹ = 1$

• Field size is the number of elements in the set.

Elliptic curves that are defined over a finite field with a prime field size have interesting properties and are key to building of elliptic curve cryptographic protocols.

Let's define a PrimeGaloisField class that contains the intrinsic property of a finite field, prime. We also define a membership rule for a value in a given finite field, by overriding the __contains__ method.

@dataclass
class PrimeGaloisField:
    prime: int

    def __contains__(self, field_element: "FieldElement") -> bool:
        # called whenever you do: <FieldElement> in <PrimeGaloisField>
        return 0 <= field_element.value < self.prime

Let's also define a FieldElement class to make sure all mathematical operations are contained within a given PrimeGaloisField.

@dataclass
class FieldElement:
    value: int
    field: PrimeGaloisField

    def __repr__(self):
        return "0x" + f"{self.value:x}".zfill(64)
        
    @property
    def P(self) -> int:
        return self.field.prime
    
    def __add__(self, other: "FieldElement") -> "FieldElement":
        return FieldElement(
            value=(self.value + other.value) % self.P,
            field=self.field
        )
    
    def __sub__(self, other: "FieldElement") -> "FieldElement":
        return FieldElement(
            value=(self.value - other.value) % self.P,
            field=self.field
        )

    def __rmul__(self, scalar: int) -> "FieldElement":
        return FieldElement(
            value=(self.value * scalar) % self.P,
            field=self.field
        )

    def __mul__(self, other: "FieldElement") -> "FieldElement":
        return FieldElement(
            value=(self.value * other.value) % self.P,
            field=self.field
        )
        
    def __pow__(self, exponent: int) -> "FieldElement":
        return FieldElement(
            value=pow(self.value, exponent, self.P),
            field=self.field
        )

    def __truediv__(self, other: "FieldElement") -> "FieldElement":
        other_inv = other ** -1
        return self * other_inv

All parameters in an elliptic curve equation are actually elements in a given prime Galois field. This includes a,b,x, andy.