Galois Fields

Galois Fields (aka Finite Fields)

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

    • Closure: if aa and bb are in the set, a+ba + b and aร—ba \times b are also in the set.

    • Additive identity: a+0=aa + 0 = aโ€‹

    • Multiplicative identity: aร—1=aa \times 1 = aโ€‹

    • Additive inverse: a+(โˆ’a)=0a + (-a) = 0

    • Multiplicative inverse: aร—aโˆ’ยน=1a \times a^-ยน = 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.

from dataclasses import dataclass
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@dataclass
class PrimeGaloisField:
prime: int
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def __contains__(self, field_value: "FieldElement") -> bool:
# called whenever you do: <FieldElement> in <PrimeGaloisField>
return 0 <= field_value.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
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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
)
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def __rmul__(self, scalar: int) -> "FieldValue":
return FieldElement(
value=(self.value * scalar) % self.P,
field=self.field
)
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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
)
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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.