A polynomial field, also called a finite field or Galois field, is a field constructed by taking polynomials over a smaller field and reducing them modulo an irreducible polynomial. It is used to create fields with more elements than the base field.
What is a Polynomial Field?
A polynomial field, also called a finite field or Galois field, is a field constructed by taking polynomials over a smaller field and reducing them modulo an irreducible polynomial. It is used to create fields with more elements than the base field.
Key Properties of a Polynomial Field
- Closure Under Operations: The field is closed under addition, subtraction, multiplication, and division (except by zero).
- Finite Number of Elements: The number of elements is \( q^m \), where \( q \) is the size of the base field and \( m \) is the degree of the irreducible polynomial.
- Irreducibility: The irreducible polynomial \( p(x) \) ensures that the field elements are unique and every nonzero element has a multiplicative inverse.
- Basis Representation: Each element can be represented as a polynomial of degree less than \( m \) with coefficients from \( \mathbb{F}_q \).
How is a Polynomial Field Constructed?
- Base Field: Start with a base field \( \mathbb{F}_q \), where \( q \) is typically a prime power \( p^k \).
- Polynomial Ring: Construct the set of all polynomials with coefficients in \( \mathbb{F}_q \), denoted \( \mathbb{F}_q[x] \).
- Irreducible Polynomial: Choose an irreducible polynomial \( p(x) \) of degree \( m \) from \( \mathbb{F}_q[x] \). The irreducibility ensures the resulting structure is a field.
- Modulo Reduction: Define the field \( \mathbb{F}_{q^m} \) as the set of polynomials of degree less than \( m \) with coefficients in \( \mathbb{F}_q \), where all arithmetic is performed modulo \( p(x) \).
Example: Polynomial Field Construction
Base Field: \( \mathbb{F}_2 = \{0, 1\} \)
Irreducible Polynomial: \( p(x) = x^2 + x + 1 \)
Field Elements:
\( \mathbb{F}_{2^2} = \{ 0, 1, x, 1 + x \} \),
where \( x \) satisfies the relation \( x^2 + x + 1 = 0 \) (modulo \( p(x) \)).
Arithmetic in the Field
- Addition: \[ (1 + x) + x = 1 + 2x = 1 \quad (\text{mod } 2). \]
- Multiplication: \[ (x + 1) \cdot x = x^2 + x \quad \text{reduce modulo } x^2 + x + 1: \] \[ x^2 + x \equiv 1 \quad (\text{since } x^2 + x + 1 = 0). \]
Why Do We Need Polynomial Fields?
Polynomial fields are widely used in areas like:
- Cryptography: Fields of size \( 2^m \) (binary fields) are used in algorithms like AES and elliptic curve cryptography.
- Error-Correcting Codes: Reed-Solomon and BCH codes use polynomial fields to detect and correct errors.
- Zero-Knowledge Proofs: Polynomial fields enable efficient evaluation of large arithmetic circuits.
- Finite Field Arithmetic: Useful for designing efficient arithmetic for hardware and software implementations.