Galois Fields in AES: A Deeper Dive

By Ioannis Alexander Konstas

Advanced Encryption Standard (AES) and Galois Field Arithmetic: A Deep Dive into GF(2⁸)

Abstract

The Advanced Encryption Standard (AES), a widely adopted symmetric block cipher, is grounded in rigorous mathematical structures that ensure its security and efficiency. Central to AES is the finite field GF(2⁸), a Galois Field containing 256 unique elements. Each element is represented as an 8-bit polynomial, enabling byte-level alignment with modern digital systems. This report explores the foundations of GF(2⁸), the role of irreducible polynomials in field construction, and the method for computing multiplicative inverses using the Extended Euclidean Algorithm.

1. Introduction to GF(2⁸)

AES operates over the finite field GF(2⁸), which consists of exactly 2⁸ = 256 elements. These elements are not arbitrary; each one is expressed as a polynomial of degree at most 7 with binary coefficients (0 or 1), reflecting its compatibility with the 8-bit architecture of contemporary digital systems.

1.1 Polynomial Representation

Each element in GF(2⁸) is represented as:

  a₇x⁷ + a₆x⁶ + a₅x⁵ + … + a₁x + a₀,  where each aᵢ ∈ {0, 1}

Thus, an 8-bit binary number like 11001010 corresponds to the polynomial:

  x⁷ + x⁶ + x³ + x¹

This representation allows AES to exploit binary logic operations natively supported by digital hardware.

2. Why AES Uses GF(2⁸)

AES’s reliance on GF(2⁸) is not arbitrary—it is designed to align with both computational and cryptographic requirements.

2.1 Byte-Level Alignment

• Every 8-bit byte naturally maps to one of the 256 elements in GF(2⁸).
• This allows direct mapping between binary data and polynomial arithmetic, facilitating efficient processing.

2.2 Closed Arithmetic Operations

• GF(2⁸) is a closed algebraic system. All basic arithmetic operations—addition, subtraction, multiplication, and division—stay within the field.
• This ensures consistency and predictability, which are vital in cryptographic transformations.

3. Constructing GF(2⁸) Using an Irreducible Polynomial

To create the field GF(2⁸), we define it over GF(2), using an irreducible polynomial of degree 8 as a modulus. The polynomial chosen by AES is:

  P(x) = x⁸ + x⁴ + x³ + x + 1

This polynomial satisfies two critical conditions:

1. It is irreducible over GF(2), meaning it cannot be factored into lower-degree polynomials with coefficients in GF(2).
2. It ensures that any product of polynomials, when reduced modulo P(x), remains within GF(2⁸).

3.1 Modular Reduction

When two elements of GF(2⁸) are multiplied, their product may exceed degree 7. To keep results within the field, the product is reduced modulo P(x), similar to modular arithmetic with integers:

  If A(x)·B(x) = Q(x)·P(x) + R(x), then the product in GF(2⁸) is R(x).

This modular reduction ensures the result fits within the 8-bit constraint.

4. Inversion in GF(2⁸) via the Extended Euclidean Algorithm

One of the most essential operations in AES, particularly during the SubBytes and MixColumns transformations, is computing the multiplicative inverse of a non-zero element in GF(2⁸).

4.1 Objective

Given a non-zero element A(x) in GF(2⁸), the goal is to find A⁻¹(x) such that:

  A⁻¹(x) × A(x) ≡ 1 mod P(x)

This is solved using the Extended Euclidean Algorithm (EEA) for polynomials.

5. Extended Euclidean Algorithm for Polynomial Inversion

The EEA computes polynomials U(x) and V(x) satisfying:

  U(x)·A(x) + V(x)·P(x) = gcd(A(x), P(x))

Since P(x) is irreducible and A(x) ≠ 0, the greatest common divisor is:

  gcd(A(x), P(x)) = 1

Hence:

  U(x)·A(x) ≡ 1 mod P(x)

Which implies: U(x) = A⁻¹(x)

6. Worked Example: Inversion in GF(2³)

To illustrate the process, consider the smaller field GF(2³), constructed using the irreducible polynomial:

  P(x) = x³ + x + 1

Let us find the inverse of A(x) = x² + 1

6.1 Initialization

• r₀ = P(x) = x³ + x + 1
• r₁ = A(x) = x² + 1
• s₀ = 1, s₁ = 0
• t₀ = 0, t₁ = 1

6.2 First Division

Divide r₀ by r₁:

• Quotient = x
• Remainder = x + 1

6.3 Update

• r₂ = x + 1
• s₂ = s₀ − x·s₁ = 1 − x·0 = 1
• t₂ = t₀ − x·t₁ = 0 − x·1 = x  (note: −x ≡ x in GF(2))

6.4 Second Division

Divide r₁ by r₂:

• Quotient = x + 1
• Remainder = 0 ⇒ algorithm terminates

6.5 Final Result

• The last non-zero remainder is 1 ⇒ inverse exists
• t₂ = x is the multiplicative inverse of x² + 1 in GF(2³)

7. Conclusion

AES’s use of GF(2⁸) underpins its ability to process data efficiently and securely. By mapping 8-bit bytes to polynomials, employing irreducible polynomials to define finite fields, and using robust algorithms like the Extended Euclidean Algorithm, AES ensures:

• Strong mathematical closure
• Efficient invertibility for encryption/decryption cycles
• Seamless hardware/software implementation

Understanding the arithmetic of Galois Fields, particularly GF(2⁸), provides deeper insight into the elegant and resilient design of AES, reaffirming its status as a cornerstone of modern cryptographic standards.