What are the properties of a field?

/ / Published in Cybersecurity, EITC/IS/CCF Classical Cryptography Fundamentals, AES block cipher cryptosystem, Introduction to Galois Fields for the AES

In the context of the Advanced Encryption Standard (AES) block cipher cryptosystem, understanding the properties of a field, particularly a Galois Field (GF), is important. A Galois Field, also known as a finite field, is a field that contains a finite number of elements. The properties of such fields are foundational to many cryptographic algorithms, including AES.

A field in mathematics is an algebraic structure consisting of a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. These operations must satisfy several properties, which are divided into two main categories: the properties of the operations themselves and the properties of the field elements under these operations.

Properties of Operations in a Field

1. Closure: For any two elements a and b in the field, both a + b and a \cdot b are also in the field. This means that performing either operation on any elements of the field will always yield another element within the same field.

2. Associativity: For all elements a, b, and c in the field, the following are true:
– (a + b) + c = a + (b + c)
– (a \cdot b) \cdot c = a \cdot (b \cdot c)
Associativity ensures that the grouping of elements does not affect the result of the operations.

3. Commutativity: For all elements a and b in the field, the following are true:
– a + b = b + a
– a \cdot b = b \cdot a
Commutativity ensures that the order in which two elements are combined does not affect the result.

4. Identity Elements: There exist two distinct elements in the field, typically denoted as 0 and 1, such that for any element a in the field:
– a + 0 = a
– a \cdot 1 = a
These are known as the additive identity and the multiplicative identity, respectively.

5. Inverse Elements: For every element a in the field, there exist elements -a and a^{-1} in the field such that:
– a + (-a) = 0
– a \cdot a^{-1} = 1 (for a \neq 0)
These are known as the additive inverse and the multiplicative inverse, respectively.

6. Distributivity: For all elements a, b, and c in the field, the following is true:
– a \cdot (b + c) = (a \cdot b) + (a \cdot c)
This property links the two operations of addition and multiplication.

Properties of Field Elements in Galois Fields

When dealing with Galois Fields, particularly GF(2^8), which is used in AES, the field elements are represented as polynomials over a binary field GF(2). This means each element can be expressed as a polynomial with coefficients in \{0, 1\}.

1. Finite Number of Elements: A Galois Field GF(2^m) contains exactly 2^m elements. For AES, m = 8, so GF(2^8) contains 256 elements.

2. Polynomial Representation: Each element in GF(2^8) can be represented as a polynomial of degree less than 8 with coefficients in \{0, 1\}. For example, the element 5 in decimal can be represented as the binary number 00000101, which corresponds to the polynomial x^2 + 1.

3. Irreducible Polynomial: The field GF(2^8) is constructed using an irreducible polynomial of degree 8. An irreducible polynomial cannot be factored into the product of two lower-degree polynomials over GF(2). For AES, the irreducible polynomial used is typically x^8 + x^4 + x^3 + x + 1.

4. Addition and Multiplication: Addition in GF(2^8) is performed by adding the corresponding coefficients of the polynomials modulo 2 (which is equivalent to the XOR operation in binary). Multiplication is more complex and involves polynomial multiplication modulo the irreducible polynomial. For example, to multiply two elements a(x) and b(x) in GF(2^8), you compute the product a(x) \cdot b(x) and then take the remainder when divided by the irreducible polynomial.

5. Multiplicative Inverses: Every non-zero element in GF(2^8) has a unique multiplicative inverse. This means for any non-zero element a, there exists an element b such that a \cdot b = 1. Finding the multiplicative inverse is essential in cryptographic algorithms for operations like the AES MixColumns step.

Examples and Applications in AES

The AES algorithm makes extensive use of the properties of GF(2^8). Here are some specific examples of how these properties are applied:

1. SubBytes Step: In this step, each byte of the state matrix is replaced with its corresponding value from a substitution box (S-box). The S-box is constructed using the multiplicative inverse in GF(2^8), followed by an affine transformation. The use of the multiplicative inverse ensures non-linearity, which is important for the security of the cipher.

2. MixColumns Step: This step involves matrix multiplication over GF(2^8). Each column of the state matrix is treated as a polynomial and multiplied by a fixed polynomial modulo the irreducible polynomial. This operation ensures diffusion, spreading the influence of each input byte over multiple output bytes.

3. Key Expansion: The AES key schedule involves generating round keys from the initial key. This process includes operations in GF(2^8), such as the application of the S-box and polynomial multiplication.

Mathematical Foundation

The rigorous mathematical foundation of Galois Fields ensures that the cryptographic operations are both efficient and secure. The finite nature of these fields allows for precise and predictable behavior, which is essential for the deterministic nature of encryption and decryption processes. The properties of closure, associativity, commutativity, identity elements, inverse elements, and distributivity provide a robust framework for these operations.

Additionally, the polynomial representation and the use of an irreducible polynomial ensure that every element in the field can be uniquely represented and manipulated. This uniqueness is critical for the bijective nature of the S-box and the invertibility of the MixColumns transformation.

Understanding the properties of a field, particularly a Galois Field, is essential for grasping the inner workings of the AES block cipher cryptosystem. The mathematical rigor and structure provided by these properties ensure the security and efficiency of the cryptographic operations. By leveraging the finite nature of GF(2^8) and the algebraic properties of fields, AES achieves a high level of security against various cryptographic attacks.

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