Linear Algebra Over GF(2) for Binary Operations

This article explores how vector spaces defined over the Galois Field of two elements, \(\text{GF}(2)\), provide a rigorous algebraic framework for standard binary operations. By viewing bit strings as coordinate vectors and fundamental logic gates as algebraic field operations, abstract linear algebra formalizes concepts such as bitwise manipulations, matrix transformations, and error correction. This mathematical structure underpins modern digital systems, cryptography, and coding theory by turning low-level boolean algorithms into tractable matrix equations.

The Foundation: The Field \(\text{GF}(2)\)

At the core of binary mathematics is \(\text{GF}(2)\) (also denoted as \(\mathbb{F}_2\)), the smallest finite field. It consists of the set \(\{0, 1\}\) equipped with two basic operations: addition and multiplication performed modulo 2.

• Addition (\(+\)): Corresponds directly to the logical XOR (exclusive OR) operation (\(0+0=0\), \(0+1=1\), \(1+0=1\), \(1+1=0\)). Under this operation, every element is its own additive inverse (\(-x = x\)).
• Multiplication (\(\cdot\)): Corresponds directly to the logical AND operation (\(0\cdot0=0\), \(0\cdot1=0\), \(1\cdot0=0\), \(1\cdot1=1\)).

Because \(\text{GF}(2)\) satisfies all field axioms—associativity, commutativity, distributivity, and the existence of identities and inverses—it serves as a valid scalar field for linear algebra.

The Vector Space \(\text{GF}(2)^n\)

When binary data is grouped into \(n\)-bit words, bytes, or registers, these sequences can be modeled as vectors in the vector space \(V = \text{GF}(2)^n\). An \(n\)-bit register \((b_1, b_2, \dots, b_n)\) represents a vector where each component \(b_i \in \{0, 1\}\).

1. Vector Addition: Adding two vectors \(\mathbf{u}, \mathbf{v} \in \text{GF}(2)^n\) corresponds to the bitwise XOR operation: \[\mathbf{u} + \mathbf{v} = (u_1 \oplus v_1, u_2 \oplus v_2, \dots, u_n \oplus v_n)\]
2. Scalar Multiplication: Multiplying a vector \(\mathbf{v}\) by a scalar \(c \in \{0, 1\}\) evaluates to either the zero vector (if \(c = 0\)) or the vector \(\mathbf{v}\) itself (if \(c = 1\)).
3. Standard Basis: The canonical basis consists of unit vectors \(e_i\) containing a single 1 at index \(i\) and 0 elsewhere, matching standard bitmasking primitives.

Linear Transformations and Matrices

Operations on binary words that distribute over XOR are linear transformations over \(\text{GF}(2)^n\). These can be represented as binary matrices acting on binary vectors via matrix multiplication modulo 2.

• Matrix-Vector Multiplication: If \(A\) is an \(m \times n\) binary matrix and \(\mathbf{x} \in \text{GF}(2)^n\), the product \(\mathbf{y} = A\mathbf{x} \pmod 2\) produces an \(m\)-bit output where each bit is a linear combination (parity sum) of the input bits.
• Permutations and Shifts: Circular shifts, bit reversals, and register shuffling are expressed through permutation matrices.
• Linear Feedback Shift Registers (LFSRs): LFSR state updates, widely used in pseudo-random number generators and stream ciphers, are simply iterations of a companion matrix over \(\text{GF}(2)\).

Geometric Properties and the Hamming Metric

The vector space structure over \(\text{GF}(2)\) also introduces a geometric interpretation of binary words:

• Hamming Weight: The number of non-zero coordinates in a vector \(\mathbf{v}\), acting as the \(L_1\) norm in this space.
• Hamming Distance: The metric \(d(\mathbf{u}, \mathbf{v}) = \text{wt}(\mathbf{u} - \mathbf{v}) = \text{wt}(\mathbf{u} + \mathbf{v})\), measuring the number of bit positions in which two binary vectors differ.
• Hypercube Geometry: The space \(\text{GF}(2)^n\) corresponds to the vertices of an \(n\)-dimensional unit hypercube, where linear subspaces correspond to sub-cubes.

Practical Implications

Formalizing binary arithmetic as a vector space over \(\text{GF}(2)\) enables high-level mathematical tools to solve low-level computing problems:

• Linear Error-Correcting Codes: Codes such as Hamming, BCH, and Reed-Muller codes define valid codewords as subspaces of \(\text{GF}(2)^n\). Parity checks become null-space computations (\(H\mathbf{x}^T = \mathbf{0}\)).
• Cryptography: Linear cryptanalysis uses linear approximations over \(\text{GF}(2)\) to find weaknesses in block ciphers like DES and AES.
• Parallel Bit Manipulation: Matrix representations allow hardware and compilers to optimize complex bitwise logic using standard linear systems algorithms, such as Gaussian elimination over \(\text{GF}(2)\).