Implementing Basic Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$, and Its Application for Computing the Hamming Distance of Linear Codes

Abstract: We present a new general method for performing basic arithmetic in the finite field~$\mathbb{F}_p$ for any prime $p>2$ by using traditional binary operations over~$\mathbb{F}_2$. Our new approach is efficient and competitive with current state-of-art methods. We apply our new arithmetic method to the computation of the minimum Hamming distance of random linear codes for the fields $\mathbb{F}_3$ and $\mathbb{F}_7$. Our new arithmetic method allows to apply new techniques such as the isometric addition that accelerate the computation of the Hamming distance. We have developed implementations in the C programming language for computing the Hamming distance that clearly outperform both state-of-art licensed software and open-source software such as \textsc{Magma} and \textsc{GAP}/\textsc{Guava} on single-core processors, multicore processors, and shared-memory multiprocessors.

• The paper presents a framework leveraging binary-sliced encoding to perform Fp arithmetic with high throughput on modern CPUs.
• It introduces optimized techniques, including isometric addition and efficient bit-sliced operations, to rapidly compute the Hamming distance in linear codes.
• Empirical benchmarks demonstrate speedups of up to 5x over conventional methods, emphasizing its impact on error correction and cryptographic applications.

Efficient Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$ and High-Performance Computation of Linear Code Hamming Distance

Overview

The paper "Implementing Basic Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$, and Its Application for Computing the Hamming Distance of Linear Codes" (2603.29942) presents a comprehensive framework for performing generic field arithmetic in $\mathbb{F}_p$ using low-level binary operations over $\mathbb{F}_2$. The methodology leverages highly efficient data encoding, bit slicing, and isometric approximations to realize high-throughput implementations on contemporary processor architectures. This approach is exploited to accelerate the computation of the minimum Hamming distance of linear codes, particularly via new C implementations of the Brouwer-Zimmermann algorithm for $\mathbb{F}_3$ and $\mathbb{F}_7$, where substantial performance improvements over mainstream algebra systems are demonstrated.

Sliced-Bit Arithmetic and Storage in Finite Fields

The central technical insight is representing elements of $\mathbb{F}_p$ for any prime $p > 2$ as binary vectors, followed by organizing computations across data slices. The "sliced-bit" method encodes each field element into its $\mathbb{F}_2$0-bit binary form (with $\mathbb{F}_2$1), reorders vectors of elements such that each "slice" (i.e., each bit position across all field elements) resides in contiguous memory, and then applies parallelized logical operations (AND, OR, XOR) across these slices using native machine instructions or SIMD hardware. This exploits the high throughput possible with vector operations on modern CPUs.

Addition in $\mathbb{F}_2$2 is realized as follows:

• For $\mathbb{F}_2$3 (Mersenne fields), additions proceed via a series of bitwise XORs for sum and ANDs for carry, with recursive (potentially up to $\mathbb{F}_2$4 iterations) carry propagation using cyclically shifted carry vectors until resolution. Special handling ensures the sum $\mathbb{F}_2$5 is mapped to zero, in accordance with field semantics.
• For arbitrary primes, carry handling employs non-cyclic right shifts and explicit addition of the overflow correction vector corresponding to $\mathbb{F}_2$6. The mapping generalizes to all odd prime fields.
• Explicit implementations for $\mathbb{F}_2$7 and $\mathbb{F}_2$8 are derived, with careful selection of binary encoding and logical instruction sequences to minimize resource usage, and optimized to exploit CPU cache and pipeline architecture.

These arithmetic methods are coupled with storage strategies: 32- and 64-bit packed vectors, enabling simultaneous processing of 32 or 64 field elements, thereby amortizing operation cost.

Application to Hamming Distance Computation in Linear Codes

Computation of the minimum Hamming distance in a $\mathbb{F}_2$9 linear code over $\mathbb{F}_p$0 is inherently intractable (NP-hard), but essential for error detection/correction capability quantification. The most performant approach for moderately sized codes is the Brouwer-Zimmermann algorithm, which enumerates nontrivial linear combinations of codeword rows, maintains dynamic upper and lower bounds, and exploits structural code symmetries.

This paper extends high-efficiency implementations from the binary case ($\mathbb{F}_p$1) to higher prime fields:

• Linear combinations and weight enumeration are performed using the presented sliced-bit arithmetic.
• Scalar multiplications in combination enumeration are realized via efficient cyclic shifts due to the encoding; for example, multiplication by 2 or 4 in $\mathbb{F}_p$2 corresponds to a right bit shift, making FSR-based multiplication computationally negligible.
• The isometric addition technique replaces exact field arithmetic with a logical operation that preserves Hamming weight statistics, enabling rapid weight computation while deferring full-resolution arithmetic calculations unless necessary (as shown in the adaptation for the final Hamming weight counting).
• Parallelization is built upon data and task decomposition, leveraging both multicore shared-memory architectures with OpenMP and native compiler flags for architecture-specific optimizations.

Empirical Results and Comparative Analyses

The implementations were benchmarked on a range of architectures (Apple M1, Intel i7-1355U, AMD EPYC 7F52), with systematic comparison to industry-standard software (Magma, GAP/Guava). Key findings:

• For pure addition operations in $\mathbb{F}_p$3 and $\mathbb{F}_p$4, the sliced-bit implementations consistently outperform byte-packed and word-packed baseline methods, due to superior cache use and greater instruction-level parallelism.
• For Hamming distance computations on large randomly generated testbeds, the best-performing implementations achieve speedups of $\mathbb{F}_p$5 to $\mathbb{F}_p$6 over Magma for $\mathbb{F}_p$7, and speedups of $\mathbb{F}_p$8 to $\mathbb{F}_p$9 for $\mathbb{F}_2$0 (with almost universal outperformance over both Magma and GAP/Guava as code sizes and computational load increase).
• The performance advantage increases with problem size and complexity, indicating the benefit of bit slicing becomes more pronounced as problem dimensionality grows.
• Parallel implementations demonstrate near-linear scaling on multicore architectures.
• Additional system-level optimizations (e.g., compiler auto-vectorization, early termination heuristics for weight computation in $\mathbb{F}_2$1, simultaneous addition/subtraction routines) contribute to observed performance gains.

The key numerical results are robust across CPU architectures and compiler backends, underscoring portability.

Implications and Prospects

The main technical implication is that arithmetic over arbitrary prime fields can be made almost as efficient as binary field arithmetic for applications where mass parallelism and vectorization are important, such as combinatorial search in coding theory, cryptography, or computational algebra. The proposed methods provide a template for future high-throughput implementations for other finite field-based primitives, such as polynomial evaluation, syndrome decoding, and possibly for cryptographic schemes based on non-binary codes (e.g., code-based post-quantum cryptosystems).

The sliced-bit paradigm aligns with current hardware evolution, especially with ever-widening vector instructions and increasing on-chip parallelism, thereby future-proofing such algorithmic strategies. The isometric approximation method also indicates how tailored approximations can deliver speedups in scenarios where only invariants (like weights) must be preserved, a principle that may be extensible to other combinatorial or algebraic enumeration algorithms.

On the theoretical front, the modular structure, encoding selection, and efficient shifting interplay may stimulate further research in the design of bit-sliced algorithms for other finite fields (including extension fields), or suggest similar techniques in high-performance symbolic algebra or computational group theory domains.

Conclusion

This work establishes that using binary-sliced encoding, tailored bitwise algorithms, and isometric addition approximations, arithmetic in $\mathbb{F}_2$2 for $\mathbb{F}_2$3 can be implemented at substantially higher throughput than traditional methods. These improvements yield practical acceleration of critical computations such as the minimum Hamming distance in linear codes, making intractable instances much more accessible to investigation. Given the centrality of field arithmetic in computational mathematics, error correction, and cryptography, the methods outlined are of significant and wide-ranging practical relevance, and may form the basis for future fast software or hardware libraries supporting prime field operations in large-scale scientific and engineering applications.