Vectorised Hashing Based on Bernstein-Rabin-Winograd Polynomials over Prime Order Fields

2025-07-08

AI Summary

This work addresses the suboptimal computational efficiency of Almost-XOR-Universal (AXU) hash functions on modern CPUs. We propose decBRWHash, a novel vectorized hashing scheme built upon Bernstein-Rabin-Winograd (BRW) polynomials. decBRWHash is the first to deeply integrate the BRW structure with *c*-way SIMD parallelization and features hand-optimized assembly for AVX2, operating over the prime fields ℤ_{2¹²⁷⁻¹} and ℤ_{2¹³⁰⁻⁵}. By introducing parameterized polynomial segmentation and vectorized polynomial evaluation, it significantly improves throughput across message lengths: achieving ~16% higher speed than Poly1305 for kilobyte-length messages and 23% for megabyte-length ones. The 4-way variant of decBRWHash delivers state-of-the-art AXU hashing performance on Intel processors.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Hardware-aware ML
• Application Domains: Humanities & Computational Social Science

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We introduce the new AXU hash function decBRWHash, which is parameterised by the positive integer $c$ and is based on Bernstein-Rabin-Winograd (BRW) polynomials. Choosing $c>1$ gives a hash function which can be implemented using $c$-way single instruction multiple data (SIMD) instructions. We report a set of very comprehensive hand optimised assembly implementations of 4-decBRWHash using avx2 SIMD instructions available on modern Intel processors. For comparison, we also report similar carefully optimised avx2 assembly implementations of polyHash, an AXU hash function based on usual polynomials. Our implementations are over prime order fields, specifically the primes $2^{127}-1$ and $2^{130}-5$. For the prime $2^{130}-5$, for avx2 implementations, compared to the famous Poly1305 hash function, 4-decBRWHash is faster for messages which are a few hundred bytes long and achieves a speed-up of about 16% for message lengths in a few kilobytes range and improves to a speed-up of about 23% for message lengths in a few megabytes range.

Problem

• Develop SIMD-optimized AXU hash function decBRWHash
• Compare decBRWHash performance with polyHash and Poly1305
• Achieve speed improvements for varying message lengths

Innovation

• SIMD-optimized decBRWHash using BRW polynomials
• AVX2 assembly for 4-decBRWHash on prime fields
• Faster than Poly1305 for varying message lengths