By Lina Hope 
A collaborative team of researchers from Sichuan University, the Huazhong University of Science and Technology, and the Institute of Software at the Chinese Academy of Sciences has released a new study detailing an automated, dynamic programming approach to optimizing polynomial multiplication over binary finite fields ($mathbbF_2[x]$). The paper, designated as Paper 2026/2371 on the Cryptology ePrint Archive, addresses a fundamental bottleneck in both classical and post-quantum cryptographic software by replacing tedious hand-tuning methods with a systematic, cross-platform optimization framework.
Authored by Junyu Zhou, Xiao Lan, Jing Wang, Hao Ren, Weiran Liu, and Si Gao, the research addresses the growing complexity of deploying cryptographic libraries across diverse hardware architectures. As open-source security libraries such as OpenSSL and liboqs increasingly rely on ARM and RISC-V platforms alongside traditional x86-64 processors, developers face the daunting task of retuning low-level arithmetic kernels for vastly different instruction set architectures, commonly known as ISAs.
At the heart of modern cryptographic software lies the need for efficient multiplication over $mathbbF_2[x]$, often referred to as carry-less multiplication. High-performance arithmetic libraries typically achieve speed by recursively applying decomposition rules pioneered by mathematicians like Karatsuba and Toom. These algorithms function by splitting large operands into smaller subproblems, performing fundamental operations, and then recombining the resulting products.
The sequence of these recursive choices—stretching from the top-level multiplication down to the foundational base kernels—forms what the researchers call a decomposition schedule. Finding the optimal schedule is critical for maximizing performance, particularly as operand sizes scale up. However, the research notes that hand-tuned schedules frequently fail to identify optimal decompositions for larger input sizes. Furthermore, the distinct performance profiles and primitive costs associated with different ISAs make cross-platform retuning an arduous and time-consuming manual endeavor.
To resolve these challenges, the research team formulated recursive $mathbbF_2[x]$ multiplication as a dedicated schedule-selection problem. The core breakthrough of their work relies on the observation that the recursive rule space possesses optimal substructure. This mathematical property enables the application of dynamic programming—a computer science optimization technique—over a compact ISA profile.
This ISA profile is designed to accurately account for a variety of low-level hardware factors, including base multiplication costs, exclusive-or (XOR) operations, memory traffic, and the overhead introduced by recursion. By feeding these hardware-specific metrics into the dynamic programming algorithm, the system can automatically generate optimal decomposition schedules. These resulting schedules are then materialized directly as dispatch tables or fixed-size kernels, ready for integration into cryptographic software.
The researchers put their framework to the test across a broad spectrum of hardware platforms, encompassing x86-64, ARM64, RISC-V64, and the microcontroller-class RP2040. Their empirical evaluations demonstrated that the dynamic programming costs closely track actual runtime trends across varying input sizes. When benchmarked against existing solutions, the full rule set generated by their framework achieved geometric mean speedups ranging from $1.04$ to $1.18$ times over gf2x, a widely used software library for arithmetic over $mathbbF_2[x]$.
Beyond isolated arithmetic benchmarks, the team evaluated the practical impact of their dynamic programming framework by integrating it into several prominent post-quantum cryptographic schemes, namely HQC, BIKE, and Classic McEliece. In integrations involving HQC and BIKE across three distinct 64-bit platforms, the optimized schedule selection yielded significant end-to-end performance gains. HQC implementations saw platform-dependent speedups ranging from $1.06$ to $1.21$ times, while BIKE integrations achieved even more pronounced improvements, ranging from $1.01$ to $1.70$ times faster than existing platform baselines.
A notable practical advantage of the newly proposed method is its efficiency during the development and compilation phase. With pre-prepared ISA profiles, the schedule selection process for HQC and BIKE requires a mere $0.075$ to $0.762$ seconds. This rapid evaluation phase successfully eliminates the need for exhaustive candidate compilation and empirical benchmarking during the search process, which has traditionally been a major bottleneck in performance engineering.
Ultimately, the research provides a systematic, highly efficient alternative to traditional hand-tuning and measurement-driven empirical searches that rely on testing every conceivable input size. By streamlining the process of porting and retuning carry-less multiplication kernels, the dynamic programming framework developed by Zhou and his colleagues offers a scalable path forward for optimizing cryptographic software in an increasingly diverse hardware landscape.
Sagoh 
Neng Nana