ByteFF-Pol: Fast Binary Polynomial Multiplication

• The paper introduces ByteFF-Pol, a framework for fast multiplication of long binary polynomials using additive FFT algorithms and tower field constructions.
• It employs a novel polynomial basis and recursive butterfly method to reduce field multiplications compared to traditional ternary FFT approaches.
• Empirical benchmarks demonstrate 10–40% speedups on large-scale (2^28-bit) polynomial multiplications through efficient subfield and SIMD optimizations.

ByteFF-Pol is a framework and implementation methodology for fast multiplication of long binary polynomials over finite fields of characteristic two, with a primary focus on practical efficiency leveraging additive Fast Fourier Transform (FFT) algorithms, tower field constructions, and modern hardware-specific optimizations. ByteFF-Pol distinguishes itself from previous approaches by replacing the classic multiplicative FFTs (often based on roots of unity and ternary butterflies) with an additive, binary-structured FFT (specifically, the Lin-Chung-Huang (LCH) additive FFT), and by exploiting the subfield structure of field multipliers via a tower field representation. The benchmarked results demonstrate significant reductions in computation time—often 10–40%—for very large polynomials as compared to established state-of-the-art methods.

1. Algorithmic Basis: Additive FFT and the Novelpoly Basis

ByteFF-Pol builds upon the recent advancements in additive FFT algorithms for binary fields, most notably the LCH FFT. Unlike multiplicative FFTs, which depend on the existence of large cyclic subgroups and roots of unity, additive FFTs operate on vector spaces over $\mathbb{F}_2$ structured by bases such as the Cantor basis. This approach allows the design to fully exploit the binary (additive) structure of the field and the subspaces within it.

Construction

• Subspace Vanishing Polynomials: For a sequence of basis vectors $\{v_1, \ldots, v_n\}$, define $V_i$ as the span of the first $i$ basis vectors. The subspace vanishing polynomial is

$s_i(x) := \prod_{a \in V_i} (x - a).$

• Novelpoly Basis: Each coefficient in the binary polynomial is represented in the basis

$$X_k(x) := \prod_{i: b_i=1} s_i(x), \quad k = \sum b_i 2^i.$$

Thus, a polynomial $f(x)$ can be written as

$$f(x) = g_0 + g_1 X_1(x) + \cdots + g_{n-1} X_{n-1}(x).$$

• Recursive Butterfly: The FFT recursively splits the evaluation set via a binary butterfly, computing

$g(X) = p_0(X) + X_{2^k} \cdot p_1(X),$

then,

$$h_0(X) \gets p_0(X) + s_k(\alpha) p_1(X), \quad h_1(X) \gets h_0(X) + s_k(\beta_k) p_1(X),$$

where $s_k(\beta_k)=1$ in the Cantor basis, simplifying one multiplication per split.

• Multiplicative Complexity: This structure achieves $\frac{1}{2} n \log n$ field multiplications, as opposed to the $\frac{4}{3} n \log_3 n$ count of ternary FFTs used in the classic Schönhage algorithm.

2. Comparison with Multiplicative FFT Approaches

Traditional binary polynomial multiplication methods used multiplicative FFTs, which evaluate polynomials at roots of unity in suitable field extensions. However, in characteristic 2, the lack of rich multiplicative subgroup structure often necessitates complicated mixed-radix or ternary algorithms and leads to increased computation and algorithmic irregularity.

In contrast, ByteFF-Pol's additive FFT leverages:

• Regular, binary structure in both the butterfly computation and data layout.
• Simpler control logic and lower algorithmic overhead.
• Reduced field multiplication count, especially at the butterfly layer.

Benchmarks indicate that even a straightforward implementation of the additive FFT brings a 10–20% speedup over optimized multiplicative FFT software on the same hardware and field settings.

3. Tower Field Construction and SIMD Optimization

A major optimization within ByteFF-Pol is the use of a tower field construction. Tower fields are built as a sequence of extensions, such as:

Tower Level	Field	Defining Polynomial
1	$\mathbb{F}_4$	$\mathbb{F}_2[x_1]/(x_1^2 + x_1 + 1)$
2	$\mathbb{F}_{16}$	$\mathbb{F}_4[x_2]/(x_2^2 + x_2 + x_1)$
3	$\mathbb{F}_{256}$	$\mathbb{F}_{16}[x_3]/(x_3^2 + x_3 + x_2 x_1)$
...	...	...

Many multipliers encountered in the butterfly steps of the additive FFT are guaranteed to reside in small subfields (e.g., when multiplying a $\mathbb{F}_{128}$ element by a $\mathbb{F}_{32}$ element).

• SIMD Table Lookup: By precomputing multiplication tables for these small subfields, ByteFF-Pol exploits modern CPU SIMD instructions, such as VPSHUFB (shuffle byte lookup) in AVX2, for low-latency multiplication by subfield elements.
• Compatibility: The tower extension and Cantor basis are shown to be structurally compatible, ensuring that basis conversions and vanishing polynomials function identically regardless of the representation.
• Field Multiplication Reduction: Multiplication by subfield elements reduces to multiple, independent multiplications in the smaller field, which are highly amenable to vectorization.

4. Empirical Performance and Benchmarks

Measured on contemporary platforms such as Intel Haswell processors, ByteFF-Pol achieves significant speedups in multiplying long binary polynomials:

• For $2^{28}$-bit polynomials, simple additive FFT implementations exhibit a 10–20% improvement over multiplicative FFT (gf2x, Harvey et al.) implementations.
• With full tower field and SIMD-optimized implementations in $\mathbb{F}_{256}$, the improvement reaches nearly 40% for problems of this size.
• Performance profiling identifies the primary sources of computational savings in accelerated subfield multiplication and the avoidance of bottleneck conversions at the butterfly steps.

This empirically positions ByteFF-Pol as a state-of-the-art methodology for large-scale, high-throughput binary polynomial multiplication.

5. Applications in Coding Theory, Cryptography, and Symbolic Computation

The practical impact of ByteFF-Pol is significant across several fields that require efficient binary polynomial arithmetic:

• Cryptographic Factorization: Polynomial arithmetic is central to factorization methods such as the Block Wiedemann algorithm in the Number Field Sieve, relevant for RSA key attacks.
• Multivariate Cryptosystems: Efficient multiplication is critical in algorithms attacking multivariate cryptosystems (e.g., XL).
• Error Correcting Codes: Both encoding and decoding in many error-correcting codes (e.g., Reed-Solomon, LDPC codes) are highly dependent on fast arithmetic in $\mathbb{F}_2$.
• Block and Stream Ciphers: Protocols such as AES-GCM offload finite field multiplications of large polynomials into performance-critical code paths.
• Symbolic Algebra Systems: Faster backbone polynomial arithmetic directly accelerates GCD computations, factorization, and modular composition.

6. Summary of Core Formulas and Structural Features

ByteFF-Pol's optimizations are crystallized in the following mathematical relations:

Purpose	Formula/Operation
Vanishing polynomial	$s_i(x) := \prod_{a \in V_i} (x-a)$
Novelpoly expansion	$X_k(x) := \prod_{i: b_i=1} s_i(x)$, $k = \sum b_i 2^i$
Additive FFT step	$g(X) = p_0(X) + X_{2^k}p_1(X)$; $h_0(X) = p_0(X) + s_k(\alpha)p_1(X)$
Tower multiplication	Small subfield element via table lookup/register shuffle (VPSHUFB)

The additive FFT's $\frac{1}{2} n \log n$ complexity and its tower field/SIMD enablement give ByteFF-Pol its empirical edge.

7. Implications and Research Directions

ByteFF-Pol's empirical and structural improvements establish a new standard baseline for binary polynomial arithmetic in large-scale and high-performance systems. By combining the algorithmic regularity of additive FFTs with hardware-level optimization in tower fields and SIMD, it showcases how mathematical insights can be tightly coupled with computer architecture. The approach's applicability to cryptographic protocols, coding theory, and symbolic computation further underscores its broad impact.

A plausible implication is that future algorithmic research in finite field and polynomial arithmetic will increasingly focus on additive structures and subfield-aware optimizations, particularly as vectorized and hardware-parallel computation become the norm across platforms. ByteFF-Pol's design paradigm anticipates such trends and provides a concrete roadmap for both algorithm designers and systems implementers working at the intersection of algebra and high-performance computing.