An Exact 56-Addition, Rank-23 Scheme for General 3*3 Matrix Multiplication

Abstract: We present a rank-$23$ algorithm for general $3\times3$ matrix multiplication that uses $56$ additions/subtractions and $23$ multiplications, for a total of $79$ scalar operations in the standard bilinear straight-line model. This improves the recent sequence of $60$-, $59$-, and $58$-addition rank-$23$ schemes. The algorithm works over arbitrary associative, possibly noncommutative, coefficient rings. Its tensor coefficients are ternary, meaning that every coefficient lies in ${-1,0,1}$. Correctness is certified by the $729$ Brent equations over $\mathbb{Z}$, and the verifier also expands the straight-line program and performs additional finite-field and noncommutative implementation tests.

• The paper presents an explicit rank-23 bilinear algorithm for 3x3 matrix multiplication that reduces the additive complexity from 58 to 56.
• It leverages tensor automorphisms and ternary coefficients to optimize the addition schedule while ensuring noncommutative generality.
• The approach is rigorously verified via 729 Brent equations and offers practical computational benefits along with a template for further algorithmic improvements.

Exact 56-Addition, Rank-23 Scheme for $3\times3$ Matrix Multiplication

Background and Context

Matrix multiplication remains a central computational primitive in numerical linear algebra and numerous algorithmic applications. While Strassen’s algorithm first reduced the asymptotic complexity by minimizing bilinear multiplications for $2\times2$ matrices, analogous efforts for $3\times3$ matrices have focused on reducing the rank and the total scalar operation count, including both multiplications and additions/subtractions, in the bilinear straight-line model. Laderman’s 1976 algorithm established the currently smallest known rank ($23$) for multiplying two general $3\times3$ matrices over any associative, possibly noncommutative, ring, but with high additive complexity (98 additions) [Laderman 1976]. Successive improvements have optimized the additive cost for rank-23 schemes without increasing the count of bilinear multiplications, progressively tightening the gap between theoretical lower bounds and practical schemes [Smirnov 2013, Stapleton 2025, Martensson–Wagner–Stapleton 2025, Perminov 2025].

Main Contributions

This work introduces an explicit rank-23 bilinear algorithm for $3\times3$ matrix multiplication requiring only 56 additions/subtractions and 23 multiplications, for a total of 79 scalar operations in the standard model (2604.27645). This scheme surpasses the recent sequence of improvements, which previously reduced the additive count from 60 to 59 and then to 58, setting a new state-of-the-art for additive efficiency at fixed rank.

Notably, the construction satisfies several essential properties:

• Coefficients are ternary: All tensor coefficients are limited to $\{-1, 0, 1\}$, enhancing both interpretability and implementation efficiency.
• Noncommutative generality: The decomposition is valid over arbitrary associative (possibly noncommutative) rings, broadening its applicability in algebraic computation beyond commutative domains.
• Straight-line program: The algorithm is provided as a self-contained straight-line program, obviating reliance on external certificates and simplifying reproducibility and code generation.
• Rigorous verification: The correctness of the construction is attested by certifying all 729 Brent equations over $\mathbb{Z}$, with supplementary verification over finite fields and noncommutative test rings.

Technical Description

The algorithm leverages automorphisms of the matrix multiplication tensor and schedule optimization. Starting with a previously verified rank-23 factorization of 58 additions [Perminov 2025], the authors apply the cyclic automorphism $(U, V, W) \mapsto (V, W^T, U^T)$—which preserves the tensor but permutes the algebraic roles of the factors. This reorientation admits further reduction in additions through output and input schedule optimization.

The scalar workflow decomposes as follows:

1. Linear Combination Staging: Constructs 13 left- and 13 right-side intermediates from $A$ and $2\times2$0 entries with 13 additions each.
2. Bilinear Multiplication: Computes 23 products $2\times2$1 from these intermediates — all multiplications use only ternary coefficients.
3. Output Assembly: Produces the 9 $2\times2$2 outputs via an optimized combination of 7 shared additive intermediates and a final 23 additions, realizing minimal reuse and cancellation.

Throughout, input and output schedule design is driven by vector-chain discovery and shared intermediate exploitation, critical for compressing additive complexity.

Numerical Results and Comparisons

The presented scheme achieves the lowest reported additive complexity (56 additions) among all known rank-23 $2\times2$3 multiplication algorithms, as summarized here:

Scheme	Rank	Additions	Coefficient Type	Verification
Laderman (1976) [Laderman 1976]	23	98	Integer	Noncommutative Alg.
Smirnov (2013) [Smirnov 2013]	23	84*	Integer	Tensor/Unoptimized
Stapleton (2025) [Stapleton 2025]	23	60	Ternary	Script/Printed
Mårtennson–Wagner–Stapleton (2025)	23	59	Ternary	Factor File
Perminov (2025) [Perminov 2025]	23	58	Ternary	Computed
This Paper (2604.27645)	23	56	Ternary	Full Certificate

* denotes additive cost prior to schedule optimization.

Implications and Prospects

From a theoretical standpoint, this result confirms that meaningful additive complexity reductions for small fixed-rank matrix multiplication are still attainable through algebraic exploitation of tensor symmetries and schedule design, even as rank-23 appears to be a persistent lower bound for $2\times2$4 multiplication over noncommutative rings.

Practically, further minimizing the number of additions can yield observable computational benefits, especially in hardware or software settings where addition is not negligible compared to multiplication (e.g., embedded systems, non-floating point domains). The restriction to ternary coefficients further suggests that efficient implementations with low memory overhead and potential bitwise optimization are viable.

This methodology—tensor automorphism exploitation followed by schedule optimization—hints at analogous opportunities for improvement in other small-size matrix multiplication problems and possibly even for larger $2\times2$5 instances where additive cost can become a primary bottleneck.

Future Directions

An open challenge remains in proving whether the 56-addition count is globally optimal among all rank-23 schemes or whether further reductions are possible via alternate automorphisms, schedule designs, or symbolic optimization. There is also scope to extend automated schedule search tools and tensor symmetry methods to larger matrices and other bilinear operations to approach theoretical lower bounds more closely.

Moreover, exploring the real-world impact of such fine-grained algorithmic improvements on modern hardware architectures, particularly in contexts with high concurrency, in-place computation, or ring-based operations, constitutes a productive direction for future research.

Conclusion

This paper establishes a new additive complexity record for $2\times2$6 matrix multiplication at rank 23, achieving only 56 additions through novel use of tensor automorphism and optimization techniques (2604.27645). The approach reinforces the value of algebraic insights and explicit straight-line programs in pushing the boundaries of fast matrix multiplication. The result offers both immediate utility for highly optimized software/hardware routines and a methodological template for analogous improvements in related bilinear algorithmic settings.