Improved wire reduction sharpens integer multiplication time bound
Bipartite Carrier Compression and Machine-Certified Bounds in Schönhage Trilinear Multiplication Tensors
Computational ComplexityData Structures and Algorithms
Summary
Multiplying very large numbers quickly is a fundamental problem in computing. This paper improves a complex mathematical method related to how components are connected in a multiplication process, reducing the number of connections needed. The authors used exact computer verification to confirm their improvements without errors. This result slightly improves the known theoretical speed limits for multiplying large numbers under certain assumptions.
What this means in practice
- •For high-performance computing engineers: Optimize hardware and algorithms for very large integer multiplication by leveraging improved structural wire reductions in multiplication tensor designs.
- •For cryptography hardware designers: Implement more efficient multiplication circuits essential for cryptographic computations by applying verified wire reduction techniques.
Authors
Volkan Dağlı, Zerrin Dağlı, Dağhan Dağlı
Abstract
Under the conditional framework breaking the longstanding O(n log n) integer multiplication barrier (originating from OpenAI Preprint #109), the asymptotic time complexity exponent T(n) = O(n (log n)^{1-κ}) is governed by bounding the second moment E[X^2] < 1 of recursive branching processes over trilinear tensor carrier graphs. In this work, we present a constructive combinatorial optimization over the alternating carrier bipartite matching graphs of the Schönhage trilinear tensor decomposition at dimensions h=23 and h=25. Through an alternating augmenting path search, we uncover a non-trivial cluster-unlocking cascade that eliminates 185 carrier roles (95 roles in h=23 and 90 roles in h=25), reducing the global physical wire width by 377,890 wires from the reviewed baseline W = 177,284,805 down to W = 176,906,915. Evaluated over the exact 10^{-14} rational grid, this wire reduction certifies an improved second-moment coefficient AB = 5,164,059 / 125,000,000,000 and establishes the sharpened conditional exponent κ= 826,215,307 / 20,000,000,000,000 \approx 4.131076535 \times 10^{-5}, strictly surpassing the preceding record on this architecture (κ\approx 4.123864 \times 10^{-5}). All 47 polynomial constraints and 7 rational security margins are audited in exact Q-arithmetic with zero floating-point error, and the fundamental wire reduction identity is formally verified in the Lean 4 interactive theorem prover without unproven axioms.