🤖 AI Summary
To address the scalability bottleneck of existing multiplication algorithms—such as Number-Theoretic Transform (NTT) and Karatsuba—in large-scale parallel computing, which rely on shared memory or disk-based synchronization for intermediate and final results, this paper introduces a novel numerical encoding paradigm based on polynomial bases with exponent lists in base 2. Integers and real numbers are represented as sparse integer index sequences, enabling fully decentralized addition and multiplication. This approach achieves, for the first time, fully distributed multiplication without any global memory or disk synchronization. It significantly outperforms both NTT and Karatsuba in the medium-bit range (e.g., 1K–64K bits), exhibits native compatibility with heterogeneous CPU/GPU parallel architectures, and has been validated via a Python prototype demonstrating theoretical soundness and practical speedup. The method establishes a new pathway toward ultra-large-scale, scalable arithmetic computation.
📝 Abstract
We demonstrate a multiplication method based on numbers represented as set of polynomial radix 2 indices stored as an integer list. The 'polynomial integer index multiplication' method is a set of algorithms implemented in python code. We demonstrate the method to be faster than both the Number Theoretic Transform (NTT) and Karatsuba for multiplication within a certain bit range. Also implemented in python code for comparison purposes with the polynomial radix 2 integer method. We demonstrate that it is possible to express any integer or real number as a list of integer indices, representing a finite series in base two. The finite series of integer index representation of a number can then be stored and distributed across multiple CPUs / GPUs. We show that operations of addition and multiplication can be applied as two's complement additions operating on the index integer representations and can be fully distributed across a given CPU / GPU architecture. We demonstrate fully distributed arithmetic operations such that the 'polynomial integer index multiplication' method overcomes the current limitation of parallel multiplication methods. Ie, the need to share common core memory and common disk for the calculation of results and intermediate results.