VEDIC MULTIPLIER IN PYTHON A SOFTWARE APPROACH TO REDUCED PARTIAL PRODUCTS AND DELAY

Abstract

Traditional multiplication algorithms impose significant computational overhead, particularly when operating on large integers within software environments. Although hardware implementations of Vedic multipliers have demonstrated substantial reductions in propagation delay and partial product complexity, their software-level applicability remains insufficiently explored. This investigation presents a Python-based implementation of the Urdhva Tiryagbhyam multiplication algorithm, conducting rigorous performance benchmarking against Python's native multiplication operator across operand sizes ranging from 8 to 2048 bits. Experimental results reveal that while Python's native implementation, leveraging Karatsuba decomposition and C-level optimization, achieves execution times between 93 and 25,092 times faster than the Vedic approach, the latter offers theoretical advantages in partial product organization and algorithmic clarity. Complexity analysis demonstrates that Vedic multiplication reduces addition operations by 18-1,720 operations for 16–128-bit operands compared to schoolbook methods, though both maintain O(n²) asymptotic complexity. The crossover at which Python transitions to sub-quadratic algorithms (70 decimal digits) establishes a critical threshold where Karatsuba's O(n^1.585) complexity surpasses quadratic approaches. This work contributes empirical evidence quantifying the software-hardware performance dichotomy in Vedic arithmetic, providing insights applicable to educational computing platforms, domain-specific acceleration opportunities, and algorithm design pedagogy. Findings suggest that while Vedic methods face substantial overhead barriers in general-purpose Python execution, their conceptual framework merits consideration for specialized numerical computing contexts and instructional environments where algorithmic transparency supersedes raw execution velocity.