[Paper Review] A Generalized Recursive Algorithm for Binary Multiplication based on Vedic Mathematics

Ajinkya Kale, Shaunak Vaidya+1 more|ArXiv.org|Oct 11, 2009

Cryptography and Residue Arithmetic2 references3 citations

TL;DR

This paper proposes a generalized recursive algorithm for binary multiplication using the Nikhilam Sutra from Vedic Mathematics, operating in a radix-2 system. By recursively applying the Nikhilam Sutra, the method reduces multiplication to simpler operations, achieving efficiency for smaller multiplicands and enabling hardware acceleration through a suggested block schematic for parallel implementation in multiprocessor environments.

ABSTRACT

A generalized algorithm for multiplication is proposed through recursive application of the Nikhilam Sutra from Vedic Mathematics, operating in radix - 2 number system environment suitable for digital platforms. Statistical analysis has been carried out based on the number of recursions profile as a function of the smaller multiplicand. The proposed algorithm is efficient for smaller multiplicands as well, unlike most of the asymptotically fast algorithms. Further, a basic block schematic of Hardware Implementation of our algorithm is suggested to exploit parallelism and speed up the implementation of the algorithm in a multiprocessor environment.

Motivation & Objective

• To develop a generalized recursive algorithm for binary multiplication that leverages principles from Vedic mathematics.
• To improve efficiency in multiplication operations for smaller multiplicands, where most asymptotically fast algorithms underperform.
• To enable hardware implementation by proposing a basic block schematic for parallel execution in multiprocessor systems.
• To analyze recursion profiles statistically based on the size of the smaller multiplicand.
• To demonstrate feasibility and performance advantages in digital and hardware contexts through theoretical and structural design.

Proposed method

• The algorithm applies the Nikhilam Sutra recursively to perform binary multiplication in a radix-2 number system.
• It transforms binary multiplication into a series of additions and subtractions using the Nikhilam principle, reducing computational complexity.
• The recursion depth is profiled based on the value of the smaller multiplicand, enabling statistical analysis of performance.
• A hardware block schematic is proposed to exploit parallelism and accelerate computation in multiprocessor environments.
• The method is designed to be scalable and efficient for small to moderate operand sizes, contrasting with traditional asymptotic algorithms.
• The algorithm operates entirely within the binary system, making it suitable for digital and embedded computing platforms.

Experimental results

Research questions

• RQ1 How can the Nikhilam Sutra from Vedic mathematics be generalized into a recursive binary multiplication algorithm?
• RQ2 What is the recursion behavior of the algorithm as a function of the smaller multiplicand's magnitude?
• RQ3 Can this recursive approach outperform conventional algorithms for small multiplicands in digital systems?
• RQ4 How can the algorithm be structured for efficient hardware implementation with parallelism?
• RQ5 What are the theoretical and practical performance characteristics of the algorithm in a radix-2 environment?

Key findings

• The proposed algorithm demonstrates superior efficiency for smaller multiplicands compared to most asymptotically fast algorithms.
• Statistical analysis of recursion profiles shows predictable and manageable depth based on the smaller multiplicand's value.
• The algorithm is well-suited for hardware implementation due to its regular recursive structure and modular operations.
• A basic block schematic is provided to guide parallel and multiprocessor implementations, enhancing speed and scalability.
• The method maintains correctness and efficiency in the binary number system, validating its applicability in digital platforms.
• The recursive application of the Nikhilam Sutra reduces complex multiplication to simpler arithmetic operations, improving computational efficiency.