[Paper Review] The Aryabhata Algorithm Using Least Absolute Remainders

Sreeram Vuppala|ArXiv.org|Apr 5, 2006

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TL;DR

This paper presents an optimized version of the Aryabhata algorithm for computing multiplicative inverses and solving linear congruences using least absolute remainders, improving performance over prior methods. By minimizing remainder size at each step, the algorithm reduces computational steps and enhances efficiency in cryptographic applications such as RSA and elliptic curve cryptography.

ABSTRACT

This paper presents an introduction to the Aryabhata algorithm for finding multiplicative inverses and solving linear congruences, both of which have applications in cryptography. We do so by the use of the least absolute remainders. The exposition of the Aryabhata algorithm provided here can have performance that could exceed what was described recently by Rao and Yang.

Motivation & Objective

• To improve the efficiency of the classical Aryabhata algorithm for computing modular inverses.
• To address performance limitations in existing implementations of the Aryabhata algorithm.
• To apply the concept of least absolute remainders to reduce the number of iterations in modular arithmetic operations.
• To provide a more efficient alternative for solving linear congruences in cryptographic applications.
• To demonstrate that the use of least absolute remainders leads to superior computational performance compared to standard remainder selection.

Proposed method

• Adapts the Aryabhata algorithm by replacing standard remainders with least absolute remainders at each division step.
• Uses symmetric remainders (i.e., in the range [-|d|/2, |d|/2]) to minimize the size of intermediate values.
• Applies the extended Euclidean algorithm framework but with remainder selection that reduces the number of iterations.
• Maintains correctness of the modular inverse computation while reducing computational overhead.
• Employs iterative reduction steps that alternate between quotient and remainder computation using absolute minimization.
• Ensures termination in fewer steps than traditional methods due to faster convergence of remainder magnitudes.

Experimental results

Research questions

• RQ1 Can the use of least absolute remainders significantly reduce the number of steps in the Aryabhata algorithm?
• RQ2 How does the performance of the Aryabhata algorithm with least absolute remainders compare to the standard version in terms of computational complexity?
• RQ3 To what extent does the least absolute remainder approach improve efficiency in cryptographic applications requiring modular inverses?
• RQ4 Does the modified algorithm maintain correctness while achieving better runtime performance?
• RQ5 What is the impact of remainder selection strategy on the convergence rate of the extended Euclidean-style algorithm?

Key findings

• The Aryabhata algorithm using least absolute remainders achieves faster convergence than the standard version.
• The method reduces the number of iterations required to compute modular inverses by minimizing remainder magnitudes at each step.
• Performance improvements are measurable, particularly in scenarios with large integers common in cryptography.
• The algorithm maintains correctness and is suitable for secure and efficient implementation in cryptographic protocols.
• The approach outperforms recent variants described by Rao and Yang in terms of computational efficiency.
• The use of symmetric remainders leads to a more predictable and stable reduction path in the algorithm’s execution.