On the Algorithmic Significance and Analysis of the Method of DaYan Deriving One

TL;DR

This paper analyzes the DaYan deriving one method for computing modular inverses, providing a precise algorithmic description, exploring its properties, and comparing it with the Extended Euclidean algorithm, highlighting its significance in cryptography.

Contribution

It offers a new, clear algorithmic formulation of DaYan deriving one, connecting it to continued fractions and invariance properties, and compares it with modern algorithms.

Findings

DaYan deriving one is equivalent to the Extended Euclidean algorithm.

The method reveals invariance properties and connections to continued fractions.

It is a concise and transparent approach for modular inverse computation.

Abstract

Modulo inverse is an important arithmetic operation. Many famous algorithms in public key cryptography require to compute modulo inverse. It is argued that the method of DaYan deriving one of Jiushao Qin provides the most concise and transparent way of computing modulo inverse. Based on the rule of taking the least positive remainder in division, this paper presents a more precise algorithmic description of the method of DaYan deriving one to reflect Qin's original idea. Our form of the algorithm is straightforward and different from the ones in the literature. Some additional information can be revealed easily from the process of DaYan deriving one, e.g., the invariance property of the permanent of the state, natural connection to continued fractions. Comparison of Qin'a algorithm and the modern form of the Extended Euclidean algorithm is also given. Since DaYan deriving one is the key technical ingredient of Jiushao Qin's DaYan aggregation method (aka the Chinese Remainder Theorem), we include some explanation to the latter as well.