The Chinese Remainder Theorem – History & Applications
Overview
This essay examines the Chinese Remainder Theorem, tracing its historical origins from Sun Zi in the 3rd century AD through its formalization by Qin Jiushao in 1247. The paper demonstrates how the theorem emerged from practical calendar calculations and explores its modern applications in cryptography, RSA encryption systems, and secret sharing protocols.
What this essay covers
- Historical development of the Chinese Remainder Theorem from ancient Chinese mathematics to formal theorem statement
- Explanation of the theorem's mathematical principles and solution methodology with numerical examples
- Applications in modern cryptography, RSA cryptosystems, and secret sharing algorithms
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The Chinese Remainder Theorem – History & Applications
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The Chinese remainder theorem is an old formula that offers the necessary conditions for specific multiple equations to have simultaneous integer solutions. Sun Zi originally found the theorem, a Chinese mathematician during the 3rd-century-AD, although a complete theorem was given in 1247 by Qin Jiushao. In the theorem, a person is asked to get a number that leaves a remainder of zero (0) when divided by five (5), the remainder six (6) when divided by seven (7), and the remaining ten (10) when divided by twelve (12). The shortest way is that the answer is three hundred and seventy (370). However, such an answer is not unique because any multiple of five times seven times twelve (5X7X12 (=420) can easily be added to the answer and still solve the problem. The theorem states that an equation system of congruence, which is linear and has pairwise relatively prime moduli, contains a specific solution modulo, the product of the system's moduli. The essay describes the history of the Chinese remainder theorem, mentions relevant problems, and applies it to them.
In historical times, the calculation of calendars is what led to the formation of the remainder theorem. Chinese astronauts around A.D. 237 called the starting point of the calendar "shangyuan," a moment similar to the new moon, winter solstice, and the midnight of the first day of the sixty (60) days cycle.