The Chinese Remainder Theorem – History and Applications

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.
The system of congruences was:
xN=r2mod y
xN=r1mod 60
The above system shows the number of years N after shangyuan. During a certain winter solstice for r2 days after the new moon and r1 days after shangyuan, the year was N 'years' after shangyuan. y represents the days in a lunar month while y represents the days during tropical years (Xiao, 2018). The computation was applied during wall building or constructing the base of a house, for instance, the construction of The Great Wall in feudal times.
The Chinese remainder theorem has multiple applications in computing algorithms and helps other theorems in advanced manipulation arrive at their conclusion. The theorem is used in encryption and cryptography. The increased cybersecurity breaches need the remainder theorem for effective formulation of algorithms. Cryptography is vital in ensuring secure communications in digital means or online. The Chinese remainder theorem is applied in the RSA cryptosystems, which are publicly used key systems for transmitting secure data. Encryption is the process that achieves all these steps because it involves encoding data so that only authorized people to access the data and understand it. The Chinese remainder theorem is used to build the decryption framework because it stores and produces values as part of a private key. The theorem applies when sharing secret information. It is impossible to recover the data from multiple shares as long as the set consists of less than a certain number of elements (Yan et al., 2018).
For example;
Let m1, m2, m3……, mt be t pairwise relatively prime positive intergers.