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But my question is: Who first noticed this modular property. Which problems led him to introduce the rule? How did he discover this rule? Sign up to join this community. The best answers are voted up and rise to the top.
Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. All one has to do is look at an analog clock. This is modular arithmetic, i. This is an important concept in the technology driven world we live in. Any time a product is purchased on the internet, cryptography comes into play. The first known publication of modular arithmetic was in the 3 rd century B. E, in the book Elements , written by Euclid. Within his book, he not only formalized the fundamentals of arithmetic, but also proved it.
In what is known as Euclids Lemma, he states that if a prime number divides the product of two different numbers x and y , then the prime number must also divide one of the numbers either x or y , but it could also be both.
Between the 3 rd and 5 th centuries a paper publish by Sun Tzu describes a modular arithmetic process known as the Chinese remainder theorem. It uses a congruent set of keys to produce the same numerical value. Imagine if there was a lock on a door that two differently cut keys could unlock and open, this is essentially how Chinese remainder theorem works.
Oil painting of mathematician and philosopher Carl Friedrich Gauss by G. Biermann Public Domain. Gauss is famous for numerous discoveries across a wide variety of fields in science and mathematics. The residue-class is represented by the remainder, which can be from 0 to M The remainder is obtained by dividing N by M. Given this fact, Gauss notices that two numbers that differ by a multiple of M are in the same residue-class.
He then discusses the role of negative numbers in modular arithmetic. The following is an excerpt from his book:. For a zero modulus, there would be infinitely many residue classes, each containing only one element. RSA is the process by which information can be passed between two parties without another individual being able to intercept the message.
The encryption must be done in such a way that decryption is only possible with knowledge of a secret decryption key. The decryption key should only be known by authorized parties. This is a high level description of how RSA encryption works. It is also called public-key encryption, because anyone can obtain a copy of the encryption key it is publically available, but the decryption key cannot be obtained. This makes RSA encryption a secure way of passing data between an individual and a web site.
Simplified view of RSA encryption. Performing this calculation encrypting and decrypting text is fairly simple. With a basic understanding of modular arithmetic it can be accomplished. First a public and private key must be produced by following the steps below:. Once those steps have been completed, a public key n, e and a private key n, d have been generated.
The public key can be distributed to anyone, but the private key must be kept safe. One could generate pairs of random numbers until a pair is found that when multiplied together, would equal the modulus n found in step two above. From there, it would be easy to find all numbers co-prime to the totient in step three. Modular arithmetic then comes into play, because it allows infinite pairs of numbers to satisfy the constraint listed in step five, but it would not allow the user to decrypt the message.
Peter Lynch. The prince Modular arithmetic was introduced by Carl Friedrich Gauss on the very first page of his magnum opus Arithmetical Investigations , or Disquisitiones Arithmeticae. More from The Irish Times Health. Home energy upgrades are now more important than ever. The Dublin start-up making the future better with an appreciation for innovation.
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