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Table of Content
- Introduction
- Mathematical Prerequisites
- Euler's Totient Function
- Coprime Integers
- Modular Multiplicative Inverse
- Euler's Theorem
- Modulo Congruence
- Operation
- Generating Public & Private Keys
- Encryption
- Key Distribution
- Decryption
- Working Example
- Alternative Solution
- Caesar Cipher
- RSA Reliability & Advantages
- Prime Generation & Integer Factorization
- Modular Exponetiation and Roots
- Advantages of RSA over Caesar Cipher
- Conclusion
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Introduction
Number theory may be one of the “purest” branches of mathematics, but it has turned out to be one of the most useful when it comes to computer security. For instance, number theory helps to protect sensitive data such as credit card numbers when you shop online. This is the result of some remarkable mathematic research from the 1970s that is now being applied worldwide.
Sensitive data exchanged between a user and a Web site needs to be encrypted to prevent it from being disclosed to or modified by unauthorized parties. The encryption must be done in such a way that decryption is only possible with the knowledge of a secret decryption key. The decryption key should be known by authorized parties.
This is the concept of public-key cryptography. The distinguishing technique used in public-key cryptography is the use of asymmetric key algorithms, where a key used by one party to perform either encryption or decryption is not the same as the key used by another in the counterpart decryption. RSA is one of the asymmetric algorithms.
RSA is one of the first practical public-key cryptosystems and is widely used for secure data transmission. In RSA, this asymmetry is based on the practical difficulty of factoring the two products of two large prime numbers, the factoring problem. RSA is made of the initial letters of the surnames of Ron Rivest, Adi Shamir, and Leonard Adleman, who first publicly described the algorithm in 1977.
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