My IB (International Baccalaureate) Extended Essay in Mathematics HL on the topic of RSA encryption
Public key encryption schemes are essential to ensuring the security of the modern internet. This extended essay explores one of the most used such schemes, RSA, through the research question of how RSA allows for secure and private communication over the internet. In the Introduction, the need for and basic idea behind public key encryption is explained. Then, in the first section, I introduce two concepts central to number theory, and therefore RSA encryption: Prime numbers and modular arithmetic. In the second section, I refer to and explain a number of primality tests which can be used for generating prime numbers for use in RSA encryption, namely, trial division, the Sieve of Eratosthenes and the Fermat Test. In the third and final section I directly address the mathematics behind RSA encryption, by providing a proof of its correctness through showing that any given message can be encrypted and then decrypted and remain intact. In the conclusion I answer the research question by showing that it is extremely difficult --- practically impossible with current technology --- to find the private key of an RSA key pair knowing only the public key. For the closing discussion of the essay, I talk about the possibility of using Mersenne prime numbers as the base primes for RSA encryption, an idea which is quickly disregarded due to the triviality of exhaustively testing every single pair of Mersenne primes, as only 49 such primes have been discovered.