# Cryptosystem: Importance of Number Theory

Number theory is one of the oldest branches of mathematics. It is the study of numbers, their properties and patterns. It is the study of prime and composite numbers. It has been, for ages, wrongly labeled useless. In today’s world of ever growing connectivity and spend, there is a greater and greater need for security and number provides powerful cryptography tools.

Cryptography is an ancient study. Secret messages have helped generals communicate with their commanders, scouts and spies. Cryptography had been a critical part of the first world war, however with the advent of the internet and inter-connectivity, cryptography has become commonplace.

One of the most widely used cryptosystem is the public key cryptosystem. In this system an encryption key is given out to the public, called  a public key, which is used to encrypt messages. Anyone can know this public key since it is almost impossible to decrypt an encrypted message using it. The key used for decryption is called the private key and it is used only by the receiver. This allows cheaper and easier ways to secure messages such as passwords.

The RSA public key cryptosystem takes advantage of an interesting property of prime numbers, it takes considerable more time to factor numbers into primes then it does to multiply prime numbers. Both the private and public key are created from two primes, but to recover the private key from the public key, one must factor the public key. Depending on the primes used, generating the public key and encryption will only takes seconds, however factoring the public key will take millenniums.

Though old number theory brings new life and security into a world of ever growing inter-connectivity and communications.