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RSA is a cryptosystem for public-key encryption, and is widely used for securing sensitive data, particularly when being sent over an insecure network such as the Internet.

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RSA's popularity: 

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RSA derives its security from the difficulty of factoring large integers that are the product of two large prime numbers.
Multiplying these two numbers is easy, but determining the original prime numbers from the total -- factoring -- is considered infeasible due to the time it would take even using today’s super computers.

The public and the private key-generation algorithm is the most complex part of RSA cryptography.
Two large prime numbers, p and q, are generated using the Rabin-Miller primality test algorithm.
A modulus n is calculated by multiplying p and q.
This number is used by both the public and private keys and provides the link between them.
Its length, usually expressed in bits, is called the key length.
The public key consists of the modulus n, and a public exponent, e, which is normally set at 65537, as it's a prime number that is not too large.
The e figure doesn't have to be a secretly selected prime number as the public key is shared with everyone.
The private key consists of the modulus n and the private exponent d, which is calculated using the Extended Euclidean algorithm to find the multiplicative inverse with respect to the totient of n.

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Example

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Alice generates her RSA keys by selecting two primes: p=11 and q=13.
The modulus n=p×q=143.
The totient of n ϕ(n)=(p−1)x(q−1)=120.
She chooses 7 for her RSA public key e and calculates her RSA private key using the Extended Euclidean Algorithm which gives her 103.

Bob wants to send Alice an encrypted message M so he obtains her RSA public key (n, e) which in this example is (143, 7).
His plain text message is just the number 9 and is encrypted into cipher text C as follows: 

C = M^e mod n = 97 mod 143 = 48 

So, cipher to be sent is 48.

When Alice receives Bob’s message she decrypts it by using her RSA private key (d, n) as follows: 

M = C^d mod n = 48103 mod 143 = 9 

Therefore, the message received is 9.

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