The RSA cryptosystem is a public-key cryptosystem that offers both encryption and digital signatures (authentication). Ronald Rivest, Adi Shamir, and Leonard Adleman developed the RSA system in 1977 .RSA stands for the first letter in each of its inventors' last names.

The RSA algorithm works as follows: take two large primes, p and q, and compute their product n = pq; n is called the modulus. Choose a number, e, less than n and relatively prime to (p-1)(q-1), which means e and (p-1)(q-1) have no common factors except 1. Find another number d such that (ed - 1) is divisible by (p-1)(q-1). The values e and d are called the public and private exponents, respectively. The public key is the pair (n, e); the private key is (n, d). The factors p and q may be destroyed or kept with the private key.

It is currently difficult to obtain the private key d from the public key (n, e). However if one could factor n into p and q, then one could obtain the private key d. Thus the security of the RSA system is based on the assumption that factoring is difficult

Here is how the RSA system can be used for encryption and digital signatures [ ]

Suppose Alice wants to send a message m to Bob. Alice creates the ciphertext c by exponentiating: c = m^{e} mod n, where e and n are Bob's public key. She sends c to Bob. To decrypt, Bob also exponentiates: m = c^{d} mod n; the relationship between e and d ensures that Bob correctly recovers m. Since only Bob knows d, only Bob can decrypt this message.

Suppose Alice wants to send a message m to Bob in such a way that Bob is assured the message is both authentic, has not been tampered with, and from Alice. Alice creates a digital signature s by exponentiating: s = m^{d} mod n, where d and n are Alice's private key. She sends m and s to Bob. To verify the signature, Bob exponentiates and checks that the message m is recovered: m = s^{e} mod n, where e and n are Alice's public key.

Thus encryption and authentication take place without any sharing of private keys: each person uses only another's public key or their own private key. Anyone can send an encrypted message or verify a signed message, but only someone in possession of the correct private key can decrypt or sign a message.

The RSA algorithm involves three steps: key generation, encryption and decryption.

RSA involves a *public key* and a *private key.* The public key can be known by everyone and is used for encrypting messages. Messages encrypted with the public key can only be decrypted in a reasonable amount of time using the private key. The keys for the RSA algorithm are generated the following way:

- Choose two distinct prime numbers
*p*and*q*.

- For security purposes, the integers*p*and*q*should be chosen at random, and should be of similar bit-length. Prime integers can be efficiently found using a primality test. - Compute
*n*=*pq*.

-*n*is used as the modulus for both the public and private keys. Its length, usually expressed in bits, is the key length. - Compute φ(
*n*) = φ(*p*)φ(*q*) = (*p*− 1)(*q*− 1) =*n*- (*p*+*q*-1), where φ is Euler's totient function. - Choose an integer
*e*such that 1 <*e*<φ(*n*) and gcd(*e*, φ(*n*)) = 1; i.e.,*e*and φ(*n*) are coprime.

-*e*is released as the public key exponent.

-*e*having a short bit-length and small Hamming weight results in more efficient encryption – most commonly 2^{16}+ 1 = 65,537. However, much smaller values of*e*(such as 3) have been shown to be less secure in some settings. - Determine
*d*as*d*≡*e*^{−1}(mod φ(*n*)); i.e.,*d*is the multiplicative inverse of*e*(modulo φ(*n*)).

- This is more clearly stated as: solve for
*d*given*d*⋅*e*≡ 1 (mod φ(*n*)) - This is often computed using the extended Euclidean algorithm. Using the pseudocode in the
*Modular integers*section, inputs*a*and*n*correspond to*e*and*φ(*n*)*, respectively. *d*is kept as the private key exponent.

The *public key* consists of the modulus *n* and the public (or encryption) exponent *e*. The *private key* consists of the modulus *n* and the private (or decryption) exponent *d*, which must be kept secret. *p*, *q*, and φ(*n*) must also be kept secret because they can be used to calculate *d*.

Alice transmits her public key *(n, e)* to Bob and keeps the private key secret. Bob then wishes to send message *M* to Alice.

He first turns *M* into an integer *m*, such that *0 ≤ m < n* by using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext*c* corresponding to

This can be done quickly using the method of exponentiation by squaring. Bob then transmits *c* to Alice.

Note that at least nine values of *m* will yield a ciphertext*c* equal to *m*, but this is very unlikely to occur in practice.

Alice can recover *m* from *c* by using her private key exponent *d* via computing

Given *m*, she can recover the original message *M* by reversing the padding scheme.

- Choose p=3 and q=11
- Compute n = p * q = 3 * 11 = 33
- Compute
*φ*(n)= (p – 1) * (q – 1) = 2 * 10 = 20 - Choose e such that 1 < e <
*φ(n)*and e and n are coprime. Let e=7 - Compute a value for d such that (d * e) %
*φ*(n)=1. One solutions is d = 3 [3(3 * 7) %20=1] - Public key is (e,n) => (7,33)
- Private key is (d,n) => (3,33)
- The encryption of
*m*= 2 is*c*= 2^{7}% 33 = 29 - The decryption of
*c*= 29 id*m*=29^{3}% 33 = 2

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