Steps • Explanation • Viability • Practical Implementation • Encryption • Decryption

---

Steps

• First select two distinct prime numbers, say p and q. (preferably big to score a decent level of security)

Can be generated using a primality test.
Fermat’s test can be used, although practically Miller Rabin’s test is more sought after.

• Take their product as n i.e. n = p*q, which is the modulus of both the keys.

• Then calculate the Euler’s totient for the above, given by ϕ(n) = (p-1)*(q-1).

• Proceed by choosing a public key e such that e > 2 and coprime to the totient i.e., gcd(e, totient) must be equal to 1.

• Choose a corresponding private key d such that it satisfies the equation or e*d mod ϕ(n) = 1.

d is the multiplicative inverse of e modϕ(n).
Public key comprises of (e, n) and private key comprises of (d).
As e is much smaller than d, encrypting a message using RSA is much faster than decrypting it.

• Ciphertext is calculated using the equation c = m^emodn, where m is the message to be encrypted.

• With the help of c and d thus obtained, we can decrypt the message using m = c^dmodn.

Explanation

The Euler’s totient φ(n) of a positive integer n greater than 1 is defined to be the number of positive integers less than n that are coprime (only positive divisor being 1 or the gcd between the two numbers being 1) to n. (φ(1) is defined to be 1)
When n is prime, ϕ(n) = n-1, as in the case of Fermat’s theorem. Plaintext and ciphertext are integers between 0 and n-1 for some n.

• Encryption and decryption are of the following form, for some plaintext M and ciphertext C:
  

  • C = M^e(mod n)
  • M = C^d(mod n) = (M^e)^d(mod n) = M^ed(mod n)

• Each communicating entity has one public-key (e, n) or private-key (d, n) pair, where both e and d are in fact the multiplicative inverse (modϕ(n)) of the other.

  • e and d are inverses (modϕ(n)), or ed ≡ 1 (modϕ(n))
  • ed = 1 + kϕ(n), for some k.
  • M^ed(mod n) = M^{1 + kϕ(n)}(mod n) = M¹ x (M ^ϕ(n))^k(mod ϕ(n)) = M¹ x 1^k(mod n) = M. (Euler’s theorem)

Therefore, to decrypt a ciphertext C = M^ed(mod n), we only need to calculate C^d(mod n), since we know C = M^e(mod n) => C^d(mod n) = M^ed(mod n) => C^d(mod n) = M. (with M^ed(mod n) being M)

Viability

The RSA cryptosystem is based on the theorem which implies that the inverse of the function a->a^emodn (where e is the public encryption exponent) is the function b->b^dmodn, (where d is the private decryption exponent) which provides the difficulty of computing ϕ(n) without knowing the factorization of n. (and thus the difficulty of computing d arises in addition)

This can only be solved by factorizing n (since every number is essentially a product of primes) and only the owner of the private key knows the factorization (primes p and q whose product yields n). This ‘factoring problem’ is the security point, with greater chances of the encryption to be secure for large values of n, or for large primes considered. The fact that only n is publicly disclosed, along with the given difficulty to factor large numbers (it is computationally infeasible to factor a large value of n to get d) gives the guarantee that no one else knows the factorization and the encrypted message, thus making it viable.

Practical Implementation

Implementations in a few languages can be found in this directory. Rosetta Code is another source for implementations in multiple languages. These are for the sole purpose of educating oneself via reference, and must not be used in codebases where the implementations are critical. Given that there are multiple factors to be considered in modern-day cryptographic routines in order to establish a deeper level of security, its best to create specific implementations than to use the generalized ones or the toy implementations.

For general use, the length for the 2 primes considered (p and q) should be preferably around 2¹¹ bits (2048) or more, which results in values (n) greater than 2¹² (4096) bits upon their multiplication (p*q). This ensures a tight encryption which is seemingly impossible to decrypt even when provided with massive computing resources (although completely possible to breach/bruteforce every combination if we were to leverage the power of supercomputers) for a large span of time.

Hence, the RSA algorithm is quite feasible in general, apart from its primary downside of being much slower than symmetric cryptosystems. There are better alternatives (faster and more secure encryption schemes) such as elliptic curve cryptosystems, but then again, quantum computing could overcome that as well. Eventually nothing seems that secure, or has a counter-measure in the long run.

---

Summary addon for encryption/decryption: (22/06/19)

🔒Encryption

Input: RSA public key (n,e), plaintext m ∈ [0,n-1]	Output: Ciphertext c, (Compute c = me(mod n), return c)

🔓Decryption

Input: RSA public key (n,e), RSA private key d, ciphertext c	Output: Plaintext m, (Compute m = cd(mod n), return m)

---

Setting up SSH using RSA keys for Github: (03/08/19)

For SSH authentication, it is rather typical that the users arrange the key pair themselves. Using the SSH protocol, one can connect and authenticate to remote servers and services, and with the SSH keys, one can connect to GitHub without supplying the username or password at each visit - which is a convenience indeed.

For setting up SSH locally within your system using RSA keys, follow along:

• First check if files '(/.ssh/)id_rsa' and '(/.ssh/)id_rsa.pub' exist in your system. If not, create such public/private keys by opening a terminal and typing:

$ ssh-keygen -t rsa -C "[email protected]"

• Next, copy the public key (contents of the freshly created 'id_rsa.pub') into your clipboard. To follow on a Mac, type:

$ pbcopy < ~/.ssh/id_rsa.pub

• Paste the obtained SSH public key into a new slot for SSH keys in github via the settings tab, accessible via your account. (i.e. click “SSH and GPG Keys” from the list of options towards the left of the settings tab, click “Add SSH Key” towards the top right, add a label and paste the public key into the text box)

• Type the following in your terminal session to test it:

$ ssh -T [email protected]

If 'You've successfully authenticated' is included in the follow-up message, you're good to go!