16.5.1. Overview of RSA

The race for coming up with the first instantiation of public key cryptosystem was won by another Turing Award winner triplet namely, RSA, Rivest, Shamir and Adleman.

RSA stands for Rivest, Shamir, and Adleman, the three researchers who developed this public key cryptosystem. They were awarded a Turing Award for their work. This means that RSA is a recognized and essential algorithm in cryptographic practices, specifically in securing communications over the internet.

Think of RSA as a secure locker designed by three brilliant engineers. Just like these engineers created a special locker that anyone can use to safely store their possessions, RSA allows people to protect their messages so they can send them over the internet without worry.

RSA is based on several interesting results from number theory that we have discussed. Recall the group Z∗N as the collection of all values in the range 1 to N - 1 which are co-prime to your modulus N.

The RSA algorithm relies on concepts from number theory, specifically the properties of prime numbers and co-primality. The group Z∗N includes numbers that share no common factors with N other than 1. This property is essential for ensuring that the encryption and decryption processes work correctly, forming the backbone of the security RSA offers.

Imagine you have a club with a strict entry policy that only allows certain members. The club's security system is similar to how RSA ensures that only authorized people can access the information, using membership (co-primality) as a key criterion.

To generate the parameters, we randomly pick some n-bit prime numbers p and q, and then compute modulus N as the product of p and q.

In RSA, key generation involves choosing two large prime numbers (p and q) at random. The product of these primes yields the modulus N, which is used in both the public and private keys. The security of RSA relies on the difficulty of factoring this large number back into its prime components.

Consider picking two large, mysterious boxes (the prime numbers). When you multiply them, you create a larger, mysterious box (the modulus). The challenge lies in figuring out what is inside the large box, similar to how factoring N back into p and q is challenging!

Set pk or the encryption key to be (N, e) and the decryption key to (N, d).

In RSA, after computing the values of N, the public exponent e, and the private exponent d, these values are used to form the public and private keys. The public key, which contains N and e, can be shared openly, while the private key, containing N and d, must be kept secret. This distinction is crucial for secure communication.

Imagine the public key as a mailbox that anyone can drop a letter into (the public key), while the private key is the key that only you have to unlock and read those letters (the private key). This way, anyone can send you messages, but only you can read them.

If there is a sender who has a plain text m and wants to encrypt it, the encryption of m is nothing but computing m^e mod N.

The encryption process in RSA involves taking the plaintext (m), raising it to the power of the public exponent (e), and then taking the result modulo N. This produces a ciphertext that can be safely sent over the open channel. The decryption process is the reverse, using the private exponent (d) to retrieve the original message.

Think of the encryption process like sealing a letter in a special envelop that only the intended recipient can open. The message sent is hidden in a way that only the recipient with the right key can decode it back into its original form.

The RSA problem is when I give you the modulus but not its prime factors and ask you to compute the inverse function without knowing d.

The security of RSA comes from the RSA problem, which states that it is computationally infeasible to determine the private key from the public key without factoring the modulus N. This difficulty is what safeguards the encryption from being broken by unauthorized parties.

Imagine trying to crack a safe without knowing the combination, where the only way to get the combination is to dismantle the safe (factor N into p and q). This demanding task represents the complexity that protects the information encrypted with RSA.

This is not precisely the way we use RSA public key cryptosystem in practice due to its determinism.

While the theoretical framework of RSA is sound, in practice, it is often enhanced to mitigate issues like determinism. This means that encrypting the same message multiple times would yield the same ciphertext, which can reveal patterns to potential attackers. Thus, additional techniques are employed to ensure randomness and security in each encryption.

Think of this as a bank that always sends the same deposit slip every time you deposit the same amount. This could alert a thief that you’re depositing the same amount repeatedly, so banks vary their slips to cover your transactions and keep your financial activities private.

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