16.5.2. RSA Function and Operations

Now, let me introduce RSA function which forms the basis of RSA public key cryptosystem. This is a function from the set $Z^* \rightarrow Z^*$. Then, how exactly is this function defined? So, imagine you have a public exponent $e$, this is not identity element, this is some notation, this is an exponent $e$ which is we are going to use in the function. And this exponent $e$ is relatively prime to $\phi(N)$, I stress it is not relatively prime to $N$, it is relatively prime to $\phi(N)$.

The RSA function is central to the RSA public key cryptosystem, which enables secure communication. This function takes an element from the set of integers that are coprime to a number $N$ (denoted as $Z^*$) and outputs another element from the same set. The public exponent $e$ plays a crucial role; it must be a number that does not share any factors with $\phi(N)$, where $\phi$ is Euler’s totient function. This condition ensures that a multiplicative inverse $d$ exists, allowing for the proper function of the cryptosystem.

Think of the RSA function like a unique lock and key system where the public exponent $e$ is like a special key that fits into a very specific lock. This lock (or function) is designed based on the properties of numbers. Just like not every key fits every lock, not every number can serve as an exponent in RSA. The right combinations ensure that the lock can only be opened with the correct key, keeping the message secure.

Now, the RSA function in the forward direction is the following. If I want to compute the output of the RSA function for x then that is same as computing $x^e \mod N$. So, remember x is a member of $Z^*$ and hence $x^e \mod N$ will also be an element of $Z^*$. Because my underlying operation is multiplication modulo N and x power e is like performing the group exponentiation.

In RSA, when we want to encrypt a message, we use the 'forward function.' This process involves raising a plaintext message, represented as $x$, to the power of the public exponent $e$ and then taking the result modulo $N$. This ensures that the result remains a valid number within our defined set. The reverse function is equally important: to decrypt, we raise the encrypted message (ciphertext) to the power of the secret exponent $d$ and apply the same modulo operation. The relationship between encryption and decryption showcases how RSA secures the message.

Imagine you have a complex code that has a specific way of turning a message into a number (encryption). To decode that number back into the original message, you need to know a secret technique that only a select few can use (decryption). The act of raising a number to an exponent and then transforming it through a modulo operation is like applying a unique transformation that ensures the message gets locked in a specific way. Only someone with the right secret can unlock it and retrieve the original message.

So, now based on whatever theory we have discussed, let us see the concrete steps of the RSA public key cryptosystem. So, remember, we have a function in the forward direction from $Z^* \rightarrow Z^*$ namely, $x^e \mod N$ and a reverse function is $y^d \mod N$. The RSA problem is the following. If I give you the modulus, but not its prime factors and if I give you the public exponent in that sense, it is public, it will be known to you and it will be known that how exactly this parameters N and e are generated.

In generating RSA parameters, a user chooses two large prime numbers, $p$ and $q$, then calculates their product $N = p \times q$. After calculating $\phi(N)$, which is crucial for finding the public and private keys, a public exponent $e$ is selected that is co-prime to $\phi(N)$. The secret exponent $d$ is then computed as the multiplicative inverse of $e$ under modulo $\phi(N)$. This process ensures that the keys are securely generated and can be used for encryption and decryption.

Generating the RSA parameters is like setting up a special vault. First, you find two strong, unique materials (the prime numbers). Then, you put those materials together to create a solid structure (the modulus $N$). Choosing the right access codes (the public exponent) is crucial to ensure that only authorized individuals can open the vault. The secret access code ensures that you can lock and unlock it safely, maintaining the confidentiality of what's stored inside.

Now, the problem instance is the following. I will be giving you a random element from my group $Z^*$ and your goal will be to compute the inverse function, output of the inverse function that we had just seen, for the randomly chosen y. Namely, your goal will be to compute $y^{1/e} \mod N$ where $1/e$ is not numeric 1 over e but it is actually the multiplicative inverse of d.

The RSA problem highlights the difficulty of decrypting messages without the secret key. Given the ciphertext and the public elements ($N$ and $e$), a potential attacker would face the task of determining the secret exponent $d$ without knowing the factors corresponding to $N$. The assumption is that factoring $N$ into its prime components is computationally difficult for large numbers, making RSA a secure encryption method.

The RSA problem can be likened to trying to unlock a safe without having the combination. You can see the external mechanism (the public key), but without knowing its internal combination (the secret key), you're stuck. As the safe locks larger items (like larger prime factors), it becomes increasingly challenging to break into without the right access.

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