Understanding RSA Cryptosystem Equations

The RSA cryptosystem relies on specific mathematical relationships between keys and the encryption/decryption process. Let's analyze the given equations to determine which ones accurately represent the RSA algorithm.

RSA Encryption and Decryption Process

The core operations in RSA are encryption and decryption. Given a plaintext message $M$ (where $0 < M < n$), the ciphertext $M'$ is calculated using the public key $(e, n)$ as follows:

Equation I (Encryption Part): \( M' \equiv M^e \pmod{n} \)

To recover the original message $M$ from the ciphertext $M'$, the private key $(d, n)$ is used:

Equation I (Decryption Part): \( M \equiv (M')^d \pmod{n} \)

These two equations correctly define the fundamental encryption and decryption steps in the RSA algorithm. The modulus used for these operations is $n$. Therefore, Equation I is a correct representation.

Key Relationship: Exponents and Euler's Totient Function

For the decryption process to successfully recover the original message, the private exponent $d$ must be related to the public exponent $e$ and the number $n$. This relationship involves Euler's totient function, denoted as $\phi(n)$. It is known that $\phi(n) = (p-1)(q-1)$ for $n = p \times q$ where $p$ and $q$ are prime numbers.

The condition that ensures $M \equiv (M'^d) \equiv (M^e)^d \equiv M^{ed} \equiv M \pmod{n}$ is:

Equation III: \( ed \equiv 1 \pmod{\phi(n)} \)

This means that $d$ is the modular multiplicative inverse of $e$ modulo $\phi(n)$. This equation correctly describes the relationship required between $e$ and $d$ for RSA to work. Therefore, Equation III is also a correct representation.

Why Other Equations Are Incorrect

Let's examine why Equations II and IV are not correct representations of the standard RSA cryptosystem:

• Equation II: \( ed \equiv 1 \pmod{n} \) This equation suggests that the relationship between $e$ and $d$ should hold modulo $n$. However, the correct condition for RSA relies on the modular inverse property related to $\phi(n)$, not $n$ itself. Using $n$ as the modulus here is incorrect.
• Equation IV: \( M' \equiv M^e \pmod{f(n)} \) and \( M \equiv (M')^d \pmod{f(n)} \) This equation incorrectly uses $f(n)$ (which usually represents $\phi(n)$ in this context) as the modulus for the encryption and decryption operations. The actual encryption and decryption must be performed modulo $n$, the public modulus. While the relationship $ed \equiv 1 \pmod{f(n)}$ (or $\phi(n)$) is correct, applying the exponentiation modulo $f(n)$ is wrong.

Conclusion

Based on the analysis of the RSA cryptosystem's requirements:

• Equation I correctly defines the encryption ($M' \equiv M^e \pmod{n}$) and decryption ($M \equiv (M')^d \pmod{n}$) processes.
• Equation III correctly states the necessary relationship between the exponents ($ed \equiv 1 \pmod{\phi(n)}$).

Therefore, the equations that correctly represent the RSA cryptosystem are I and III.