How do you calculate RSA encryption step by step?
RSA encryption is based on the mathematical difficulty of factoring large numbers.
It relies on the fact that while multiplying two large prime numbers is easy, factoring their product back into the original primes is computationally hard.
To begin calculating RSA encryption, you first need to select two distinct prime numbers, typically denoted as \( p \) and \( q \).
These primes should be large enough to ensure security, often hundreds of digits long.
The product of these two primes, \( N = p \times q \), is called the modulus.
This modulus is used in both the public and private keys.
For example, if \( p = 61 \) and \( q = 53 \), then \( N = 61 \times 53 = 3233 \).
Next, you calculate Euler’s totient function, \( \phi(N) = (p-1)(q-1) \).
This function counts the integers up to \( N \) that are coprime with \( N \).
For our earlier example, \( \phi(3233) = (61-1)(53-1) = 3120 \).
Choose a public exponent \( e \), which must be an integer that is coprime to \( \phi(N) \) and typically small.
A common choice for \( e \) is 65537, as it strikes a balance between security and computational efficiency.
To check if \( e \) is coprime to \( \phi(N) \), you can use the Euclidean algorithm.
If the greatest common divisor (GCD) of \( e \) and \( \phi(N) \) is 1, then they are coprime.
The next step is to compute the private exponent \( d \) using the modular multiplicative inverse of \( e \) modulo \( \phi(N) \).
This means finding \( d \) such that \( (d \cdot e) \mod \phi(N) = 1 \).
The Extended Euclidean Algorithm is typically used for this.
With \( p \), \( q \), \( N \), \( e \), and \( d \) calculated, you now have a pair of keys: the public key \( (N, e) \) and the private key \( (N, d) \).
To encrypt a message \( M \) (which must be less than \( N \)), you use the public key.
The encrypted message \( C \) is computed as \( C = M^e \mod N \).
For decryption, the recipient uses their private key.
The original message is recovered using \( M = C^d \mod N \).
RSA encryption can be made more secure by using padding schemes like OAEP (Optimal Asymmetric Encryption Padding), which helps prevent various attacks, such as chosen ciphertext attacks.
RSA is widely used in secure communications, such as HTTPS connections, email encryption, and digital signatures, highlighting its importance in modern internet security.
The size of the numbers used in RSA directly affects its security; as computational power increases, the recommended key sizes have also increased, with 2048 bits being a common standard as of 2025.
RSA is also used in digital signatures, which authenticate a message's origin and ensure that it has not been altered during transmission.
The RSA algorithm can be vulnerable to certain types of attacks, such as factorization attacks, where an attacker tries to factor \( N \) back into \( p \) and \( q \) to break the encryption.
The strength of RSA encryption relies heavily on the selection of the prime numbers \( p \) and \( q \).
Randomly selecting large primes is crucial, as predictable primes can make the system vulnerable to attack.
As of 2025, post-quantum cryptography is a major area of research due to potential future capabilities of quantum computers to break RSA encryption by efficiently solving the factoring problem using algorithms like Shor's algorithm.
RSA encryption is not suitable for encrypting large amounts of data directly; instead, it is often used in conjunction with symmetric encryption methods, where RSA encrypts a symmetric key used for encrypting the actual data.
Despite its widespread use, RSA is slower than symmetric-key algorithms, making it less efficient for encrypting large datasets or streaming data.
The RSA algorithm was first introduced in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman, paving the way for modern public key cryptography and significantly influencing the field of cryptography and information security.