Data in a computer is essentially a string of bytes. When encrypting, this data is divided into blocks of fixed length, with each block represented as a binary number, denoted as 'd'.
The RSA algorithm is an asymmetric encryption method that uses two keys: a public key for encryption and a private key for decryption. The public key consists of two values: 'e' (the exponent) and 'N' (the modulus), while the private key is represented by 'd', also known as the private exponent.
In practice, the size of 'N' determines the security level of the system. It's typically at least 1024 bits long, but modern standards recommend 2048 bits or more to ensure robust protection against attacks.
The encryption process involves raising the plaintext value 'd' to the power of 'e', then taking the modulus with 'N' to produce the ciphertext 'c': c = d^e mod N.
To decrypt, the receiver raises the ciphertext 'c' to the power of 'd', again using modulus 'N' to recover the original message: d = c^d mod N.
The security of RSA relies on the difficulty of factoring large numbers. While it's easy to compute 'N' from its prime factors 'p' and 'q', reversing this process is computationally infeasible with current technology. However, if the random number generators used to create keys are not truly random, vulnerabilities could be introduced. This is why secure implementations must use high-quality entropy sources.
**2. Description of the RSA Encryption Algorithm:**RSA is a widely used public-key cryptographic algorithm based on number theory. The process begins by selecting two large prime numbers, 'p' and 'q'. These are multiplied to form 'n = p * q', which serves as the modulus for both encryption and decryption.
Next, Euler’s totient function φ(n) is calculated as φ(n) = (p-1)(q-1). A public exponent 'e' is chosen such that it is coprime with φ(n), meaning gcd(e, φ(n)) = 1. Then, a private exponent 'd' is computed such that e * d ≡ 1 mod φ(n). This ensures that encryption and decryption are inverses of each other.
For example, if we choose p = 11 and q = 13, then n = 143 and φ(n) = 120. Selecting e = 7 (which is coprime with 120), we find d = 103, since 7 * 103 = 721, and 721 mod 120 = 1. The public key is (n, e) = (143, 7), and the private key is (n, d) = (143, 103).
To encrypt a message m = 85, the sender computes c = m^e mod n = 85^7 mod 143 = 123. The ciphertext c = 123 is sent to the receiver. Upon receiving, the receiver decrypts it using their private key: m = c^d mod n = 123^103 mod 143 = 85.
In real-world applications, messages are often split into blocks smaller than 'n' for efficient encryption. Each block is processed individually, ensuring that even large messages can be securely transmitted. This approach also helps maintain the integrity and confidentiality of the data throughout the communication process.
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