RSA Algorithm and its implementation in C++.pptx
- 1. RSA Algorithm: Pseudocode and Implementation in C++ Name: Anees ur Rehman Reg#: BCB242011 Design And Analysis Of Algorithm
- 2. Introduction •What is RSA? • RSA (Rivest-Shamir-Adleman) is a widely used public-key cryptosystem for secure data transmission. • It is based on the mathematical properties of large prime numbers. •Key Concepts: • Public Key: Used for encryption. • Private Key: Used for decryption. •Applications: • Secure communication (e.g., HTTPS, SSL/TLS). • Digital signatures. • Data encryption.
- 3. Common Problems Related to RSA •Key Generation Challenges: • Finding large prime numbers. • Ensuring the security of the private key. •Computational Complexity: • Modular exponentiation is computationally expensive. •Security Concerns: • Vulnerable to brute-force attacks if key size is small. • Vulnerable to quantum computing attacks (Shor’s algorithm). •Implementation Issues: • Handling large integers in programming languages.
- 4. RSA Algorithm Overview •Steps in RSA: • Key Generation: • Choose two large prime numbers, pp and qq. • Compute. • Choose public key ee such that and . • Compute private key . • Encryption: • , where M is the plaintext message. • Decryption: • , where C is the ciphertext.
- 5. “1. Generate two large prime numbers, p and q. 2. Compute n = p * q. 3. Compute φ(n) = (p-1) * (q-1). 4. Choose e such that 1 < e < φ(n) and gcd(e, φ(n)) = 1. 5. Compute d such that d ≡ e^(-1) mod φ(n). 6. Public Key: (e, n). 7. Private Key: (d, n). Pseudocode for RSA Key Generation
- 6. “1. Input: Plaintext message M, Public Key (e, n). 2. Compute ciphertext C = M^e mod n. 3. Output: Ciphertext C. Pseudocode for RSA Encryption
- 7. “1. Input: Plaintext message M, Public Key (e, n). 2. Compute ciphertext C = M^e mod n. 3. Output: Ciphertext C. Pseudocode for RSA Encryption
- 8. “1. Input: Ciphertext C, Private Key (d, n). 2. Compute plaintext M = C^d mod n. 3. Output: Plaintext M. Pseudocode for RSA Decryption
- 9. Working Example of RSA Step 1: Key Generation Step 2: Encryption • Let M=65. • Compute Step 3: Decryption • Compute
- 10. Time and Space Complexity •Key Generation: • Time Complexity: , where k is the number of bits in n. •Encryption/Decryption: • Time Complexity: for modular exponentiation. •Space Complexity: • O(k)O(k) for storing keys and intermediate results.
- 14. Summary • RSA is a robust public-key cryptosystem based on the difficulty of factoring large prime numbers. • It involves key generation, encryption, and decryption using modular arithmetic. • Proper implementation requires handling large integers and ensuring security through appropriate key sizes. • Applications include secure communication and digital signatures.
- 15. References •Books: • "Introduction to Algorithms" by Cormen, Leiserson, Rivest, and Stein. • "Applied Cryptography" by Bruce Schneier. •Online Resources: • Wikipedia: RSA Algorithm • GeeksforGeeks: RSA Algorithm
- 16. Thank you Anees ur Rehman Reg#: BCB242011