This ppt is showing details of RSA and implementation of RSA in C++.

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.

11. Implementation in C++

12. Implementation in C++

13. Implementation in C++

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