Introduction to RSA Algorithm

Study Notes

Introduction to RSA Algorithm

RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem widely used for secure data transmission.
It's based on the computational difficulty of factoring large integers. This difficulty makes RSA secure.
RSA is crucial for digital signatures, data encryption, and secure communication protocols.

Key Generation

To generate public and private keys:
- Select two large prime numbers, p and q, with similar bit lengths.
- Calculate n = p * q.
- Calculate Euler's totient function, φ(n) = (p-1)(q-1).
- Choose an integer e such that 1 < e < φ(n) and gcd(e, φ(n)) = 1. e is the public exponent.
- Compute the modular multiplicative inverse of e modulo φ(n). This is the private exponent, d.
- The public key is (n, e) and the private key is (n, d).
Prime numbers p and q are kept secret.
Larger values of p and q enhance security.

Encryption Process

To encrypt a message m, the receiver's public key (n, e) is used. Plaintext m must be an integer between 0 and n-1.
The ciphertext c is calculated as c = m^e mod n.

Decryption Process

To decrypt the ciphertext c, the receiver's private key (n, d) is used.
The plaintext m is calculated as m = c^d mod n.

Security Considerations

RSA's robustness comes from the difficulty of factoring large numbers; faster factorization algorithms would weaken RSA.
The size of the modulus n directly affects security. Larger values (often 2048 bits or greater) provide increased security.
The public exponent e is often kept small (like 65537) for efficiency in encryption. Smaller e saves computational time.
Strong random number generators are crucial in generating p and q to prevent attacks.

Limitations

RSA is computationally intensive, particularly with large numbers, which affects performance with large datasets.
RSA's strength depends on the difficulty of integer factorization—significant advancements in factorization techniques would compromise RSA.
Improper implementation or parameter selection can leave RSA vulnerable to attacks.

Applications

Digital signatures verify message senders.
Secure data transmission (e.g., HTTPS).
Key exchange protocols (e.g., Diffie-Hellman, often with RSA).
Digital certificates authenticate communication endpoints and validate digital certificates.

Summary of RSA Algorithm

RSA is a public-key cryptosystem, secure due to the difficulty of factoring large integers.
It generates distinct public and private keys.
Encryption uses the public key, and decryption uses the private key.
Security relies on keeping primes secret and using large integers.
Performance considerations influence implementation.
Applications are numerous, encompassing digital signatures, encryption, and certificate validation.

Description

This quiz covers the fundamentals of the RSA algorithm, a public-key cryptosystem essential for secure data transmission. It includes key generation steps and the mathematical principles that make RSA secure. Test your knowledge on digital signatures and encrypting data using RSA.