4.1 RSA cryptosystem

RSA is a cornerstone of public key cryptography, using prime numbers and modular arithmetic to secure digital communications. It's based on the difficulty of factoring large numbers, allowing for secure key exchange, encryption, and digital signatures.

RSA's strength lies in its mathematical foundations, but it requires careful implementation to avoid vulnerabilities. It's widely used in everyday secure communications, from web browsing to email, making it a crucial part of modern cybersecurity infrastructure.

Mathematical Foundations of RSA

Prime Numbers and Modular Arithmetic

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• RSA cryptosystem relies on mathematical principles of modular arithmetic and number theory
• Prime numbers form the cornerstone of RSA security due to the difficulty of factoring products of large primes
• Modular arithmetic underpins key RSA operations (encryption, decryption, key generation)
• Multiplicative inverses in modular arithmetic play a fundamental role in RSA key generation and decryption processes
• Euler's totient function φ(n) calculates integers coprime to n, crucial for RSA key generation
  • For a prime p, φ(p) = p - 1
  • For n = pq (product of two primes), φ(n) = (p-1)(q-1)

Theorems and Optimizations

• Chinese Remainder Theorem optimizes RSA decryption operations, especially for large moduli
  • Allows computations to be performed separately modulo p and q, then combined
• Fermat's Little Theorem provides a basis for RSA correctness
  • States that for prime p and integer a not divisible by p: $a^{p-1} \equiv 1 \pmod{p}$
• Euler's Theorem generalizes Fermat's Little Theorem for any positive integer n
  • States that for coprime a and n: $a^{\phi(n)} \equiv 1 \pmod{n}$
• Fast modular exponentiation algorithms (square-and-multiply) enhance RSA efficiency
  • Reduces number of multiplications needed for large exponents

RSA Encryption and Decryption

Key Generation Process

• Select two large prime numbers p and q (typically hundreds of digits long)
• Compute modulus n = pq and φ(n) = (p-1)(q-1)
• Choose public exponent e coprime to φ(n)
  • Common choices include 65537 (balance of security and efficiency)
• Calculate private exponent d as modular multiplicative inverse of e modulo φ(n)
  • Solve equation: $ed \equiv 1 \pmod{\phi(n)}$
• Public key consists of (e, n), private key consists of (d, n)

Encryption and Decryption Processes

• RSA encryption transforms plaintext m to ciphertext c using public key (e, n)
  • Compute: $c \equiv m^e \pmod{n}$
• RSA decryption recovers plaintext m from ciphertext c using private key (d, n)
  • Compute: $m \equiv c^d \pmod{n}$
• Correctness of RSA relies on Euler's Theorem and properties of modular exponentiation
• Padding schemes (OAEP) enhance RSA encryption security
  • Prevent attacks like padding oracle attacks and chosen-ciphertext attacks
• Chinese Remainder Theorem optimizes decryption and signing processes
  • Compute $m_p \equiv c^d \pmod{p}$ and $m_q \equiv c^d \pmod{q}$
  • Combine results to obtain m mod n

Security of RSA

Computational Hardness and Algorithms

• RSA security relies on difficulty of integer factorization problem
• Best classical factoring algorithms (General Number Field Sieve) have subexponential time complexity
  • Running time approximately $\exp((c+o(1))(\ln n)^{1/3}(\ln \ln n)^{2/3})$
• Quantum computers using Shor's algorithm could theoretically break RSA in polynomial time
  • Poses significant threat to long-term RSA security
• RSA modulus size directly impacts system security
  • Current recommendations suggest minimum 2048 bits for adequate protection
  • Larger key sizes (3072 or 4096 bits) provide stronger security margins

Practical Security Considerations

• Side-channel attacks can compromise RSA security if not properly mitigated
  • Timing attacks exploit variations in operation time
  • Power analysis attacks analyze power consumption patterns
• Choice of public exponent e affects performance and security
  • Small values (65537) common for efficiency
  • Very small values (3) can lead to vulnerabilities in certain implementations
• Proper key management crucial for maintaining RSA security
  • Secure generation of keys using reliable random number generators
  • Safe storage of private keys (hardware security modules)
  • Timely destruction of compromised or outdated keys

Applications of RSA

Secure Communication and Key Exchange

• RSA often used to encrypt symmetric keys in hybrid cryptosystems
  • Combines efficiency of symmetric encryption with security of asymmetric encryption
• Widely implemented in secure communication protocols
  • TLS/SSL for secure web browsing (HTTPS)
  • Secure email systems (S/MIME, PGP)
• Used in various authentication mechanisms
  • SSH for secure remote access
  • VPN systems for secure network connections

Digital Signatures and Certificates

• RSA digital signatures provide integrity and non-repudiation
  • "Encrypt" message digest with private key
  • Verify signature using corresponding public key
• Signature process requires secure hash function (SHA-256, SHA-3)
  • Creates fixed-size message digest before signing
• Padding schemes (PSS) essential for secure RSA signatures
  • Prevent existential forgery attacks
• Certificate Authorities use RSA to sign digital certificates
  • Forms crucial component of Public Key Infrastructure (PKI)
• X.509 certificates commonly use RSA for key exchange and signatures
  • Used in HTTPS, email security, code signing