RSA and Elliptic Curve Cryptography are key players in modern encryption. RSA uses prime factorization and modular arithmetic, while ECC leverages the math of elliptic curves. Both create secure public-private key pairs for encryption and digital signatures.

These systems form the backbone of asymmetric cryptography. RSA offers robust security but requires larger keys, while ECC provides comparable protection with smaller keys. Understanding their strengths and applications is crucial for implementing secure communication systems.

RSA Cryptosystem

Fundamental Principles of RSA

• RSA algorithm forms the foundation of public key cryptography utilizing prime factorization and modular arithmetic
• Prime factorization involves breaking down a composite number into its prime factors, a computationally difficult task for large numbers
• Modular arithmetic performs calculations with a fixed modulus, resulting in remainders within a specific range
• Euler's totient function φ(n) calculates the count of numbers coprime to n, crucial for RSA key generation
• RSA security relies on the difficulty of factoring large composite numbers into their prime factors

RSA Key Generation and Usage

• Key generation process creates public and private key pairs for secure communication
• Public key consists of two components: modulus n (product of two large primes) and public exponent e
• Private key includes modulus n and private exponent d, calculated using the extended Euclidean algorithm
• Encryption transforms plaintext message m into ciphertext c using the formula $c = m^e \mod n$
• Decryption recovers the original message m from ciphertext c using the formula $m = c^d \mod n$
• RSA algorithm ensures that only the holder of the private key can decrypt messages encrypted with the corresponding public key

RSA Implementation and Considerations

• Key size significantly impacts RSA security, with larger keys providing stronger protection against attacks
• Common RSA key sizes range from 2048 to 4096 bits, balancing security and computational efficiency
• RSA padding schemes (OAEP, PSS) enhance security by adding randomness to messages before encryption
• Side-channel attacks exploit physical implementations of RSA, requiring countermeasures in hardware and software
• RSA operations can be optimized using the Chinese Remainder Theorem for faster decryption and signing
• Quantum computers pose a theoretical threat to RSA security, prompting research into quantum-resistant alternatives

Elliptic Curve Cryptography (ECC)

Fundamentals of Elliptic Curves

• Elliptic Curve Cryptography utilizes mathematical properties of elliptic curves for secure key exchange and digital signatures
• Elliptic curves consist of points satisfying the equation $y^2 = x^3 + ax + b$ over a finite field
• ECC operations involve point addition and scalar multiplication on the elliptic curve
• Discrete logarithm problem on elliptic curves forms the basis of ECC security, making it computationally infeasible to determine the scalar given a point and its multiple
• ECC offers comparable security to RSA with significantly smaller key sizes, reducing computational and storage requirements

ECC Algorithms and Applications

• ECDSA (Elliptic Curve Digital Signature Algorithm) provides digital signature functionality using elliptic curves
• ECDSA signing process involves generating a random value k and computing two components of the signature (r, s)
• ECDSA verification confirms the authenticity of a signature using the signer's public key and the message hash
• ECDH (Elliptic Curve Diffie-Hellman) enables secure key exchange between parties using elliptic curve operations
• ECC finds applications in secure messaging apps (Signal), cryptocurrencies (Bitcoin), and TLS/SSL protocols

ECC Security and Performance Comparisons

• Key size comparison between ECC and RSA shows ECC requires significantly smaller keys for equivalent security levels
• 256-bit ECC key provides comparable security to a 3072-bit RSA key, resulting in faster computations and lower bandwidth usage
• ECC demonstrates superior performance in resource-constrained environments (mobile devices, IoT)
• Standardized elliptic curves (NIST curves, Curve25519) offer vetted parameters for secure ECC implementations
• Side-channel attacks on ECC implementations necessitate constant-time algorithms and other countermeasures
• Post-quantum cryptography research explores ECC variants resistant to quantum computer attacks (supersingular isogeny-based cryptography)