3.2 Public-key cryptography and RSA

Public-key cryptography revolutionized secure communication by using key pairs: public for encryption and private for decryption. This solved the key distribution problem of symmetric-key systems and enabled digital signatures for authenticity and integrity verification.

RSA, a popular public-key system, relies on the difficulty of factoring large numbers. It uses a public key (n,e) for encryption and a private key (n,d) for decryption. RSA supports encryption, decryption, and digital signatures, balancing security and performance through key size choices.

Public-key cryptography: Concepts and advantages

Key concepts and properties

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• Public-key cryptography uses a pair of keys: a public key for encryption and a private key for decryption
• The public key is freely distributed, while the private key is kept secret by the owner
• Enables secure communication without the need for a shared secret key (eliminates key distribution problem faced by symmetric-key cryptography)
• Allows for the creation of digital signatures using the private key, enabling verification of authenticity and integrity using the public key

Applications and benefits

• Solves the key distribution problem by eliminating the need for a secure channel to exchange secret keys
• Enables key exchange protocols (Diffie-Hellman) that allow parties to establish a shared secret key over an insecure channel
• Provides the foundation for secure communication protocols (SSL/TLS) and public key infrastructure (PKI)
• Offers non-repudiation through digital signatures, as the signer cannot deny having signed the message
• Facilitates secure communication and transactions in various domains (e-commerce, email, cryptocurrencies)

Mathematical foundations of RSA

Integer factorization problem

• RSA security relies on the difficulty of factoring the product of two large prime numbers (RSA modulus)
• Factoring large numbers is believed to be computationally infeasible, providing the basis for RSA's security
• The integer factorization problem states that given a positive integer n, find its prime factors
• As the size of the modulus increases, the difficulty of factoring grows exponentially, making RSA more secure

Key generation and components

• Key generation involves selecting two distinct prime numbers, p and q, and calculating their product n = pq (modulus for public and private keys)
• The public key consists of the modulus n and a public exponent e, typically chosen as a small odd integer coprime to (p-1)(q-1)
• The private key consists of the modulus n and a private exponent d, calculated as the modular multiplicative inverse of e modulo (p-1)(q-1)
• The security assumption is that an adversary cannot efficiently factor the modulus n to obtain p and q, preventing them from calculating the private key
• Proper selection of key components (p, q, e, d) is crucial for ensuring the security and efficiency of RSA

RSA applications: Encryption, decryption, and signatures

Encryption and decryption process

• RSA encryption converts the plaintext message into an integer m (0 ≤ m < n) and computes the ciphertext c ≡ m^e (mod n) using the public key (n, e)
• RSA decryption recovers the original plaintext message m by computing m ≡ c^d (mod n) using the private key (n, d)
• The encryption and decryption processes are based on the mathematical properties of modular exponentiation and the relationship between the public and private exponents

Digital signatures and verification

• To create an RSA digital signature, the signer computes a hash of the message using a secure hash function (SHA-256) and encrypts the hash value using their private key to obtain the signature s ≡ hash(m)^d (mod n)
• To verify an RSA digital signature, the recipient decrypts the signature using the signer's public key to obtain the hash value and compares it with the independently computed hash of the received message
• Digital signatures provide authentication, integrity, and non-repudiation, ensuring the message originated from the claimed signer and has not been tampered with

Combining encryption and signatures

• RSA encryption and digital signatures can be combined to provide both confidentiality and authentication in secure communication protocols (SSL/TLS)
• The sender encrypts the message using the recipient's public key and signs the encrypted message using their own private key
• The recipient verifies the signature using the sender's public key and decrypts the message using their own private key
• This combination ensures that only the intended recipient can read the message and that the message originated from the claimed sender

RSA security vs performance trade-offs

Key size and security

• RSA security depends on the size of the modulus n, with larger key sizes providing higher levels of security but increasing computational overhead
• Common RSA key sizes range from 1024 to 4096 bits, with 2048 bits being the recommended minimum for long-term security as of 2021
• As computing power advances, larger key sizes may be necessary to maintain adequate security against potential attacks (factoring algorithms, quantum computing)

Performance considerations

• RSA is computationally intensive compared to symmetric-key algorithms, making it unsuitable for directly encrypting large amounts of data
• Hybrid encryption is often used, where RSA encrypts a symmetric key (AES), which is then used to encrypt the actual data
• The choice of public exponent e affects both security and performance, with smaller values (e = 3 or e = 65537) being more efficient but potentially more vulnerable if not implemented correctly
• Proper implementation and optimization techniques (Chinese Remainder Theorem, Montgomery multiplication) can improve RSA performance

Side-channel attacks and countermeasures

• Side-channel attacks (timing attacks, power analysis) can potentially leak information about the private key during RSA operations
• Careful implementation and the use of countermeasures (constant-time operations, blinding) are necessary to mitigate these risks
• Proper padding schemes (OAEP) should be used when encrypting with RSA to prevent attacks (padding oracle attack) and ensure the security of the encryption scheme
• Regular security audits and adherence to best practices are essential for maintaining the security of RSA implementations in real-world applications