11.3 Public-key cryptography and information theory
2 min read•july 25, 2024
Public-key cryptography revolutionizes secure communication by using two keys: public for encryption and private for decryption. This system eliminates the need for secure key exchange and enhances scalability, making it ideal for large networks like the internet.
While public-key systems can't achieve perfect secrecy due to the openly available encryption key, they offer computational security. This security is based on hard mathematical problems, making it infeasible for attackers to break the encryption without significant computational resources.
Public-Key Cryptography Fundamentals
Public-key vs symmetric-key cryptography
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- Public-key cryptography employs two distinct keys (public and private) enabling secure communication without prior key exchange
- Public key distributed openly allows anyone to encrypt messages or verify signatures
- Private key kept confidential by owner decrypts messages or signs documents
- Asymmetric encryption process encrypts with public key, decrypts only with corresponding private key
- Digital signatures signed with private key, verified using public key
- Advantages include eliminating secure key exchange, improved scalability for large networks (internet), enhanced security for digital signatures, non-repudiation capability
Information-theoretic security in public-key systems
- Perfect secrecy unattainable in public-key systems due to openly available encryption key
- Information leakage occurs as reveals some details about , bounded by public key size and message length
- Semantic security computationally equivalent to perfect secrecy based on hardness assumptions
- Probabilistic encryption introduces randomness enhancing security, reducing information leakage in repeated encryptions (padding schemes)
Complexity and Security in Public-Key Cryptography
Computational complexity for cryptosystem security
- One-way functions easily computed forward but computationally infeasible to reverse (hash functions)
- Trapdoor one-way functions allow easy reversal with additional information (private key)
- Hard mathematical problems underpin security (integer factorization, discrete logarithm, elliptic curve discrete logarithm)
- Cryptosystem strength directly tied to difficulty of underlying mathematical problem
- Computational security considers system secure if breaking requires infeasible computational resources (years of supercomputer time)
Key length and security relationship
- correlates with longer keys generally provide higher security
- Security increases exponentially with linear key length increase
- Information content of keys measured in bits of affects attacker's search space
- Brute-force attacks time complexity grows exponentially with key length for n-bit keys
- Quantum computing considerations may require longer keys for equivalent security (Shor's algorithm)
- Key length recommendations based on current computational capabilities updated by standards organizations (NIST)
- Trade-offs between security and efficiency balance needed as longer keys increase computational overhead
Key Terms to Review (18)
Certificate authority: A certificate authority (CA) is a trusted entity that issues digital certificates, which are used to verify the identity of individuals, organizations, or devices in digital communications. By acting as a mediator in public-key cryptography, CAs play a crucial role in ensuring secure online transactions and communications, enabling users to trust the identities of the parties they interact with.
Ciphertext: Ciphertext is the result of encryption performed on plaintext, transforming readable data into an unreadable format to ensure confidentiality and security. It plays a crucial role in cryptography, particularly in public-key cryptography, where it allows users to send secure messages over insecure channels. Ciphertext is designed so that only authorized parties with the correct decryption key can revert it back to its original readable form.
Ciphertext-only attack: A ciphertext-only attack is a type of cryptanalysis where an attacker only has access to the ciphertext, which is the encrypted version of the plaintext, without any knowledge of the corresponding plaintext or the key used for encryption. This form of attack is crucial in assessing the security of cryptographic algorithms, particularly in the context of public-key cryptography, as it tests the system's ability to withstand attacks that do not rely on prior knowledge of the encryption parameters.
Claude Shannon: Claude Shannon was an American mathematician and electrical engineer, widely regarded as the father of Information Theory, who developed key concepts that quantify information, enabling efficient communication systems. His pioneering work laid the foundation for understanding how to measure and transmit information in the presence of noise, connecting directly to fundamental principles that drive modern telecommunications and data processing.
Diffie-Hellman Key Exchange: Diffie-Hellman Key Exchange is a method used to securely share cryptographic keys over a public channel. This technique allows two parties to establish a shared secret key, which can then be used for encrypted communication, without having to send the key itself over the network. The strength of this method lies in its reliance on the mathematical properties of modular arithmetic and the difficulty of solving discrete logarithm problems.
Digital signature: A digital signature is a cryptographic technique used to validate the authenticity and integrity of a digital message or document. It employs a pair of keys, one private and one public, which ensure that the sender's identity can be confirmed, and that the message has not been altered during transmission. This makes digital signatures essential for secure communication in various applications, ensuring non-repudiation and trust in electronic transactions.
Elliptic Curve Cryptography: Elliptic Curve Cryptography (ECC) is a form of public-key cryptography based on the algebraic structure of elliptic curves over finite fields. This method allows for smaller key sizes compared to traditional cryptosystems, while still providing a high level of security. The efficiency and strength of ECC make it particularly attractive for securing communications and data in a world that increasingly demands robust encryption.
Entropy: Entropy is a measure of uncertainty or randomness in a set of data, reflecting the amount of information that is missing when predicting the value of a random variable. In various contexts, entropy quantifies the average amount of information produced by a stochastic source of data, thus providing insights into the efficiency of coding schemes and the capacity of communication systems.
Key Distribution: Key distribution refers to the process of sharing cryptographic keys among users in a secure manner. It is crucial for enabling secure communication and ensuring that only authorized parties can access the shared information. The effectiveness of key distribution impacts various applications in technology, enhances security in data transmission, and serves as a foundational element in cryptographic systems.
Key Length: Key length refers to the size of a cryptographic key, usually measured in bits, that is used in encryption algorithms to secure data. A longer key length generally increases the security of the encryption, making it harder for unauthorized parties to decrypt the information without the correct key. In the context of information-theoretic security and public-key cryptography, the choice of key length plays a crucial role in determining the overall strength of the encryption method employed.
Man-in-the-middle attack: A man-in-the-middle attack is a form of cyber-attack where an attacker secretly intercepts and relays messages between two parties who believe they are communicating directly with each other. This type of attack is particularly relevant in the context of public-key cryptography and information theory, as it can compromise the confidentiality and integrity of communication by allowing the attacker to read, alter, or inject messages without the victims' knowledge.
Mutual Information: Mutual information is a measure of the amount of information that one random variable contains about another random variable. It quantifies the reduction in uncertainty about one variable given knowledge of the other, connecting closely to concepts like joint and conditional entropy as well as the fundamental principles of information theory.
PGP: PGP, or Pretty Good Privacy, is an encryption program that provides cryptographic privacy and authentication for data communication. It uses a combination of symmetric-key cryptography for speed and public-key cryptography for secure key exchange, making it a powerful tool for ensuring the confidentiality and integrity of messages in various digital communications.
Plaintext: Plaintext refers to the original, unencrypted data or message that is readable without any special processing. It serves as the input for encryption algorithms, transforming it into ciphertext, which is unintelligible to anyone without the appropriate decryption key. The significance of plaintext is highlighted in cryptographic systems, where its security relies on the strength of the encryption method used, such as the one-time pad or public-key cryptography.
RSA: RSA is an asymmetric cryptographic algorithm used for secure data transmission, relying on the mathematical properties of prime numbers. It is one of the first public-key cryptosystems and is widely used for secure communication over the internet, ensuring that data can be encrypted and decrypted using two different keys: a public key and a private key. The security of RSA hinges on the difficulty of factoring large composite numbers, making it a fundamental component in public-key cryptography.
Security level: The security level refers to the measure of strength or robustness of a cryptographic system in protecting data from unauthorized access or tampering. It encompasses various aspects such as the difficulty of breaking the encryption, the complexity of key management, and the potential vulnerabilities that might be exploited. The security level is critical in ensuring that communication remains confidential and authentic, especially in systems that utilize asymmetric encryption methods.
SSL/TLS: SSL (Secure Sockets Layer) and TLS (Transport Layer Security) are cryptographic protocols designed to provide secure communication over a computer network. These protocols ensure the privacy and integrity of data transmitted between clients and servers by using encryption, authentication, and message integrity checks. As a foundation for secure online transactions, SSL/TLS has become essential in protecting sensitive information such as credit card numbers, personal data, and login credentials.
Whitfield Diffie: Whitfield Diffie is a pioneering cryptographer best known for his groundbreaking work in public-key cryptography, which revolutionized secure communications over the internet. His introduction of the Diffie-Hellman key exchange protocol laid the foundation for how information is securely transmitted between parties without needing to share secret keys in advance. This innovation has had significant implications for securing online transactions and protecting sensitive information in various fields.