Glossary

11.2 Perfect secrecy and one-time pad

2 min read•july 25, 2024

, introduced by Claude Shannon, is the ultimate form of cryptographic security. It ensures that ciphertext reveals nothing about the , making it unbreakable even with unlimited computational power. This concept sets the gold standard for encryption.

The is a practical implementation of perfect secrecy. It uses a random key as long as the message, XORed with the plaintext. While theoretically unbreakable, it faces challenges in key management and practical implementation, limiting its real-world use.

Perfect Secrecy and One-Time Pad

Definition of perfect secrecy

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  • Perfect secrecy concept introduced by Claude Shannon establishes strongest notion of security in cryptography
  • Ciphertext reveals no information about plaintext meaning probability of message given ciphertext equals probability of message without ciphertext
  • Mathematically represented as P(M=m∣C=c)=P(M=m)P(M=m|C=c) = P(M=m) for all messages m and ciphertexts c
  • Implications create unbreakable encryption immune to any computational power giving attackers no advantage over random guessing
  • Achieving perfect secrecy requires key length at least as long as message and each key used only once

One-time pad encryption scheme

  • algorithm invented by Gilbert Vernam in 1917
  • Encryption process generates random key of same length as plaintext and performs bitwise between key and plaintext
  • Decryption process performs bitwise XOR operation between ciphertext and key
  • Key properties include truly random nature, same length as plaintext, and single-use requirement
  • Security properties achieve perfect secrecy and unconditional security
  • Mathematical representation for encryption: C=M⊕KC = M \oplus K and decryption: M=C⊕KM = C \oplus K

Proof of one-time pad's perfect secrecy

  • Information-theoretic approach based on probability theory and
  • Proof demonstrates uniform distribution of ciphertexts and independence between plaintext and ciphertext
  • Entropy concepts include H(M)H(M) for plaintext, H(K)H(K) for key, and H(C)H(C) for ciphertext
  • Entropy relationships show H(K)≥H(M)H(K) \geq H(M) and H(C)=H(M)H(C) = H(M)
  • Mutual information I(M;C)=0I(M;C) = 0 indicates no information leakage

Limitations of one-time pad implementation

  • Key management issues involve generation of truly random keys, secure distribution, and storage of large key volumes
  • Key length requirement must be at least as long as plaintext limiting practical message length
  • Single-use key constraint requires frequent key exchanges
  • Synchronization challenges ensure sender and receiver use same key and maintain key order for multiple messages
  • Vulnerability to modification attacks due to lack of protection makes it susceptible to malicious changes in ciphertext
  • Limited applicability makes it impractical for most real-world scenarios mainly used in highly sensitive communications (diplomatic channels, intelligence agencies)
  • Implementation complexities include ensuring true randomness in key generation and protecting against side-channel attacks (timing attacks, power analysis)

Key Terms to Review (16)

Asymmetric encryption: Asymmetric encryption is a type of encryption that uses a pair of keys: a public key, which can be shared with anyone, and a private key, which is kept secret by the owner. This method allows for secure communication and data transfer, as the public key encrypts the data while only the private key can decrypt it. This setup provides a foundational framework for secure digital communication, especially in the context of data integrity and authentication.
Conditional Entropy: Conditional entropy measures the amount of uncertainty remaining about a random variable given the knowledge of another random variable. It quantifies how much additional information is needed to describe one random variable when another is known, and helps in understanding relationships between variables in various contexts, such as communication and data analysis.
Confidentiality: Confidentiality refers to the principle of keeping sensitive information private and secure from unauthorized access or disclosure. It is essential in maintaining trust and protecting personal, corporate, or governmental data. In cryptography, confidentiality is achieved through encryption methods, which ensure that only authorized users can access the information, making it a vital aspect of secure communication.
Information Entropy: Information entropy is a measure of the unpredictability or uncertainty associated with a random variable. It quantifies the amount of information that is produced when one outcome of a random process occurs, and is essential for understanding how much information can be transmitted or stored in a system. This concept is foundational for analyzing data compression, coding schemes, and secure communication protocols.
Information theory: Information theory is a mathematical framework that deals with the quantification, storage, and communication of information. It provides tools for measuring the amount of information in messages and understanding how this information can be transmitted efficiently and securely over various channels. One of its key principles is the concept of entropy, which measures the uncertainty or randomness in a set of outcomes, directly linking it to ideas like perfect secrecy and methods such as the one-time pad.
Integrity: Integrity refers to the assurance that information is accurate, consistent, and protected from unauthorized modification. It is a crucial aspect of data security that ensures the integrity of data during storage, transmission, and processing. In the context of perfect secrecy and the one-time pad, integrity ensures that messages remain unaltered and trustworthy, providing confidence in the communication process.
Key reuse: Key reuse refers to the practice of using the same cryptographic key multiple times for encrypting different pieces of data. This practice can significantly compromise security, especially when considering the principles of perfect secrecy and the one-time pad method. The one-time pad is designed to be used with a key that is as long as the message, used only once, and kept completely secret. When keys are reused, it opens up potential vulnerabilities that can be exploited by attackers.
Key Space: Key space refers to the total number of possible keys that can be used in a cryptographic algorithm. The larger the key space, the more secure the encryption is, as it becomes increasingly difficult for an attacker to guess or brute-force the key. This concept is particularly important in the context of achieving perfect secrecy with techniques like the one-time pad, where each key must be unique and as long as the message itself.
Modular arithmetic: Modular arithmetic is a system of arithmetic for integers where numbers wrap around after reaching a certain value, known as the modulus. This concept is essential in cryptography, especially when it comes to ensuring secure communication by manipulating numbers in a way that can only be understood by those who possess the correct key. It plays a crucial role in various encryption techniques, allowing operations to stay within a limited range, thus facilitating operations like addition, subtraction, and multiplication under a modulus.
One-time pad: A one-time pad is an encryption technique that uses a single-use, random key that is as long as the message itself. This method ensures perfect secrecy because each character of the plaintext is combined with a character from the key, making it impossible to decrypt without having the exact key. The unique nature of this approach highlights its applications in secure communications, information-theoretic security principles, and its role in achieving perfect secrecy in cryptographic systems.
Perfect secrecy: Perfect secrecy is a concept in cryptography that ensures a message remains completely secure from an adversary, regardless of their computational power. This means that even with unlimited resources, an eavesdropper cannot gain any information about the plaintext from the ciphertext without the key. The idea is fundamentally linked to information-theoretic security principles and is most famously exemplified by the one-time pad encryption method, which achieves this level of security under specific conditions.
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.
Shannon's Theorem: Shannon's Theorem, also known as the Channel Capacity Theorem, defines the maximum rate at which information can be transmitted over a communication channel without error. This fundamental principle not only laid the groundwork for modern digital communication but also influences various fields like cryptography and data security, revealing the limits and capabilities of communication systems.
Symmetric encryption: Symmetric encryption is a cryptographic technique where the same key is used for both encryption and decryption of data. This method ensures that the data remains confidential, as only those with access to the secret key can decrypt the information. The strength of symmetric encryption relies heavily on the secrecy and complexity of the key, making it essential for secure communication.
Unbreakability: Unbreakability refers to the characteristic of a cryptographic system that ensures it cannot be deciphered or broken by any means, provided that certain conditions are met. This property is a cornerstone of perfect secrecy, where the information remains secure against any potential adversaries, and is notably achieved through the one-time pad method, which uses a key that is as long as the message itself and is never reused.
Xor operation: The xor operation, or exclusive or, is a binary operation that outputs true or 1 only when the inputs differ. This operation is crucial in cryptography, especially in achieving perfect secrecy when using techniques like the one-time pad. The xor operation ensures that each bit of the plaintext is combined with a corresponding bit of the key, creating a ciphertext that is unpredictable if the key is truly random and used only once.
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