Information-Theoretic Security

The defining feature of information-theoretic security is unconditional security. Unlike computational security (e.g., RSA, ECC), which relies on the assumed difficulty of mathematical problems, this model guarantees security even against an adversary with infinite computational resources. The proof of security is based on probability theory and information entropy, not on unproven complexity assumptions.

The One-Time Pad is the canonical example of an information-theoretically secure cipher. For perfect secrecy, it requires:

• A truly random key as long as the plaintext.
• The key is used only once (hence 'one-time').
• The key is kept completely secret. When these conditions are met, the ciphertext reveals zero information about the plaintext, as proven by Claude Shannon's work.

Formalized by Claude Shannon, perfect secrecy is achieved when the ciphertext provides no information about the plaintext. Mathematically, the probability of a plaintext P given a ciphertext C is equal to the probability of the plaintext alone: Pr[P|C] = Pr[P]. This means observing the ciphertext does not change an attacker's knowledge about the original message. The One-Time Pad is the only system proven to meet this criterion.

The major practical limitation is the key distribution problem. To achieve perfect secrecy, a secret key as long as the total message volume must be established in advance over a perfectly secure channel. This requirement makes it impractical for most general communication, leading to the development of computational cryptography and quantum key distribution (QKD) as potential solutions for scalable secure key exchange.

Information-theoretic security is crucial for secret sharing schemes, like Shamir's Secret Sharing. A secret is split into n shares, where any k shares (the threshold) can reconstruct it, but any k-1 shares reveal absolutely no information about the secret. This property is information-theoretic; it doesn't rely on an adversary's computational limits, making it future-proof against advances in computing.

This highlights the core trade-off between the two main cryptographic models:

• Information-Theoretic: Unconditional security, proven. Burden is on key management (size, distribution).
• Computational Security: Security based on computational hardness (e.g., factoring, discrete log). Enables efficient public-key crypto and small keys, but is vulnerable to algorithmic breakthroughs (e.g., quantum computers). Most modern cryptography (blockchains, TLS) uses computational security for practicality.

The one-time pad is the only cryptosystem proven to be information-theoretically secure. It requires a pre-shared random key that is:

• As long as the message.
• Truly random.
• Used only once.

Any deviation (key reuse, non-randomness) breaks the perfect secrecy property. While impractical for most modern applications due to key distribution challenges, it is the foundational example of information-theoretic security.

A threshold secret-sharing scheme where a secret (like a private key) is split into n shares. The scheme is information-theoretically secure because:

• Possessing fewer than the threshold k shares reveals zero information about the original secret.
• Security relies on polynomial mathematics, not computational limits.

It is widely used for secure, decentralized custody of cryptographic keys in multi-signature wallets and DAO treasuries.

Protocols like BB84 use quantum mechanics to establish a shared secret key between two parties. Security is based on the laws of physics (e.g., the no-cloning theorem), making it information-theoretically secure against eavesdropping during the key exchange phase. Any interception attempt introduces detectable errors. The final key is then used in a one-time pad for communication.

A protocol that allows a client to retrieve an item from a database server without the server learning which item was retrieved. Information-theoretically secure PIR schemes achieve this by distributing the database across multiple non-colluding servers. The client's query privacy is unconditional, not based on computational assumptions.

In the theoretical framework of MPC, security is defined by comparing a real protocol execution to an ideal functionality where a trusted party computes the function. Information-theoretic MPC protocols (secure against passive adversaries) achieve this ideal with unconditional security for a honest majority of participants, using secret sharing and verifiable secret sharing as core building blocks.

Information-theoretic security has inherent limitations:

• Often requires trusted setup or pre-shared keys.
• Can be impractical for bandwidth or storage (e.g., one-time pad).
• Contrast with computational security (used in Bitcoin, Ethereum), which assumes an adversary's computational power is bounded. Most blockchain cryptography (ECDSA, SHA-256) is computationally secure, relying on the hardness of problems like discrete logarithms.

The canonical example of information-theoretic security. A message is encrypted by combining it with a random key of equal length using XOR. Security is perfect if the key is:

• Truly random
• Used only once
• Kept completely secret While impractical for most blockchain data due to key distribution, it's the benchmark for unconditional security.

Methods like Shamir's Secret Sharing split a secret (e.g., a private key) into multiple shares. The secret can only be reconstructed with a minimum threshold of shares. The scheme is information-theoretically secure because possessing fewer than the threshold shares reveals zero information about the original secret. This is foundational for multi-party computation (MPC) and decentralized custody solutions.

The primary barrier to using information-theoretic security in decentralized networks. Systems like the OTP require a pre-shared secret key as long as the total communication. In a trustless, peer-to-peer environment like a blockchain, establishing and managing these keys for all participants is infeasible, leading to the dominance of computational security models (e.g., RSA, ECC) which rely on computational hardness assumptions.

An enhancement to secret sharing that adds cryptographic proofs. It allows participants to verify that their shares are consistent and derived from a single secret, even if the dealer is malicious. This prevents the dealer from sending invalid shares. VSS is crucial for Byzantine Fault Tolerant protocols and secure distributed key generation in validator setups.

Highlights the fundamental trade-off in cryptography:

• Information-Theoretic: Security against unlimited computational power. Relies on probability and information theory.
• Computational (Practical): Security based on the computational hardness of problems (e.g., factoring, discrete log). Assumes adversaries have bounded resources. Blockchains almost universally adopt computational security for efficiency, making the safety of assets like Bitcoin contingent on the difficulty of solving cryptographic puzzles.

Information-theoretic schemes are inherently secure against quantum computers because their security doesn't rely on computational problems a quantum machine could solve. As quantum computing advances, principles from information-theoretic security are being adapted into post-quantum cryptography, though most proposals remain computationally secure but based on different mathematical problems.

Information-theoretic schemes, such as Shamir's Secret Sharing or one-time pads, require perfect key management. The secret key must be:

• Generated with true randomness.
• Distributed to all parties over a perfectly secure channel.
• As long as the message itself.
• Used only once and then destroyed. Any flaw in this process, such as a compromised random number generator or a key reuse, collapses the security to a computational level.

The guarantees come at a massive cost in resources. For example, a one-time pad requires a pre-shared key exactly as long as all future communication, making it impractical for ongoing systems like blockchains that process gigabytes of data daily. Verifiable Secret Sharing (VSS) and Multi-Party Computation (MPC) protocols with information-theoretic security involve intense communication rounds and cryptographic operations, creating latency and scalability bottlenecks.

These proofs assume an adversary with unlimited computational power but limited to specific attack models. In practice, security can fail if:

• The adversary has additional capabilities (e.g., side-channel attacks, physical tampering).
• The model is incorrect (e.g., assuming a passive adversary when an active adversary exists).
• There are implementation bugs outside the mathematical model. Thus, a theoretically secure protocol can be broken in practice by attacks its model didn't consider.

Modern cryptography, including blockchain foundations like ECDSA and SHA-256, relies on computational security. It assumes an adversary's computational resources are bounded (e.g., cannot solve the elliptic curve discrete logarithm problem in feasible time). This trade-off is necessary for efficiency. Information-theoretic security is often reserved for critical, small-scale components (e.g., distributing a master key seed) within a larger computationally-secure system.

Many efficient cryptographic schemes used in practice are proven secure in the Random Oracle Model. This model treats a hash function (like SHA-256) as a perfect, public random function—an information-theoretic idealization. While no real hash function can be a true random oracle, this model provides a useful bridge, allowing for proofs of schemes that would otherwise be impractical under purely information-theoretic or standard computational models.

Pure information-theoretic security is rare in end-to-end systems. Its primary modern applications are in specialized, high-assurance components:

• Secure Multi-Party Computation (MPC) for threshold signatures.
• Quantum Key Distribution (QKD) for key exchange.
• The theoretical security proofs of commitment schemes and zero-knowledge arguments. These are often hybridized with computational assumptions to create systems that are practically efficient while retaining strong security properties for core secrets.

The most common paradigm in modern cryptography, where security relies on the computational hardness of a problem (e.g., factoring large integers). It is assumed an adversary has bounded computational resources (polynomial time). This is in direct contrast to information-theoretic security, which makes no such assumptions. Most blockchain primitives, like digital signatures and public-key encryption, are computationally secure.

The foundational concept of information-theoretic security for encryption, defined by Claude Shannon. A cipher achieves perfect secrecy if the ciphertext reveals zero information about the plaintext, even to an adversary with unlimited computing power. Formally, the probability of a plaintext given a ciphertext is equal to its prior probability: P(M|C) = P(M). The One-Time Pad is the only known system to provably achieve this, requiring a key as long as the message and never reused.

The canonical example of an information-theoretically secure cipher. It operates by combining a plaintext message with a random key of equal length using bitwise XOR. Security relies on the key being:

• Truly random
• As long as the message
• Used only once If these conditions are met, the ciphertext is mathematically indistinguishable from random noise and provides perfect secrecy. Its impractical key management makes it unsuitable for most systems.

A synonym for information-theoretic security. A system is unconditionally secure if it cannot be broken even with infinite computational resources. The security proof relies solely on information theory and probability, not on unproven computational assumptions. This is the strongest form of security but is extremely difficult to achieve for complex functionalities beyond simple encryption and secret sharing.

A primary application of information-theoretic security. Schemes like Shamir's Secret Sharing allow a secret (e.g., a private key) to be split into n shares. The secret can only be reconstructed when a threshold t of shares are combined. Crucially, possessing fewer than t shares provides zero information about the secret, even with unlimited computing power. This is foundational for secure multi-party computation (MPC) and distributed key generation.

The core resource for information-theoretic security. Entropy is a measure of uncertainty or randomness. For a key to be information-theoretically secure, it must possess full entropy—it must be truly unpredictable. The security of systems like the One-Time Pad collapses if the key is generated by a pseudorandom process. This highlights the critical role of cryptographically secure random number generators in bridging theory and practice.