In blockchain and cryptographic systems, correlated randomness is a foundational technique for protocols requiring secure collaboration, such as threshold signatures, secure multi-party computation (MPC), and distributed key generation (DKG). Unlike independent randomness, where each party's random value is completely secret and unrelated, correlated randomness ensures the values are linked by a public correlation function—like summing to zero or being points on a shared polynomial—while keeping each party's specific share private. This allows a group to perform a joint computation where the outcome depends on their collective secret inputs, without any single party learning the others' secrets or controlling the final result.
LABS
Glossary
Correlated Randomness
Correlated randomness is pre-distributed random data shared among parties in a Multi-Party Computation (MPC) protocol, which is correlated in a specific way to improve the efficiency of the online computation phase.
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definition
CRYPTOGRAPHIC PRIMITIVE
What is Correlated Randomness?
Correlated randomness refers to a cryptographic protocol where multiple parties generate random values that are individually unpredictable but share a predetermined mathematical relationship, enabling secure multi-party computation without a trusted dealer.
The generation of correlated randomness typically relies on commitment schemes and verifiable secret sharing (VSS). In a common setup, each party generates a secret random value and commits to it by broadcasting a cryptographic commitment, such as a Pedersen commitment. They then engage in a protocol to verifiably share a function of these secrets, like a sum or a product. The correlation is established by the mathematical properties of the sharing protocol, ensuring that even if some parties are malicious, they cannot bias the final correlated output beyond the protocol's security guarantees. This process eliminates the need for a trusted third party to distribute the correlated shares.
A primary application is in threshold cryptography, where a private key is split into shares among multiple parties. Using correlated randomness, these parties can collaboratively generate a public/private key pair in a distributed manner, with the private key never existing in one place. Later, they can produce a valid signature—a threshold signature—only when a threshold number of parties cooperate, with the signature being indistinguishable from one made by a single key. This enhances security for wallets and validator nodes by removing single points of failure. Other use cases include random beacons for leader election in consensus and private set intersection protocols.
Implementing correlated randomness securely requires careful defense against active adversaries who may deviate from the protocol. Robust constructions use information-theoretic or cryptographic security models to ensure that the correlation holds even if a subset of parties is corrupted. Protocols like DKG based on Feldman or Pedersen VSS, and MPC frameworks like SPDZ, employ correlated randomness as a setup phase. The security proofs demonstrate that the correlated values appear uniformly random to any coalition of adversaries below the corruption threshold, preserving the unpredictability and fairness of the subsequent computation.
how-it-works
MECHANISM
How Correlated Randomness Works
An explanation of the cryptographic protocol that allows multiple parties to generate a single, unpredictable random value from their individual secret shares.
Correlated randomness is a cryptographic primitive where multiple participants, or parties, each generate a secret random value, or share, which collectively determines a final, unpredictable random output that none could have predetermined alone. This process, often implemented via a Verifiable Random Function (VRF) or commit-reveal scheme, ensures the result is both unbiased and publicly verifiable. The core property is that while each party's contribution is independent and secret, the final output is a deterministic function of all contributions, creating a shared random beacon resistant to manipulation by any single participant or a colluding minority.
The protocol typically operates in distinct phases to guarantee security and fairness. First, in the commit phase, each party generates a random secret and publishes a cryptographic commitment, such as a hash of their secret, locking in their choice without revealing it. Next, in the reveal phase, all parties disclose their original secrets. The final random value is then computed by aggregating all revealed secrets, for example, by hashing their concatenation. If a party fails to reveal a valid secret matching their commitment, they are penalized, and their deposit may be slashed, ensuring protocol adherence.
This mechanism is foundational for blockchain applications requiring decentralized and trustless randomness. Key use cases include: - Validator/Leader Election in Proof-of-Stake networks, where the next block proposer is chosen fairly. - Shard Allocation in sharded blockchains, assigning nodes to committees unpredictably. - Gaming and NFTs, for provably fair loot box mechanics or trait generation. By removing the need for a trusted third-party oracle, correlated randomness enables these systems to be both secure and decentralized, as the random output is generated by the network for the network.
The security model hinges on the assumption of an honest majority among participants. An adversary controlling more than a threshold of participants (e.g., >50% in many schemes) could potentially bias the outcome. Advanced implementations, like Drand, utilize threshold cryptography to distribute trust even further, requiring a threshold of participants to collaborate to produce the randomness, thereby increasing resilience against attacks. This makes correlated randomness not just a tool for fairness, but a critical component for the security infrastructure of decentralized systems.
key-features
CORRELATED RANDOMNESS
Key Features and Purpose
Correlated randomness is a cryptographic primitive that allows multiple parties to generate random values that are individually unpredictable but share a pre-defined mathematical relationship, enabling secure coordination in decentralized systems.
01
Verifiable Random Functions (VRFs)
A core cryptographic tool for generating correlated randomness. A Verifiable Random Function (VRF) produces a random output and a cryptographic proof that the output was correctly generated from a specific input and secret key. This allows the result to be publicly verified by anyone, ensuring unpredictability and public verifiability without revealing the secret.
- Key Property: The output appears random, even to the party who generated it, until the proof is published.
- Use Case: Used in blockchain consensus (e.g., Algorand's leader election) and on-chain lotteries.
02
Threshold Cryptography
A method to distribute the generation of correlated randomness across a group. In threshold cryptography, a secret (like a random number) is split into shares among n participants. The secret can only be reconstructed when a threshold number t of participants collaborate.
- Security: No single party or small coalition (less than
t) can predict or control the final random value. - Application: Essential for Distributed Key Generation (DKG) in validator committees for protocols like Drand, ensuring the beacon output is unbiased and resilient to malicious actors.
03
Randomness Beacons
Public services that emit a continuous stream of verifiable, unpredictable random values at regular intervals. Beacons like Drand use threshold cryptography among a network of nodes to produce publicly verifiable, unbiasable randomness.
- Function: Provides a common, agreed-upon random seed for decentralized applications.
- Properties: Liveness (values are produced on schedule), Unpredictability, Public Verifiability.
- Example: Drand's beacon powers randomness for Filecoin's storage proof allocation and other Web3 protocols.
04
Commit-Reveal Schemes
A two-phase protocol for generating fair random numbers in a decentralized setting where participants may act maliciously. In the commit phase, each participant publishes a cryptographic commitment (hash) to their secret random value. In the reveal phase, they later disclose the original value.
- Purpose: Prevents participants from changing their submitted random number after seeing others' submissions.
- Limitation: Vulnerable to last-revealer attacks and requires all participants to eventually reveal, which can cause delays.
05
Application: On-Chain Gaming & NFTs
Correlated randomness is critical for fairness in decentralized applications where outcomes must be random and provably fair.
- NFT Traits & Minting: Randomly assigning rarity traits to NFTs upon minting in a way that is verifiable and cannot be manipulated by the project team.
- Game Mechanics: Determining combat outcomes, loot drops, or matchmaking in blockchain-based games.
- Requirement: The randomness source must be tamper-proof and unpredictable to prevent exploits and ensure user trust.
06
Application: Blockchain Consensus & Security
Used within consensus mechanisms to select leaders, committees, or shards in a secure and unbiased manner.
- Leader Election: Protocols like Algorand use VRFs to secretly and randomly select the next block proposer, reducing attack surfaces.
- Shard Allocation: In sharded blockchains, validators are randomly assigned to shards using correlated randomness to prevent collusion.
- Security Goal: Ensures cryptographic sortition—the selection process is random, private until necessary, and verifiable after the fact.
common-types
MECHANISMS
Common Types of Correlated Randomness
Correlated randomness describes cryptographic protocols where multiple parties generate random values that are individually unpredictable but share a known, verifiable mathematical relationship. These are foundational for secure multi-party computation and decentralized systems.
02
Threshold Signatures
A threshold signature scheme allows a group of N parties to collaboratively generate a single, valid digital signature, but only if a threshold T of them participate. The signature is correlated randomness because each party contributes a random share, and the final signature is a deterministic function of these shares.
- Decentralized Trust: No single party ever knows the full private key.
- Application: Used in distributed key generation (DKG) for validator sets in protocols like Dfinity's Internet Computer and various multi-sig wallets to produce a common, verifiable random beacon output.
03
Commit-Reveal Schemes
A commit-reveal scheme is a two-phase protocol where parties first commit to a secret value (e.g., by publishing its hash) and later reveal the original value. The final correlated random value is typically a function (like XOR or addition) of all revealed secrets.
- Prevents Manipulation: The commitment phase locks in choices before the reveal, preventing last-second changes based on others' inputs.
- Limitation: Requires two rounds of communication and is vulnerable to block withholding if a participant refuses to reveal.
- Common Use: Found in early blockchain randomness beacons and simple on-chain games.
05
Multi-Party Computation (MPC) for Randomness
Secure Multi-Party Computation (MPC) protocols allow a group of parties to jointly compute a function (like generating a random number) over their private inputs without revealing those inputs. The final random output is correlated through the cryptographic computation.
- Information-Theoretic Security: Some MPC protocols provide security even against computationally unbounded adversaries.
- Process: Parties generate and secret-share random values, perform computations on shares, and then reconstruct the final result.
- Use Case: Underpins advanced threshold signature schemes and distributed random beacon constructions.
06
Proof-of-Stake Lotteries
In Proof-of-Stake (PoS) blockchains, validator selection for block production is often a weighted lottery based on the size of a validator's stake. The randomness for this lottery must be unpredictable, unbiased, and verifiable to prevent manipulation.
- Correlation Mechanism: The random seed is typically derived from block hashes from previous blocks, combined with validator-specific data, creating a randomness chain.
- Challenges: Susceptible to grinding attacks where a validator influences future randomness. Solutions often incorporate VRFs (Algorand) or RANDAO / VDFs (Ethereum) to add resilience.
security-model
SECURITY MODEL AND GUARANTEES
Correlated Randomness
Correlated randomness refers to a cryptographic technique where multiple parties generate random values that are individually unpredictable but share a known, verifiable mathematical relationship, enabling secure protocols like distributed key generation and verifiable secret sharing.
In cryptographic protocols, correlated randomness is a foundational concept where participants receive random values that are not independent. Instead, these values are linked by a predefined correlation, such as a shared secret or a commitment to a common polynomial. This structure allows the group to perform computations—like generating a collective signature or decrypting a ciphertext—without any single party learning the complete secret. The correlation is established through a secure distributed key generation (DKG) protocol or similar mechanism, ensuring that even if some participants are malicious, the overall security properties of the system are preserved.
The security guarantees of correlated randomness protocols are critical for Byzantine Fault Tolerance (BFT). A well-designed scheme ensures that honest participants can detect and, in some cases, recover from the actions of adversarial nodes that attempt to deviate from the protocol. For example, in a threshold signature scheme using correlated randomness, the correlation allows participants to verify that the shares they receive are consistent with a single, valid secret key, preventing a malicious actor from causing the group to generate invalid signatures. This property is often formalized as verifiability, a core guarantee that underpins trust in decentralized systems.
Implementing correlated randomness securely requires careful protocol design to avoid common pitfalls. A primary challenge is preventing adaptive adversaries from influencing the correlation after learning partial information. Techniques like commit-and-reveal schemes, non-interactive zero-knowledge proofs (NIZKs), and verifiable random functions (VRFs) are employed to bind participants to their random values before the correlation is established. Furthermore, the randomness must be unpredictable and unbiasable, meaning no party, including the protocol designers, can control or predict the final correlated output, which is essential for fairness in applications like blockchain consensus and lotteries.
In practice, correlated randomness is a key enabler for modern Multi-Party Computation (MPC) and threshold cryptography. For instance, the Drand network generates publicly verifiable, unbiased random beacons by having a decentralized set of nodes each contribute a share to a correlated randomness setup, resulting in a collective random number that no single node could have predicted. Similarly, distributed validators in Ethereum 2.0's consensus rely on correlated randomness (through a DKG) to securely split a validator's private key among multiple operators, enhancing security and reducing single points of failure.
ecosystem-usage
CORRELATED RANDOMNESS
Ecosystem Usage and Protocols
Correlated randomness is a cryptographic primitive that enables multiple parties to generate a shared random value, ensuring fairness and unpredictability in decentralized applications. Its primary use cases span gaming, governance, and secure computation.
benefits-impact
CORRELATED RANDOMNESS
Benefits and Performance Impact
Correlated randomness enables multiple parties to generate and share a common random seed without a trusted third party, providing cryptographic guarantees for fairness and coordination in decentralized applications.
01
Verifiable Fairness
Correlated randomness provides cryptographic proofs that a random outcome was generated fairly and was not manipulated by any single participant. This is critical for applications like on-chain gaming, lotteries, and leader election where trust is paramount. The process ensures all parties receive the same random value and can independently verify its integrity.
02
Reduced On-Chain Computation
By generating randomness off-chain through protocols like DKG (Distributed Key Generation) or VRF (Verifiable Random Function) commitments, the heavy cryptographic computation is moved off the main chain. Only the final proof and result are submitted on-chain, significantly reducing gas costs and blockchain bloat for applications requiring frequent random draws.
03
Enhanced Protocol Coordination
A shared, unpredictable random seed allows decentralized protocols to synchronize actions without communication rounds. Examples include:
- Shuffling validator committees in proof-of-stake networks.
- Assigning tasks in decentralized oracle networks.
- Seeding cryptographic sortition for anonymous voting systems. This eliminates the need for complex consensus on the sequence of events.
04
Resistance to MEV and Front-Running
When future random seeds are commit-revealed (e.g., using a hash commitment), they are unpredictable until revealed. This prevents Maximal Extractable Value (MEV) bots and validators from front-running transactions whose outcome depends on the random value, securing auctions, NFT mints, and reward distributions from manipulation.
05
Scalability for Mass Participation
Correlated randomness schemes like random beacons can produce a continuous stream of public randomness for thousands of applications simultaneously. This creates a public good resource, allowing many dApps to leverage the same trustless random source without each building their own costly system, enabling scalable, fair randomness for the entire ecosystem.
06
Latency and Finality Considerations
The performance impact depends on the underlying cryptographic protocol. VRF-based systems offer low latency but may require a relay. DKG-based beacons (e.g., drand) have higher latency due to multi-party computation rounds but provide strong, bias-resistant randomness. The trade-off is between speed and the level of decentralization/security required by the application.
security-considerations
CORRELATED RANDOMNESS
Security Considerations and Challenges
Correlated randomness refers to a critical security flaw where multiple participants in a protocol (e.g., validators, oracles, or players in a game) generate random values that are not independently unpredictable, potentially allowing collusion or manipulation of outcomes.
01
The Core Vulnerability
The fundamental risk is that if the randomness source is predictable or can be influenced by a subset of participants, it ceases to be random for the protocol. This can lead to:
- Front-running in DeFi lotteries or NFT mints.
- Validator manipulation in Proof-of-Stake leader election.
- Biased outcomes in on-chain gaming and gambling applications. The security of the entire application depends on the independence and unpredictability of each random number generated.
02
Common Attack Vectors
Several methods can lead to or exploit correlated randomness:
- Validator Collusion: In a blockchain using a random beacon, if a supermajority of validators collude, they can bias the output.
- Oracle Manipulation: Relying on a single, potentially compromisable off-chain oracle for randomness.
- Block Data Dependence: Using easily influenced on-chain data like block hash, timestamp, or gas price as a randomness source, which miners/validators can slightly manipulate.
- Sequential Submission: In commit-reveal schemes, if participants reveal in sequence, later players can adapt their reveals based on earlier ones.
03
Commit-Reveal Schemes & Their Limits
A classic mitigation where participants first submit a cryptographic commitment (hash) to their secret random value, then later reveal it. Challenges include:
- High Latency: Requires two rounds of communication, unsuitable for fast applications.
- Liveness Issues: If a participant refuses to reveal, the protocol may stall, requiring complex timeout and slashing mechanisms.
- Last-Revealer Advantage: The last participant to reveal has full information before committing their own value, which can be mitigated by threshold cryptography or verifiable random functions (VRFs).
04
Verifiable Random Functions (VRF)
A cryptographic primitive that provides a gold standard for on-chain randomness. A VRF allows a prover to generate a random number and a cryptographic proof that it was generated correctly from a secret key and a public input. Key properties:
- Uniqueness & Unpredictability: The output is deterministic yet pseudorandom and cannot be predicted without the secret key.
- Public Verifiability: Anyone can verify the proof, ensuring the random number was generated honestly.
- Examples: Used by Algorand for leader election and Chainlink VRF for smart contract applications. The main challenge is secure key management for the VRF producer.
05
Random Beacons & DKG
A random beacon is a continuously available source of public randomness. Decentralized beacons often use Distributed Key Generation (DKG) and threshold cryptography to eliminate single points of failure.
- Process: A group of participants collaboratively generates a shared public key and individual secret shares. Randomness is generated by aggregating signatures from a threshold of participants.
- Security: The random output is secure as long as fewer than the threshold are malicious (Byzantine fault tolerance).
- Challenges: Complex setup, significant on-chain computation for verification, and potential liveness vs. safety trade-offs. Used in protocols like Drand.
06
Economic & Game-Theoretic Attacks
Beyond technical flaws, correlated randomness enables economic attacks:
- Bribing Attacks: An attacker can bribe validators or oracle nodes to bias randomness in a profitable application (e.g., a high-value NFT mint), making collusion rational.
- MEV Extraction: Predictable randomness can be exploited for Maximal Extractable Value (MEV) by bots front-running transactions.
- Solution Approaches: Combating these requires cryptoeconomic security—designing slashing penalties, staking requirements, and reward structures that make collusion economically irrational, often in combination with robust cryptographic schemes.
COMPARISON
Correlated Randomness vs. Related Concepts
A technical comparison of correlated randomness with other cryptographic and blockchain primitives that involve randomness or verifiability.
| Feature / Property | Correlated Randomness | Verifiable Random Function (VRF) | Commit-Reveal Scheme | True Random Number Generator (TRNG) |
|---|---|---|---|---|
Cryptographic Proof of Correctness | ||||
Output is Deterministic | ||||
Output is Unpredictable | ||||
Enables Multi-Party Coordination | ||||
Primary Use Case | Multi-party protocols (e.g., leader election, lotteries) | Single-party, verifiable randomness (e.g., blockchain consensus, NFTs) | Delayed revelation of secrets (e.g., voting, bidding) | Seeding entropy pools, gaming RNG |
Trust Model | Trusted dealer or distributed key generation (DKG) | Trust in the secret key holder's honesty | Trust in participants to reveal correctly | Trust in physical process and hardware |
Latency to Usable Output | Pre-processing possible; instant at use | Instant | Two-phase (commit then reveal) | Instant |
Example Blockchain Implementation | Drand network, Obol DV | Chainlink VRF, Algorand consensus | Ethereum's early RANDAO | Provable/Oracle (off-chain source) |
CORRELATED RANDOMNESS
Frequently Asked Questions (FAQ)
Correlated randomness is a cryptographic technique that enables multiple parties to generate random values that are individually unpredictable but share a predefined mathematical relationship, enabling complex decentralized applications.
Correlated randomness is a cryptographic primitive where multiple parties generate random values that are individually unpredictable and verifiable, yet share a pre-defined mathematical correlation, such as a sum or product. This allows decentralized applications to execute protocols that depend on shared, unpredictable state without relying on a trusted third party. It is foundational for multi-party computation (MPC), threshold cryptography, and leader election in consensus mechanisms. For example, in a random beacon, validators each contribute a random share; the final beacon output is a function of all shares, but no single validator can predict or bias the final result.
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