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‎docs/crypto.md

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### How Sigma Protocols Work
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At their core, Sigma protocols provide a secure way to prove the following properties:
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At their core, [Sigma protocols](sigma.md) provide a secure way to prove the following properties:
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1. **Proof of Knowledge of Discrete Logarithm**: Prove knowledge of the discrete logarithm of a given public key without revealing the secret key.
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‎docs/dev/data-model/data-model.md

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‎docs/dev/data-model/dlog.md

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# Discrete Logarithm Proofs in Ergo
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## Overview
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Discrete logarithm proofs are a fundamental cryptographic primitive in Ergo's signature verification mechanism, based on the computational hardness of the discrete logarithm problem in elliptic curve cryptography.
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## Key Characteristics
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- **Cryptographic Foundation**: Proofs of knowledge of a discrete logarithm (DLog) verify signature authenticity without revealing the secret key
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- **Schnorr Signature Basis**: Ergo uses Schnorr signatures built on discrete logarithm proofs
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## Technical Details
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- **Proof Structure**: Demonstrate knowledge of secret exponent `w` such that `g^w = x`
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- `g`: Generator of an elliptic curve group
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- `x`: Public key point
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- `w`: Private key
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## Related Cryptographic Concepts
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- [Sigma Protocols](scs/sigma.md)
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- [Threshold Signatures](threshold.md)
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- [Ring Signatures](ring.md)
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## Implementation in ErgoScript
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In ErgoScript, discrete logarithm proofs are implemented using the `proveDlog()` predicate, which returns true if a valid proof of knowledge can be provided.
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```scala
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// DLog-based signature verification
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val pubKey = ... // Public key point
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val signature = ... // Signature proof
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proveDlog(pubKey)
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```
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## Practical Examples
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- [Schnorr Signature Verification](scs/sigma/verifying.md)
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- [Public Key Cryptography](scs/ergoscript/public-keys.md)
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## Security Considerations
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- Based on discrete logarithm problem hardness
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- Efficient and compact signature verification
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- Supports multi-signatures and ring signatures
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## Advanced Applications
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- [Cryptographic Foundations](crypto.md)
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- [ZeroJoin Privacy Protocol](uses/mixer.md)
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- [Sidechains Interoperability](uses/sidechains/sigma-chains.md)
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## References
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- [Cryptographic Primitives](crypto.md)
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- [ErgoScript Capabilities](scs/ergoscript.md)

‎docs/dev/data-model/nizk.md

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# Non-Interactive Zero-Knowledge Proofs in Ergo
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## Overview
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Non-Interactive Zero-Knowledge Proofs (NIZKs) are advanced cryptographic techniques that allow one party to prove knowledge of a secret without revealing the secret itself, and without requiring real-time interaction between the prover and verifier.
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## Theoretical Foundation
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NIZKs in Ergo are primarily implemented through **Sigma Protocols** (Σ-protocols), which provide a powerful and flexible approach to zero-knowledge proofs. These protocols are a cornerstone of Ergo's privacy and cryptographic infrastructure.
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### Key Characteristics
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- **Non-Interactive**: Proofs can be verified without direct communication
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- Unlike traditional interactive zero-knowledge proofs, NIZKs can be verified asynchronously
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- Reduces computational overhead and network complexity
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- **Zero-Knowledge**: No information about the secret is revealed
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- Cryptographically guarantees that only the validity of a statement is proven
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- Protects sensitive information while maintaining verifiability
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- **Composable**: Can be combined using logical operators like AND, OR, and THRESHOLD
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- Enables creation of complex cryptographic conditions
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- Supports advanced smart contract logic and privacy-preserving protocols
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## Cryptographic Primitives
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Ergo supports several fundamental zero-knowledge proof types:
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1. **Discrete Logarithm Proofs**
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- Prove knowledge of a secret key without revealing it
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- Fundamental to [Schnorr signature verification](schnorr.md)
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- Implemented using `proveDlog()` predicate in [ErgoScript](ergoscript.md)
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2. **Diffie-Hellman Tuple Proofs**
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- Prove equality of discrete logarithms across different generators
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- Enables privacy-preserving key exchange and contract designs
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- Critical for advanced cryptographic protocols
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## Implementation Techniques
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### Fiat-Shamir Transformation
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Ergo makes proofs non-interactive using the Fiat-Shamir transformation, which converts interactive proofs into non-interactive ones by using a cryptographic hash function.
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Key steps:
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- Transform an interactive proof into a non-interactive version
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- Use a cryptographic hash function to generate a challenge
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- Eliminates the need for real-time communication between prover and verifier
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### Proof Composition
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Sigma protocols can be combined to create complex proofs:
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```scala
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// Example of a threshold signature proof
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val thresholdProof = prove {
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atLeast(
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3, // Minimum number of signatures required
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Coll(
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PK("pubkey1"),
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PK("pubkey2"),
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PK("pubkey3"),
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PK("pubkey4"),
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PK("pubkey5")
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)
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)
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}
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```
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## Advanced Applications
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### Privacy-Preserving Techniques
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1. **Ring Signatures**
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- Prove one of multiple possible signers without revealing the exact signer
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- Enables anonymous transactions
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- Detailed in [Ring Signatures](ring.md) documentation
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2. **Threshold Signatures**
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- Require k-out-of-n participants to sign
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- Supports multi-party computational scenarios
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- Explored in [Threshold Signatures](threshold.md) documentation
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3. **Stealth Addresses**
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- Generate one-time addresses for enhanced transaction privacy
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- Prevent linking of transactions to a specific public address
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- Crucial for maintaining financial privacy
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### Mixer Protocols
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**ZeroJoin** demonstrates a practical application:
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- Uses ring signatures and Diffie-Hellman tuples
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- Restores fungibility of digital tokens
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- Provides non-interactive, trustless mixing
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- Detailed in [Mixer Protocol](mixer.md) documentation
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## Security Considerations
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- Based on the hardness of the discrete logarithm problem
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- Requires careful implementation to prevent potential vulnerabilities
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- Extensive test coverage in Ergo's cryptographic implementations
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- Relies on well-established cryptographic assumptions
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## Related Cryptographic Concepts
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- [Discrete Logarithm Proofs](dlog.md)
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- [Ring Signatures](ring.md)
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- [Threshold Signatures](threshold.md)
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- [Sigma Protocols](sigma.md)
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## Future Research Directions
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- Enhanced privacy protocol implementations
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- More efficient zero-knowledge proof constructions
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- Cross-chain interoperability using NIZKs
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- Integration with advanced cryptographic techniques
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## Performance and Scalability
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NIZKs in Ergo are designed with performance in mind:
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- Constant-time proof verification
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- Minimal computational overhead
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- Efficient serialization and deserialization
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- Support for batch verification techniques
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## References
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- [Sigma Protocols Overview](sigma.md)
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- [Cryptographic Foundations](crypto.md)
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- [Zero-Knowledge Proofs in Ergo](zkp.md)
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- Academic Papers:
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- [Sigma Protocols: A Survey](https://eprint.iacr.org/2021/1022)
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- [Non-Interactive Zero-Knowledge Proofs](https://eprint.iacr.org/2016/263)
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## Conclusion
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Ergo's Non-Interactive Zero-Knowledge Proofs represent a sophisticated approach to cryptographic privacy, enabling complex, secure, and flexible smart contract designs while maintaining user confidentiality. By leveraging advanced cryptographic techniques like Sigma Protocols and the Fiat-Shamir transformation, Ergo provides a robust framework for privacy-preserving computational techniques.

‎docs/dev/data-model/ring.md

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# Ring Signatures in Ergo
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## Overview
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Ring signatures are an advanced privacy-preserving cryptographic technique that allows a user to sign a transaction on behalf of a group without revealing which specific group member signed it.
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## Key Features
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- **Anonymity**: Provides plausible deniability by obscuring the actual signer
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- **Privacy**: Prevents tracing the origin of a signature to a specific participant
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- **Flexible Composition**: Implemented through Ergo's Sigma protocols
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## Use Cases
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1. **Anonymous Transactions**: Enabling privacy in blockchain transactions
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2. **Decentralized Mixers**:
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- [ErgoMixer Privacy Protocol](mixer.md)
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- [ZeroJoin Privacy Mechanism](zerojoin.md)
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3. **Confidential Voting**: Where the voter's identity must remain secret
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## Technical Implementation
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In Ergo, ring signatures are implemented using Sigma protocols, allowing for:
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- Proving knowledge of one secret from a set of secrets
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- Creating cryptographic proofs that obfuscate the true signer
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### Example Scenario
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```scala
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// Simplified conceptual representation
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val ringSignature = prove {
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atLeastOneOf(
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List(
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proveDlog(pubKey1),
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proveDlog(pubKey2),
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proveDlog(pubKey3)
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)
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)
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}
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```
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## Related Cryptographic Concepts
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- [Discrete Logarithm Proofs](dlog.md)
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- [Threshold Signatures](threshold.md)
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- [Sigma Protocols Overview](sigma.md)
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## Privacy Mechanisms
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- **ZeroJoin**: A privacy protocol leveraging ring signatures to restore fungibility
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- **ErgoMixer**: A non-custodial mixing service using ring signature techniques
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## Advanced Applications
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- [Cryptographic Foundations in Ergo](crypto.md)
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- [Schnorr Signatures and Privacy](schnorr.md)
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- [Sidechains and Interoperability](sigma-chains.md)
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## Security Considerations
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- Computational complexity makes tracing the original signer computationally infeasible
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- Relies on the hardness of the discrete logarithm problem
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- Provides strong privacy guarantees without compromising blockchain security
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‎docs/dev/data-model/threshold.md

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# Threshold Signatures in Ergo
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## Overview
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Threshold signatures are a cryptographic mechanism that allows a subset of a group to collectively sign a transaction, providing enhanced security and distributed trust.
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## Key Characteristics
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- **Distributed Signing**: Requires a minimum number of participants to authorize a transaction
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- **Flexible Thresholds**: Can be configured as k-out-of-n signatures (e.g., 3-out-of-5)
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- **Multi-Party Computation**: Enables complex collaborative signing scenarios
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## Detailed Examples
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### 3-out-of-5 Threshold Signature
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For a comprehensive example, refer to the dedicated tutorial:
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- [3-out-of-5 Threshold Signature](scs/sigma/3-out-of-5.md)
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### Practical Use Cases
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1. **Corporate Governance**:
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- Multi-signature wallets requiring collective approval
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- [Microcredit Scenario](scs/microcredit.md)
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2. **Cross-Chain Interoperability**:
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- [Rosen Bridge Mechanisms](eco/rosen.md)
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## Implementation Techniques
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Ergo supports threshold signatures through its Sigma protocol framework, allowing:
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- Proving knowledge of at least k secrets out of n total secrets
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- Creating multi-party computational scenarios with robust security guarantees
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## Conceptual Implementation
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```scala
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val thresholdSignature = prove {
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atLeastKOutOfN(
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k = 3, // Minimum signatures required
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n = 5, // Total possible signers
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publicKeys = List(
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pubKey1, pubKey2, pubKey3,
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pubKey4, pubKey5
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)
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)
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}
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```
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## Related Cryptographic Concepts
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- [Sigma Protocols](scs/sigma.md)
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- [Discrete Logarithm Proofs](dlog.md)
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- [Ring Signatures](ring.md)
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## Technical Advantages
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- **Reduced Single Point of Failure**: No single participant can unilaterally control funds
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- **Flexible Configuration**: Adaptable to various security requirements
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- **Privacy Preservation**: Sigma protocols ensure minimal information leakage
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## References
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- [Cryptographic Foundations](crypto.md)
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- [ErgoScript Capabilities](scs/ergoscript.md)

‎mkdocs.yml

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- Cryptographic:
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- crypto.md
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- Sigma Protocols:
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- dev/scs/sigma.md
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- Schnorr:
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- dev/scs/sigma/schnorr.md
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- Verifying Schnorr Signatures: dev/scs/sigma/verifying.md
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- Diffie:
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- dev/scs/sigma/diffie.md
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- Ring Signatures:
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- 3-out-of-5 Threshold Signature: dev/scs/sigma/3-out-of-5.md
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- Distributed Signatures: node/sigs.md
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# - Signature Scheme Internals: sig-scheme.md
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- Signature Schemes:
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- Sigma Protocols:
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- dev/scs/sigma.md
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- Schnorr:
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- dev/scs/sigma/schnorr.md
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- Verifying Schnorr Signatures: dev/scs/sigma/verifying.md
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- Diffie:
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- dev/scs/sigma/diffie.md
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- Other Signatures:
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- Ring Signatures: dev/data-model/ring.md
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- Threshold Signatures:
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- dev/data-model/threshold.md
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- 3-out-of-5 Threshold Signature: dev/scs/sigma/3-out-of-5.md
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- Distributed Signatures: node/sigs.md
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- Signature Scheme Internals: sig-scheme.md
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- Zero-Knowledge Proofs:
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- Non-Interactive ZK: dev/data-model/nizk.md
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- ZeroJoin: dev/crypto/zerojoin.md
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- Data Structures:
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- dev/data-model/data-structures.md
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- Merkle Tree:

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