# Hacker Dōjō | Project Research: Zero-Knowledge Proof——Groth16

I. Introduction

Zero-Knowledge Proof (ZKP) is a cryptographic protocol that enables one party to prove to another party that a statement is true, without revealing any information about the statement itself. In other words, ZKP allows a party to prove knowledge of a secret without revealing the secret itself.

One of the most widely used ZKP systems is Groth16, named after its inventor Jens Groth. Groth16 is a type of non-interactive zero-knowledge proof (NIZK) system, meaning that it allows a prover to generate a proof of knowledge without any interaction with the verifier. This makes it highly useful for situations where two parties may not be able to communicate with each other directly, such as in blockchain transactions.

A simple example of Groth16 in action can be found in the context of a financial transaction. Suppose Alice wants to prove to Bob that she has enough funds to pay for a particular item, without revealing exactly how much money she has in her account. Using Groth16, Alice can generate a proof of knowledge that she has at least the required amount of funds, and send that proof to Bob. Bob can then verify the proof without learning anything about Alice’s actual account balance.

This type of ZKP system has numerous applications in fields like blockchain, cybersecurity, and identity management. It allows parties to verify information without revealing sensitive data, making it an important tool for maintaining privacy and security in digital systems.

### A. Explanation of Zero-Knowledge Proof

Zero-Knowledge Proof (ZKP) is a cryptographic protocol that allows one party (the prover) to prove to another party (the verifier) that a particular statement or assertion is true, without revealing any information about the statement or assertion itself. In other words, the prover can convince the verifier that they know something without actually revealing what it is they know.

The concept of zero-knowledge proof was first introduced in 1985 by Goldwasser, Micali, and Rackoff in a paper titled “The Knowledge Complexity of Interactive Proof-Systems”. The idea was further developed by various researchers over the years, leading to the creation of a variety of ZKP systems that are widely used today.

ZKP is based on the idea of interactive proof systems, where a prover and a verifier interact with each other in order to establish the truth of a statement. In a typical interactive proof system, the prover attempts to convince the verifier that a statement is true by providing a series of arguments or proofs, and the verifier either accepts or rejects the proof based on their own criteria.

In a zero-knowledge proof, however, the prover is able to convince the verifier of the truth of a statement without providing any information that could potentially compromise the confidentiality of the statement. This is achieved by allowing the prover to generate a proof that demonstrates knowledge of the statement without revealing the statement itself.

There are several different types of ZKP systems, including interactive and non-interactive proof systems. Interactive systems require the prover and verifier to engage in a back-and-forth exchange of information, whereas non-interactive systems allow the prover to generate a proof that the verifier can verify without any further interaction.

ZKP has numerous applications in a variety of fields, including cybersecurity, blockchain technology, and identity management. For example, in blockchain transactions, ZKP allows parties to verify the validity of a transaction without revealing the identities of the parties involved or the specific details of the transaction itself. ZKP also allows for the creation of secure password authentication systems, where a user can prove their identity without actually revealing their password to the system.

Overall, ZKP is a powerful tool for maintaining privacy and security in digital systems, allowing parties to verify information without revealing sensitive data.

Here’s an example of a zero-knowledge proof in action:

Suppose Alice wants to prove to Bob that she knows the value of a secret number, without actually revealing the number to Bob. To do this, they could use a zero-knowledge proof.

- First, Alice chooses a random number (let’s call it r), and calculates the result of an equation using both the secret number and the random number. For example, the equation could be something like: 5x + 3r = 27, where x is the secret number.
- Alice sends the result of the equation to Bob (in this case, the value of 5x + 3r).
- Bob then sends Alice a random number, say s.
- Alice then calculates another result using the secret number, the random number Bob sent her, and the original random number she chose (r). For example, the new equation could be: 5s + 3r = y, where y is the new result.
- Alice sends the new result (y) to Bob.
- Bob then checks if the two results Alice provided are consistent with each other. In other words, he checks if it’s true that 5x + 3r = y and 5s + 3r = y.

If the two results are consistent, then Bob can be confident that Alice knows the value of the secret number (x), without actually knowing what it is. Alice has proven that she has knowledge of the secret number without revealing the number itself.

This is just a simple example, but zero-knowledge proofs have many more advanced and complex applications, such as in the field of cryptography and blockchain technology.

### B. Importance of Zero-Knowledge Proof

Zero-knowledge proofs (ZKPs) are an important tool for enhancing privacy, security, and trust in a variety of digital systems. Here are some of the key reasons why ZKPs are important:

- Privacy: ZKPs allow individuals to prove that they know something without revealing the actual information itself. This can be particularly useful in situations where individuals want to protect their privacy, such as in online transactions, where they may not want to reveal their identity or personal information.
- Security: By using ZKPs, individuals can verify the authenticity and integrity of data and transactions without revealing sensitive information. This can help prevent data breaches, hacking attacks, and other forms of cybercrime that threaten the security of digital systems.
- Trust: ZKPs can help establish trust between parties in digital systems. By using a ZKP, one party can prove to another party that they have certain information or knowledge, without the need for a third-party intermediary to verify the information. This can increase trust and reduce the need for trust in third-party institutions or individuals.
- Decentralization: ZKPs can also help facilitate decentralized systems, such as blockchain technology, by allowing parties to verify transactions and data without relying on a central authority or intermediary.
- Efficiency: ZKPs can improve the efficiency of digital systems by reducing the need for costly and time-consuming verification processes. By using ZKPs, parties can quickly and efficiently verify information, transactions, and other data without the need for lengthy and complex verification procedures.
- Innovation: ZKPs also have the potential to enable new types of applications and services that were previously impossible due to privacy and security concerns. For example, ZKPs can be used to create new types of decentralized finance (DeFi) applications, where parties can transact with each other without revealing their identities or financial information.

Overall, ZKPs are a powerful tool for enhancing privacy, security, and trust in digital systems. As more and more sensitive information is stored and transacted online, the importance of ZKPs is likely to continue to grow in the coming years.

### C. Brief Overview of Groth16

Groth16 is a zero-knowledge proof protocol that uses a type of proof called a zk-SNARK (Zero-Knowledge Succinct Non-Interactive Argument of Knowledge). It was introduced in a paper by Jens Groth in 2016 and has become widely used in the field of blockchain technology, particularly by the cryptocurrency platform Zcash.

The Groth16 protocol works by allowing one party (the prover) to prove to another party (the verifier) that they have certain knowledge or information, without revealing any details about the information itself. This is accomplished using a process that involves constructing a zk-SNARK, which is a succinct proof that the prover knows a certain piece of information without revealing any additional details about the information itself.

The protocol is particularly useful in the context of blockchain technology, where privacy and security are key concerns. For example, the Groth16 protocol allows users to transact with each other on a blockchain without revealing their identities or the specific details of their transactions. This can help protect the privacy of users and prevent the potential misuse of their personal information.

Another key advantage of the Groth16 protocol is that it is extremely efficient and can be computed quickly, even on low-powered devices. This makes it well-suited for use in decentralized systems, where transactions must be processed quickly and efficiently without the need for a central authority or intermediary.

Overall, the Groth16 protocol and zk-SNARKs more generally have emerged as a powerful tool for enhancing privacy, security, and trust in digital systems. As the use of blockchain technology continues to grow, it is likely that the Groth16 protocol and other zero-knowledge proof protocols will become increasingly important in the coming years.

Zero-knowledge proofs involve complex cryptographic algorithms and are typically implemented using specialized libraries or tools. However, here is a simplified example code that demonstrates how a zero-knowledge proof might work in practice:

Note that this code is for demonstration purposes only and is not secure or efficient for real-world applications. It is intended to give a basic idea of how a zero-knowledge proof might be implemented using cryptographic algorithms.

II. Background

### A. Definition of Zero-Knowledge Proof

A zero-knowledge proof is a type of proof in which one party (the prover) can prove to another party (the verifier) that they have certain knowledge or information, without revealing any details about the information itself. The proof allows the verifier to be confident that the prover possesses the necessary knowledge, but without gaining any additional information beyond that fact.

One way to think about a zero-knowledge proof is to consider a scenario in which Alice wants to prove to Bob that she knows the secret password to a locked door, but without revealing the password itself. A zero-knowledge proof allows Alice to prove to Bob that she knows the password, without revealing the password itself or any other information that might compromise the security of the system.

### C. Advantages and Disadvantages of Zero-Knowledge Proof

Zero-knowledge proofs offer several advantages over other methods of proving knowledge or authentication, but they also come with some disadvantages. Here are some of the key advantages and disadvantages of zero-knowledge proofs, along with a code example to illustrate some of these points:

Advantages:

- Privacy: One of the main advantages of zero-knowledge proofs is that they allow users to prove knowledge or authentication without revealing any sensitive information. This is especially important in scenarios where the information being proved is confidential, such as in financial transactions or medical records.
- Security: Zero-knowledge proofs offer a high level of security, as they are based on cryptographic protocols that are difficult to break. This makes them a reliable method of authentication and proof of knowledge.
- Efficiency: Zero-knowledge proofs can be performed quickly and with minimal resources, which makes them an attractive option for many applications. For example, zero-knowledge proofs can be used to provide proof of age without revealing the person’s birthdate, which can be useful in age-restricted settings like online gambling.

Disadvantages:

- Complexity: Zero-knowledge proofs can be complex to implement, and they require a high level of technical expertise to design and execute. This can make them inaccessible to many users and limit their adoption in certain applications.
- Vulnerability to side-channel attacks: Zero-knowledge proofs can be vulnerable to side-channel attacks, where an attacker is able to extract information about the proof by analyzing things like the timing or power consumption of the device performing the proof. This can be mitigated through careful design and implementation, but it is an ongoing challenge for zero-knowledge proofs.
- Lack of standardisation: Zero-knowledge proofs are still a relatively new technology, and there is a lack of standardisation in terms of protocols and algorithms. This can make it difficult for developers to build interoperable systems using zero-knowledge proofs.

III. Groth16

### A. Definition and Explanation of Groth16

Groth16 is a non-interactive zero-knowledge proof system that enables a prover to convince a verifier that they possess knowledge of a solution to a mathematical problem, without revealing any information about that solution. It was introduced by Jens Groth in 2016 and is particularly well-suited for use in blockchain applications.

Groth16 is a proof system that is based on bilinear pairings, which are a type of cryptographic function that takes two elements from two groups and returns an element from a third group. In the context of Groth16, the two groups are typically elliptic curve groups, and the third group is a multiplicative group.

The basic idea behind Groth16 is that the prover generates a proof that consists of two parts: a commitment to a set of random values, and a set of polynomial equations that are satisfied by those random values. The verifier checks that the polynomial equations are satisfied by the random values, and also checks that the commitment was generated correctly. If both checks pass, the verifier accepts the proof.

### B. Features and Advantages of Groth16

Groth16 is a proof system used in zero-knowledge proofs (ZKP) to provide privacy-preserving transactions on a blockchain network. It was introduced by Jens Groth in 2016 and is based on bilinear pairing operations over elliptic curve groups. Groth16 provides a highly efficient and secure way of generating ZKP and is widely used in several blockchain networks like Ethereum, Zcash, and Tezos.

Some of the features and advantages of Groth16 are as follows:

- Succinctness: Groth16 provides very succinct proofs that can be verified very efficiently. In other words, the proofs generated by Groth16 are very short, which makes them ideal for use in blockchain networks where block space is limited.
- Efficiency: Groth16 is highly efficient and can generate proofs much faster than other proof systems like Bulletproofs and Sonic. This is because Groth16 uses bilinear pairing operations over elliptic curves, which can be computed very efficiently.
- Security: Groth16 is a highly secure proof system and is resistant to attacks like zero-knowledge extraction, algebraic attacks, and existential forgery attacks. The security of Groth16 is based on the hardness of the discrete logarithm problem over elliptic curve groups.
- Flexibility: Groth16 is a very flexible proof system and can be used for a wide range of applications. It can be used to generate proofs for arbitrary computations and can also be used for non-interactive zero-knowledge proofs.
- Compatibility: Groth16 is compatible with several popular blockchain networks like Ethereum, Zcash, and Tezos. This means that it can be easily integrated into existing blockchain networks, which is essential for interoperability and the development of decentralized applications.

In summary, Groth16 is a highly efficient and secure proof system that is widely used in blockchain networks. Its succinctness, efficiency, security, flexibility, and compatibility make it an ideal choice for generating zero-knowledge proofs in a variety of applications.

An example of using Groth16 in a blockchain network is in Zcash. Zcash is a privacy-focused cryptocurrency that uses zero-knowledge proofs to keep transactions private.

In Zcash, when a user wants to send funds to another user, they generate a proof using Groth16 that proves they have the necessary funds without revealing any information about their actual balance or the transaction details.

The proof generated by Groth16 is then validated by the Zcash network to ensure that the transaction is valid and that the sender has the necessary funds. This validation is done without revealing any sensitive information about the transaction.

By using Groth16, Zcash is able to provide a high level of privacy to its users while still ensuring that transactions are valid and secure. Groth16’s efficiency and security make it an ideal choice for generating zero-knowledge proofs in a privacy-focused blockchain network like Zcash.

### C. Applications of Groth16

Groth16 is a zero-knowledge proof system that is used to provide security and privacy in various applications. The system is named after its creator, Jens Groth, who introduced it in 2016. Here are some of the applications of Groth16:

- Blockchain and Cryptocurrency: Groth16 is used to provide privacy and security in blockchain networks and cryptocurrency transactions. It is used in privacy-focused cryptocurrencies like Zcash and its fork, Horizen. The system allows users to verify the validity of transactions without revealing any personal information, such as the amount transferred, the sender, and the receiver.
- Identity and Access Management: Groth16 is used in identity and access management systems to verify the authenticity of users without disclosing any sensitive information. This can be used in online authentication systems, such as logging into a website or accessing a secure network.
- Secure Messaging and File Sharing: Groth16 can be used to encrypt messages and files and ensure that only the intended recipients can access them. This is done through a zero-knowledge proof, which proves the sender’s identity and the recipient’s authorization to access the message or file without revealing any content or metadata.
- Digital Voting Systems: Groth16 can be used to create secure and transparent digital voting systems that protect the privacy of voters. The system allows for the verification of the voting process and results without revealing any voter’s personal information.
- Supply Chain Management: Groth16 can be used to ensure the integrity and authenticity of products in a supply chain. It can be used to create a digital trail of products from the manufacturer to the consumer, which can be verified without disclosing any sensitive information.

In summary, Groth16 is a versatile zero-knowledge proof system that can be used to provide security and privacy in a wide range of applications, including blockchain and cryptocurrency, identity and access management, secure messaging and file sharing, digital voting systems, and supply chain management.

here’s an example to illustrate how Groth16 is used in practice:

Suppose you want to prove that you know a secret key without revealing the key itself. You can use Groth16 to create a zero-knowledge proof of this knowledge. The proof consists of a set of public parameters and a proof string that can be verified by anyone without revealing any information about the secret key.

To create the proof, you first generate a commitment to the secret key and its inverse using the public parameters. You then use a random value and the commitment to create a set of polynomials that satisfy certain constraints. These polynomials are used to construct a proof string that can be verified using the public parameters.

When someone wants to verify your proof, they use the public parameters and the proof string to check that the polynomials satisfy the constraints. If the verification succeeds, they know that you have the knowledge of the secret key without learning anything about the key itself.

This example shows how Groth16 can be used to create a zero-knowledge proof of knowledge without revealing any sensitive information. This type of proof can be used in many different applications, such as blockchain and cryptocurrency, secure messaging and file sharing, and digital voting systems, to provide security and privacy.

## IV. Technical Details

### A. Mathematical Framework of Groth16

The Groth16 proof system is a mathematical framework used for constructing zero-knowledge proofs of knowledge for algebraic problems. It was proposed by Jens Groth in 2016 and has been used in several blockchain protocols like Zcash.

The Groth16 proof system is based on the following three components:

- A bilinear pairing: This is a mathematical operation that maps two elements from a given group to an element in another group. It satisfies certain properties like bilinearity, non-degeneracy, and computability.
- A structured reference string (SRS): This is a set of public parameters that is generated during a setup phase. It consists of group elements that are used to encode the problem instance.
- A quadratic arithmetic program (QAP): This is a system of quadratic equations that describes the problem instance. It consists of three polynomials - A(x), B(x), and C(x) - that satisfy the equation A(x) * B(x) = C(x).

The Groth16 proof system consists of two algorithms:

- Setup Algorithm: The setup algorithm takes as input the problem instance and generates the structured reference string. The algorithm selects random elements from the group and computes the necessary group operations to construct the SRS.
- Prover Algorithm: The prover algorithm takes as input the problem instance and the SRS and generates a proof of knowledge. The algorithm constructs two polynomials - H(x) and K(x) - that are used to generate the proof. The polynomial H(x) is a linear combination of A(x) and B(x) that satisfies certain constraints. The polynomial K(x) is a degree-1 polynomial that is used to prove that H(x) satisfies the constraints.

VII. Conclusion

### A. Summary of Groth16

Groth16 is a zero-knowledge proof system that allows for the verification of computational integrity without revealing any information about the underlying computation. It was introduced in 2016 by Jens Groth and has since been applied in various domains, including cryptocurrency, blockchain technology, and data privacy.

Groth16 provides a powerful tool for ensuring privacy and security in a range of contexts, but there are also many opportunities for future research. Some of the areas that may be explored in future research include scalability, security, applications, interoperability, and privacy.

Overall, Groth16 represents a significant advance in the field of zero-knowledge proof systems, and it has the potential to enable new and innovative use cases that were not previously possible. As the use of these systems continues to grow, there is a significant opportunity for researchers to make a meaningful impact in a range of domains, including finance, healthcare, and data privacy.

### B. Contributions of Groth16 to Zero-Knowledge Proof

Groth16 is a zero-knowledge proof system that has made several significant contributions to the field of zero-knowledge proofs. Here are some of the key contributions of Groth16 to zero-knowledge proof systems:

- Improved efficiency: Groth16 is more efficient than previous zero-knowledge proof systems, allowing for faster verification of computational integrity without revealing any information about the computation. This efficiency is critical for real-world applications, where speed and performance are essential.
- Better security: Groth16 provides stronger security guarantees than previous zero-knowledge proof systems. It relies on the hardness of the discrete logarithm problem, making it more resistant to attacks than other systems that rely on weaker assumptions.
- Simplicity: Groth16 is simpler to implement than previous zero-knowledge proof systems, making it more accessible to a broader range of users. This simplicity also makes it easier to understand and verify, increasing trust in the system.
- Flexibility: Groth16 is a flexible system that can be applied to a wide range of use cases. It can be used to verify computations in a variety of contexts, including in blockchain technology, data privacy, and secure multi-party computation.
- Wide adoption: Groth16 has been widely adopted and used in many real-world applications. This adoption has helped to increase trust in the system and build a community of developers and researchers who are working to improve and extend the system.

Overall, Groth16 has made significant contributions to the field of zero-knowledge proof systems, improving efficiency, security, simplicity, and flexibility. Its wide adoption has also helped to increase awareness and understanding of zero-knowledge proofs, paving the way for future advances in this critical area of cryptography.

### C. Importance of Groth16 in Cryptography

Groth16 is an important development in the field of cryptography, with several significant applications and implications. Here are some of the key reasons why Groth16 is important in cryptography:

- Privacy and security: Groth16 is a zero-knowledge proof system, which means it allows for the verification of computational integrity without revealing any information about the underlying computation. This property is critical for privacy and security in many contexts, such as financial transactions, identity verification, and data privacy.
- Blockchain technology: Groth16 has been widely adopted in blockchain technology, where it is used to enable private and secure transactions without revealing sensitive information about the transaction or the parties involved. This is important for ensuring the integrity and security of blockchain-based systems, such as cryptocurrency.
- Data privacy: Groth16 has also been applied in data privacy, where it is used to enable secure computation and data sharing without revealing any sensitive information about the data being shared. This is important for protecting user privacy and preventing data breaches.
- Multi-party computation: Groth16 can be used in multi-party computation, where multiple parties collaborate on a computation without revealing their inputs. This is important for enabling secure collaborations and computations in contexts such as finance, healthcare, and scientific research.
- Advancing the field of zero-knowledge proofs: Groth16 represents a significant advance in the field of zero-knowledge proof systems, improving efficiency, security, simplicity, and flexibility. Its wide adoption has also helped to increase awareness and understanding of zero-knowledge proofs, paving the way for future advances in this critical area of cryptography.

Overall, Groth16 is an important development in cryptography, with many applications and implications for privacy, security, and secure computation. Its wide adoption in blockchain technology, data privacy, and multi-party computation is a testament to its effectiveness and potential for future applications.

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