How to Understand Zero-Knowledge Proofs
Motivation (Preface)
Before we start, I want to share why I'm doing this and what I hope to achieve.
In the last couple of years, I've been studying zero-knowledge proofs (ZKPs) and how they're used in blockchain technology. But it's been tough. The language used to explain these ideas is often very academic and hard to understand. To grasp what was going on, I had to learn a lot of new and complex concepts.
I think this is a problem. Right now, if you want to understand ZKPs, it feels like you need to have a master's degree in computer science or years of experience in academia. This means a lot of people who are interested in this technology can't get into it.
But I believe it doesn't have to be this way.
I will provide a comprehensive summary of each of the 19 chapters from the book titled "Proofs, Arguments, and Zero-Knowledge" written by researchers from Georgetown University and published in December 2022. By dissecting the information contained in each chapter, I try to present a detailed analysis and breakdown of the valuable insights and concepts covered in the book, enabling readers to gain a deep understanding of its content and specifically ZKPs.
With these summaries, I want to make ZKPs easier to understand for everyone, while also educating and learning for myself. I will try to explain this technology in a way that even a high school student can get it. I'll also introduce and explain other ideas and concepts you need to know to fully understand this technology.
The summary of every chapter will have a certain structure to amplify your gain of knowledge in the domain of zero knowledge (pun intended). The structure is as follows:
Introduction
Background Information
Examples and Illustrations
Main Points
Discussion and Analysis
Summary and Takeaways
Preview of the Next Chapter
Additional Resources
Discussion questions
So, whether you're an experienced computer scientist or a high school student, I invite you to join me on this journey. Let's explore the world of ZKPs together!
Chapter 1: Introduction
1. Introduction
Welcome to the first release and [Part 1] of my summaries of the academic paper on zero-knowledge proofs written by Justin Thaler et al. called ‘’Proofs, Arguments, and Zero-Knowledge’’.
This week, we're diving into the first chapter, which sets the stage for our journey into the world of traditional and non-traditional proofs and their applications in computer science and cryptography. This chapter introduces us to the concept of non-traditional proofs specifically, a topic that will be central to our exploration of zero-Knowledge proofs.
This article begins by defining zero-knowledge proofs (ZKPs). It then explores the key distinctions between probabilistic and deterministic proof systems. Following this background information, the article delves into the core points of the chapter, presenting important characteristics of proofs along with their definitions. Additionally, this article introduces various non-traditional proof systems and illustrates their differences through simplified examples, aiding readers in comprehending their unique aspects. Subsequently, these proof systems are thoroughly discussed and analyzed, shedding light on their differences and trade-offs.
2. Background Information
Before we go into the specifics of the chapter, let's take a moment to understand what ZKPs are. These are cryptographic methods that allow one party (the prover) to prove to another party (the verifier) that they know a value x, without conveying any information apart from the fact that they know the value x. This concept is revolutionary in the field of cryptography and has wide-ranging applications, from secure authentication to digital signatures. Don’t worry if this is still too abstract for you, this whole endeavour will take months and is aimed to explain ZKPs and all their glory.
In this context of proof systems, it's important to understand the difference between deterministic and probabilistic proof systems. In a deterministic proof system, the verification procedure is fixed and does not involve any randomness. This means that for any given statement and proof, the verification procedure will always produce the same result: either the proof is valid, or it is not. This is the traditional notion of a mathematical proof, where a proof can be written and checked line-by-line for correctness.
On the other hand, a probabilistic proof system involves randomness in the verification procedure. This means that the verification procedure might produce different results for the same statement and proof, depending on the random choices made during the verification process. However, the soundness guarantee holds with (very) high probability over those random choices. This means that there is a (very) small probability that the verification procedure will declare a false statement to be true.
3. Main Points
The chapter begins by defining what we mean by a proof and a proof system. Informally, a proof is anything that convinces someone that a statement is true, and a proof system is any procedure that decides what is and is not a convincing proof. A proof system is specified by a verification procedure that takes as input any statement and a claimed "proof" that the statement is true, and decides whether or not the proof is valid.
The chapter then outlines four properties that we ideally want in a proof system:
Completeness: Any true statement should have a convincing proof of its validity.
Soundness: No false statement should have a convincing proof.
The efficiency of Verification: Simple statements should have short (convincing) proofs that can be checked quickly.
The efficiency of Proving: Simple statements should have short (convincing) proofs that can be found quickly.
The traditional notion of a mathematical proof, which can be written and checked line-by-line for correctness, is contrasted with more general and exotic notions of proofs that are probabilistic. These probabilistic proofs, where the verification procedure makes random choices and the soundness guarantee holds with a high probability over those random choices, have transformed our understanding of what it means to prove something and have led to major advances in complexity theory and cryptography.
The chapter also introduces categories of these probabilistic proofs as they can still vary a lot within their class and have different trade-offs. The chapter discusses several types of non-traditional proofs:
Interactive Proofs (IPs): These are cryptographic protocols that allow an untrusted prover to provide a guarantee that they performed a requested computation correctly. IPs allow for interaction between the prover and the verifier, as well as a tiny but non-zero probability that an invalid proof passes verification.
Argument Systems: These are similar to IPs but permit there to be "proofs" of false statements, as long as those "proofs" require exorbitant computational power to find. Argument systems are secure against computationally bounded provers.
Multi-prover Interactive Proofs (MIPs): These are a generalization of IPs where there are multiple provers that the verifier can interact with. The provers cannot communicate with each other during the execution of the protocol. MIPs can be used to verify more complicated statements efficiently.
Probabilistically Checkable Proofs (PCPs): These are proofs that can be checked by a verifier that reads only a small random portion of the proof. PCPs are used in the construction of some argument systems and have deep connections to complexity theory.
Zero-Knowledge Proofs (ZKPs): These are proofs that reveal no information other than their validity. They have a myriad of applications in cryptography and are a major focus of the document.
The chapter emphasizes that all these notions of proof are probabilistic and have transformed computer scientists' notions of what it means to prove something, leading to major advances in complexity theory and cryptography.
4. Examples and Illustrations
The chapter uses the example of a traditional mathematical proof to explain the concept of a deterministic proof system. This traditional proof can be written and checked line-by-line for correctness, making it a straightforward and reliable method of proving a statement.
For instance, consider a simple mathematical statement like "2+2=4". A deterministic proof of this statement might involve demonstrating the calculation step-by-step:
Start with 2.
Add 2.
The result is 4.
This proof is deterministic because it follows a fixed procedure and always produces the same result.
However, the chapter then introduces the concept of probabilistic proofs, which are more complex and nuanced. In these proofs, the verification procedure makes random choices, and the soundness guarantee holds with a high probability.
Consider a simple probabilistic proof system where there's a 50% chance of a false positive per proof, and you can check it yourself a large number of times. If it keeps getting a true statement after 1000 tries, the probability of a false positive would be 0.5^1000=9.3^-302., which is extremely small. This illustrates the power of probabilistic proof systems: even though a single test might not be completely reliable, by repeating the test multiple times, we can make the probability of a false positive arbitrarily small.
These examples illustrate the key differences between deterministic and probabilistic proof systems and highlight the unique advantages and challenges of each approach. Let’s also illustrate how the proof systems in the previous chapter work with an oversimplified example:
Interactive Proofs (IPs): Let's take the example of a game where Alice has a secret number and Bob wants to guess it. Alice's number is 7, but she doesn't want to reveal it directly. Instead, she tells Bob that her number is between 1 and 10. Bob guesses 5. Alice says her number is higher. Bob guesses 8. Alice says her number is lower. Bob assumes 7, and Alice confirms that he's correct. This is a simple example of an interactive proof, where the prover (Alice) and the verifier (Bob) interact with each other, and the verifier can determine the truth through this interaction.
Argument Systems: Let's consider a scenario where Alice wants to prove to Bob that she knows the password to a secure system, but she doesn't want to reveal the password itself. Alice could use a cryptographic hash function to generate a hash of her password and send it to Bob. Bob can't reverse-engineer the password from the hash, but when Alice enters her password into the system and it's accepted, Bob can hash the password in the same way and see that the hashes match. This is a simple example of an argument system, where the prover uses a cryptographic method to convince the verifier of the truth without revealing the secret information.
Multi-Prover Interactive Proofs (MIPs): Let's imagine a scenario where Alice and Bob are trying to convince Charlie that they can communicate telepathically. Charlie isolates Alice and Bob in separate rooms and asks Alice a question. Alice gives her answer, and then Charlie asks Bob the same question. If Bob's answer matches Alice's, Charlie might be convinced of their telepathic connection. This is a simple example of a multi-prover interactive proof, where multiple provers (Alice and Bob) convince a verifier (Charlie) of a claim through interaction, where the multiple provers (Alive and Bob) can’t communicate with each other.
Probabilistically Checkable Proofs (PCPs): Suppose Alice wants to convince Bob that she has solved a complex mathematical problem. Instead of showing Bob the entire solution, Alice gives Bob proof that he can check by looking at only a few random parts. For example, Alice could provide a list of all the steps she took to solve the problem, and Bob could verify the correctness of a few random steps. If the steps he checks are correct, he can be reasonably confident that the entire solution is correct. This is a simple example of a probabilistically checkable proof.
Zero-Knowledge Proofs (ZKPs): Suppose Alice wants to prove to Bob that she knows the solution to a Sudoku puzzle without revealing the solution itself. Alice could first cover each puzzle cell with a piece of paper. Then, to prove that she knows a solution, Alice could ask Bob to point to any row, column, or box in the puzzle. Alice would then lift the pieces of paper covering the cells in that row, column, or box, showing that they contain a valid set of Sudoku numbers. Bob would learn that Alice knows a solution, but he wouldn't learn anything about the solution itself. This is a simple example of a ZKP.
5. Discussion and Analysis
The introduction of probabilistic proofs has transformed our understanding of what it means to prove something. It has also led to major advances in complexity theory and cryptography. The chapter raises the question of whether the next 5-10 years will bring a similar flood of developments and whether this will make general-purpose arguments efficient enough for routine deployment in diverse cryptographic systems. This opens up a discussion about the future of cryptography and the potential impact of further advancements in the field.
It is important to know the trade-offs and limitations of each proof system so that we can understand the development of these systems and how we got here. There are two main characteristics of proof systems which paved the way to where we are now.
Succinctness and non-interactivity are highly desirable properties in a proof system due to their significant benefits in terms of efficiency, scalability, and practicality. Let's discuss each of these properties in detail and then provide a comparison table for the five proof systems.
Succinctness: Succinctness refers to the ability of a proof system to generate compact proofs that are significantly smaller than the original statement being proven. Succinct proofs are advantageous for several reasons:
a. Efficiency: Succinct proofs require less computational and communication resources to generate, transmit, and verify. This leads to faster verification times, reduced computational complexity, and improved performance.
b. Storage and Bandwidth: Smaller proof sizes consume less storage space and bandwidth, which is crucial for applications with limited resources or when transmitting proofs over networks. Succinctness allows for efficient transmission and storage of proofs, making them more practical in various scenarios.
c. Scalability: As proof sizes grow, maintaining scalability becomes challenging. Succinctness enables handling larger-scale computations by reducing the storage and computational requirements for proofs, facilitating the verification process even for complex problems.
d. Verifiability: Succinct proofs can be verified more efficiently, enabling faster and more accessible verification. This is particularly important in distributed systems or applications where multiple parties need to independently verify proofs.
Non-interactivity: Non-interactivity refers to the property of a proof system that does not require multiple rounds of interaction between the prover and verifier. Instead, a non-interactive proof can be generated by the prover and then verified independently by the verifier. Non-interactivity offers several advantages:
a. Efficiency: Non-interactive proofs can be generated and verified more quickly since they do not require back-and-forth communication. This reduces latency and computation time, making the proof process more efficient.
b. Independence: Non-interactivity allows verifiers to independently verify proofs without relying on continuous interaction with the prover. This enhances decentralization and allows multiple verifiers to simultaneously verify the same proof.
c. Scalability: Non-interactive proofs are more scalable since they eliminate the need for repeated interaction, which can be time-consuming and resource-intensive. This scalability enables the verification of multiple proofs concurrently and facilitates the integration of proof systems into large-scale applications.
d. Flexibility: Non-interactivity enables proofs to be shared and verified across different platforms, systems, and parties without requiring real-time communication or synchronization. This flexibility is valuable for distributed systems and scenarios where real-time interaction is not feasible or desirable.
It becomes apparent why this kind of proof system with these two characteristics is preferred in blockchain systems where decentralisation is a key component and limited resources are present.
Now, let's provide a comparison table focusing on the properties of succinctness and non-interactivity for the five proof systems:
Please note that the table provides a general comparison, and the properties can vary depending on the specific implementations and protocols used within each proof system. The focus is on the typical tendencies and benefits associated with succinctness and non-interactivity.
6. Summary and Takeaways
In this part 1, we have set sail on an adventure to explore ZKPs and their applications in computer science, cryptography and blockchains. Let's summarize the key findings and takeaways from our journey so far:
ZKPs are cryptographic methods that allow one party (the prover) to prove to another party (the verifier) that they know a value or statement without revealing any additional information.
ZKPs are part of a broader landscape of proof systems, including Interactive Proofs (IPs), Argument Systems, Multi-prover Interactive Proofs (MIPs), Probabilistically Checkable Proofs (PCPs), and ZKPs.
Each type of proof system has its unique characteristics, trade-offs, and applications. Interactive Proofs allow for interaction between the prover and verifier, Argument Systems permit proofs of false statements requiring high computational power, MIPs involve multiple provers interacting with a verifier, PCPs offer efficient verification through random sampling, and ZKPs reveal no information other than their validity.
Succinctness and non-interactivity are highly desirable properties in a proof system. Succinctness refers to the ability to generate compact proofs, reducing the computational and communication resources required. Non-interactivity eliminates the need for continuous back-and-forth communication between the prover and verifier, enabling efficiency, independence, scalability, and flexibility.
ZKPs excel in providing non-interactivity and can also achieve succinctness depending on the specific scheme employed. They have widespread applications in cryptography, privacy-preserving protocols, blockchain technology, and secure computations.
Understanding the different proof systems and their properties allows us to appreciate the trade-offs and limitations involved. Succinctness and non-interactivity play a vital role in efficient and scalable solutions, making ZKPs particularly suitable for decentralized systems like blockchain where limited resources and decentralization are key considerations.
In conclusion, ZKPs are a powerful tool in cryptography, enabling efficient and relatively secure protocols while also having the ability to preserve privacy. The ongoing research and development in proof systems continue to enhance their capabilities, leading to advancements in complexity theory, cryptography, and decentralized technologies. By understanding the nuances and applications of various proof systems, we can harness their potential to build more secure, efficient, and privacy-preserving systems.
7. Preview of Chapter 2
Next chapter, we're going deeper into math. We'll look at things like multiplying matrices and polynomials. These are important for understanding two ways we can check our proof systems and the data for their validity. So, stick around for more on these cool concepts!
8. Additional Resources
For those interested in learning more about the concepts introduced in this chapter, here are a few additional resources:
9. Discussion Questions
Every week I will pose a couple of questions that I thought of when working on elaborating on the respective chapter from the research paper. I might use these later in an AMA or Twitter space to engage readers in educating about ZKPs.
How has the concept of probabilistic proofs transformed our understanding of what it means to prove something?
Will we see other proofs than ZKP in the endgame of proof systems in blockchains? If so, which proof system applies the best in what place?
Where are these different proof systems applied theoretically and practically and in what fields do they come to fruition the most?
I hope you found this summary informative and engaging. I look forward to exploring the next chapter with you!