This blog post explores methods for proving false statements within formal systems like logic and mathematics. It focuses on proof by contradiction, where you assume the statement is true and then demonstrate that this assumption leads to a logical inconsistency, thereby proving the original statement false. The post uses the example of proving the irrationality of √2, illustrating how assuming its rationality (expressibility as a fraction) ultimately contradicts the fundamental theorem of arithmetic. It highlights the importance of clearly defining the terms and axioms of the system within which the proof operates.
This blog post, titled "How to Prove False Statements? (Part 1)," delves into the fascinating realm of zero-knowledge proofs, specifically focusing on how these cryptographic constructs can be cleverly manipulated to seemingly prove the veracity of demonstrably false assertions. The author begins by establishing the fundamental principle of zero-knowledge proofs: convincing a verifier that a statement is true without revealing any information beyond the truth of the statement itself. They illustrate this with the classic example of Peggy proving to Victor that she knows the secret to opening a magic door without actually revealing the secret.
The core of the post then transitions into exploring how this seemingly ironclad system can be subverted. The author meticulously deconstructs the mechanics of a simplified Schnorr protocol, a common type of zero-knowledge proof. This protocol relies on the discrete logarithm problem, where calculating the discrete logarithm of a given value is computationally infeasible. The protocol involves a series of carefully orchestrated steps involving random numbers, cryptographic hashes, and modular arithmetic. The prover, possessing the secret, performs calculations based on these elements and sends a specific value to the verifier. The verifier then independently generates a challenge, and the prover responds with another calculated value. Through a final verification step, the verifier can be convinced of the prover's knowledge of the secret without learning the secret itself.
However, the author reveals a crucial vulnerability. By subtly altering the calculations, specifically by pre-computing certain values based on a desired outcome before receiving the verifier's challenge, a dishonest prover can effectively force the verification process to succeed even without possessing the secret. They demonstrate this with a detailed, step-by-step example, showcasing how manipulating the initial calculations allows the prover to fabricate a "proof" that satisfies the verification equation, thereby deceiving the verifier into believing a false statement. This effectively simulates possession of the secret when, in reality, no such knowledge exists.
The post concludes by emphasizing that this demonstration is not intended to expose a flaw in Schnorr protocols themselves, which remain secure when implemented correctly. Instead, it serves as a cautionary tale, highlighting the importance of meticulous protocol design and the potential for exploitation if specific steps are not rigorously adhered to. The author hints at further explorations of this theme in subsequent parts of the series, promising to delve into more sophisticated techniques and real-world implications of proving false statements. The current post serves as a foundational introduction to the concept, leaving the reader intrigued by the possibilities and potential dangers of manipulating zero-knowledge proofs.
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https://news.ycombinator.com/item?id=42939312
Hacker News users discuss the potential misuse of zero-knowledge proofs (ZKPs), expressing concern that they could be used to convincingly lie or create fraudulent attestations. Some commenters highlight the importance of distinguishing between a ZKP verifying a computation versus verifying a real-world fact. They argue that while ZKPs can prove the correct execution of a program on given inputs, they cannot inherently prove the veracity of those inputs. Others discuss the "garbage in, garbage out" principle in this context, suggesting the need for robust, real-world verification methods alongside ZKPs to prevent their misuse. The trustworthiness of the prover remains crucial, and ZKPs alone cannot bridge the gap between computation and reality. A few comments also touch upon the complexity of understanding and implementing ZKPs correctly, potentially leading to vulnerabilities.
The Hacker News post titled "How to prove false statements? (Part 1)" linking to a blog post about cryptography and zero-knowledge proofs generated several comments discussing the technical details and implications of the presented concepts.
One commenter highlights the importance of distinguishing between provably false statements within a formal system and statements that are false in the "real world". They emphasize that a formal system can only work with what's defined within its axioms and rules, and thus "false" refers to inconsistency within that system, not necessarily a reflection of external reality. This commenter also points out the challenge of bridging the gap between a formal system and the real world, especially when dealing with real-world data and measurements that might be inherently imprecise or subject to error.
Another commenter delves into the specifics of zero-knowledge proofs, particularly the concept of a "simulation trapdoor". They explain how this trapdoor allows a simulator to create convincing "proofs" even for false statements, which is crucial for demonstrating the soundness of the zero-knowledge system. This comment also mentions the use of non-interactive zero-knowledge proofs and how they enhance the efficiency and practicality of the system.
Several commenters discuss the practical applications and limitations of zero-knowledge proofs. One user raises the issue of computational complexity and the potential for proof generation or verification to be computationally expensive. Another commenter mentions the importance of trusting the setup phase of the zero-knowledge system, as a compromised setup could undermine the entire system's security.
The topic of using zero-knowledge proofs for authentication and authorization also receives attention. One commenter points out the benefits of using these proofs to selectively disclose information without revealing unnecessary details, enhancing privacy and security. However, another commenter counters that this approach relies on having agreed-upon facts in the first place, which might be challenging to establish in certain scenarios.
Finally, there's a brief discussion on the relation between these cryptographic concepts and philosophical ideas about truth and provability, with one commenter drawing parallels to Gödel's incompleteness theorems.
Overall, the comments on the Hacker News post delve into the technical intricacies of zero-knowledge proofs, their practical implications, and even their philosophical connections. They provide valuable insights and perspectives beyond the original blog post, highlighting both the potential and the limitations of this fascinating area of cryptography.