ZKP Summary

Cryptography · Axiom Academy

Let's review the fundamental concepts, protocols, and applications of zero-knowledge cryptography. Definition: A zero-knowledge proof allows one party (the prover) to convince another party (the verifier) that a statement is true without revealing any information beyond the validity of the statement itself Key Insight: Prove knowledge without revealing the knowledge Example: Prove you know a password without transmitting the password itself Why It Matters: Enables privacy-preserving authentication, confidential transactions, and verifiable computation Completeness: If the statement is true and both parties follow the protocol honestly, the verifier will be convinced Soundness: If the statement is false, no cheating prover can convince the verifier except with negligible probability Zero-Knowledge: If the statement is true, the verifier learns nothing beyond the fact that the statement is true Formal Guarantee: These properties must hold even against computationally bounded adversaries Protocol Example: Schnorr Protocol Setup: Prover knows secret x such that y = g x (discrete log problem). Goal is to prove knowledge of x without revealing it Commitment: Prover chooses random r , computes t = g r , and sends t to verifier Challenge: Verifier sends random challenge c from a large space Response: Prover computes s = r + cx and sends s to verifier Verification: Verifier checks that g s = t · y c . If true, accept; otherwise, reject

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