How zkSNARKs Actually Work: R1CS, QAP, and Pairings Explained With Examples

Technical educational tutorial explaining the mathematical foundations of zero-knowledge succinct non-interactive arguments of knowledge (zkSNARKs). Part 2 of a 4-part series. Walks through four core steps: (1) flattening computation into addition and multiplication gates; (2) converting gates to Rank-1 Constraint System (R1CS) with three-slot constraints (Left × Right = Output); (3) transforming R1CS into Quadratic Arithmetic Program (QAP) using polynomial interpolation to collapse all constraints into one polynomial identity; (4) using elliptic curve pairings to verify the polynomial identity without revealing witness values. Uses concrete example of proving knowledge of x where x³ + x + 5 = 35 without revealing x=3. Explains why only secret-to-secret multiplications require constraints (additions are free), the role of the structured reference string (SRS) and trusted setup ceremony, the 'toxic waste' problem and multi-party computation ceremonies, and how the Schwartz-Zippel lemma ensures one random point evaluation is cryptographically sufficient to verify polynomial identities. Applications mentioned include Zcash, TornadoCash, and zkEVMs. Concludes that succinctness has been achieved but zero-knowledge property requires additional blinding mechanisms explained in subsequent articles.