1. SHPLONK vs PLONK: What is the difference? SHPLONK is a more effective commitment scheme than KZG10 commitment. On the premise of Multi-poly and Multi-openings, SHPLONK can make the workload of Prover not increase with Openings. As a result, developers tend to use Permutation or Multi-shifts techniques to increase the number of polynomials when designing constraint systems. (As for the current PLONK algorithm, there are two Openings, i.e., z and zw. When there are more Openings, this algorithm will have more advantages.) 2. General Introduction to Polynomial Commitment Schemes (PCS) 2.1 One open point If f(x) - a can be divided by x  -  z, then f(z) = a For set S, there is polynomial Zs(X) = ПzϵS(X-z). If f(X) - r(X) can be divided by the polynomial Zs(X), it means that for any element z in set S, satisfying f(z) = r(z) 2.2 Multi open points A polynomial commitment scheme should consist of three parts, ϴ = (gen, com, open), satisfying: 2.3 PCS protocol a. The inputs of Prover are as follows: b. The inputs of Verifier are as follows: c. At the end of the protocol, Verifier should output either true or false and meet the following requirements: 3. First Improved PCS 3.1 Protocol a. Verifier selects a number γ ϵ F at random
b. Prover computes polynomials: and computes c. Verifier computes: d. Verifier computes: e. Verifier output is correct if and only if: Proof: 3.2 Complexity 4. Final Improved PCS 4.1 Protocol a. Verifier randomly selects random number γ b. Prover computes polynomials: c. Verifier randomly selects z and sends it to Prover d. Prover computes: Of which: It can be found: So (X-z)|L. Prover computes: e. Verifier computes: f. Verifier output is correct if and only if: 4.2 Complexity Reference SHPLONK: https://eprint.iacr.org/2020/081.pdf Plonk: https://eprint.iacr.org/2019/953.pdf KZG10: https://www.iacr.org/archive/asiacrypt2010/6477178/6477178.pdf Slonk: https://ethresear.ch/t/slonk-a-simple-universal-snark/6420

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