Zero-knowledge succinct non-interactive arguments of knowledge based on sets of polynomials

The paper discusses the basic principles of construction and the main types of zero-knowledge succinct non-interactive argument of knowledge (zk-SNARK) which is used in the model of a three-way insecure computing environment and based on sets of polynomials. A number of zk-SNARK cryptographic protocols with different algorithms for generating public parameters (Trusted Setup) are given, constructing succinct proofs of reliability calculations (Prover) and public/designated verification of proofs (Verifier). The cases of satisfying the feasibility of discrete functions (arithmetic/ Boolean circuits) using different polynomial sets are presented in quadratic arithmetic programs (QAP), square arithmetic programs (SAP), quadratic span programs (QSP), square span programs (SSP), quadratic polynomial programs (QPP), etc., also the use of authenticated data are described. The cryptographic transformations needed to build zk-SNARKs based on symmetric and asymmetric hash functions, exponential knowledge problems, digital signatures, homomorphic encryption, bilinear pairings based on elliptic curves, etc. are presented. Examples of multilateral verifiable calculations based on zk-SNARK are given.