Common use of Succinctness Clause in Contracts

Succinctness. There exists a universal polynomial p(·) such that for every compliance predicate C ∈ C, every time bound B ∈ N, and every B-bounded distributed computation transcript trans, • The computation time of PCD.Prover(σpcd, zin, πin, linp, zout) is p(n + |C| + B). • The verification algorithm PCD.Verify(τpcd, z, π) runs in time p(n + |C| + |z| + log B) • An honestly generated proof has size p(n + log B). | | ≤ |P |P Theorem A.8 ([9]). Let the size of a compliance predicate C, denoted by s(C), be the largest number of nodes in any transcript compliant with C. Assuming the existence of SNARKs with linear extraction (i.e., E ∗ c ∗ for some constant c), there exist PCD systems for logarithmic-depth and polynomial-size compliance predicates.

Succinctness. There exists a universal polynomial p(·) such that for every compliance predicate C ∈ C, every time bound B ∈ N, and every B-bounded distributed computation transcript trans, • The computation time of PCD.Prover(σpcd, zin, πin, linp, zout) is p(n p(κ + |C| + B). • The verification algorithm PCD.Verify(τpcd, z, π) runs in time p(n p(κ + |C| + |z| + log B) • An honestly generated proof has size p(n p(κ + log B). | | ≤ |P |P Theorem A.8 A.7 ([98]). Let the size of a compliance predicate C, denoted by s(C), be the largest number of nodes in any transcript compliant with C. Assuming the existence of SNARKs with linear extraction (i.e., E ∗ c ∗ for some constant c), there exist PCD systems for logarithmic-depth and polynomial-size compliance predicates.

Succinctness. There exists a universal polynomial p(·) such that for every compliance predicate C ∈ C, every time bound B ∈ N, and every B-bounded distributed computation transcript trans, • The computation time of PCD.Prover(σpcd, zin, πin, linp, zout) is p(n p(κ + |C| + B). • The verification verification algorithm PCD.Verify(τpcd, z, π) runs in time p(n p(κ + |C| + |z| + log B) • An honestly generated proof has size p(n p(κ + log B). | | ≤ |P |P Theorem A.8 A.7 ([98]). Let the size of a compliance predicate C, denoted by s(C), be the largest number of nodes in any transcript compliant with C. Assuming the existence of SNARKs with linear extraction (i.e., E ∗ c ∗ |EP∗ | ≤ c|P∗| for some constant c), there exist PCD systems for logarithmic-depth and polynomial-size compliance predicates.