The paper is devoted to the investigation of the distributed proof generation process, which makes use of recursive zk-SNARKs. Such distributed proof generation, where recursive zk-SNARK-proofs are organized in perfect Mercle trees, was for the first time proposed in Latus consensus protocol for zk-SNARKs-based sidechains. We consider two models of a such proof generation process: the simplified one, where all proofs are independent (like one level of tree), and its natural generation, where proofs are organized in partially ordered set (poset), according to tree structure. Using discrete Markov chains for modeling of corresponding proof generation process, we obtained the recurrent formulas for the expectation and variance of the number of steps needed to generate a certain number of independent proofs by a given number of provers. We asymptotically represent the expectation as a function of the one variable n/m, where n is the number of provers m is the number of proofs (leaves of tree). Using results obtained, we give numerical recommendation about the number of transactions, which should be included in the current block, idepending on the network parameters, such as time slot duration, number of provers, time needed for proof generation, etc.

Sidechains are among the most promising scalability and extended functionality solutions for blockchains. Application of zero knowledge techniques (Latus, Mina) allows for reaching high level security and general throughput, though it brings new challenges on keeping decentralization where significant effort is required for robust computation of zk-proofs. We consider a simultaneous decentralized creation of various zk-proof trees that form proof-trees sequences in sidechains in the model that combines behavior of provers, both deterministic (mutually consistent) or stochastic (independent) and types of proof trees. We define the concept of efficiency of such process, introduce its quantity measure and recommend parameters for tree creation. In deterministic cases, the sequences of published trees are ultimately periodic and ensure the highest possible efficiency (no collisions in proof creation). In stochastic cases, we obtain a universal measure of prover efficiencies given by the explicit formula in one case or calculated by a simulation model in another case. The optimal number of allowed provers’ positions for a step can be set for various sidechain parameters, such as number of provers, number of time steps within one block, etc. Benefits and restrictions for utilization of non-perfect binary proof trees are also explicitly presented.

For a finite partially ordered set I, we define an abstract polytope PI which is a cube or a globe in the cases of discrete or linear poset, respectively. For a poset P, we have built a small category ♦P with finite lower subsets in P as objects. This category ♦P = ♦P+♦P- is factorized into a product of two wide subcategories ♦P+ of faces and ♦P- of degenerations. One can imagine a degeneration from I to J ⊂ I as a projection of an abstract polytope PI to the subspace spanned by J. Morphisms in ♦P+ with fixed target I are identified with faces of PI . The composition in ♦P admits the natural geometric interpretation. On the category ♦I of presheaves on ♦I , we construct a monad of free category in two steps: for a terminal presheaf, the free category is obtained via a generalized nerve construction; in the general case, the cells of a nerve are colored by elements of the initial presheaf. Strict P-fold categories are defined as algebras over this monad. All constructions are functorial in P. The usual theory of globular and cubical higher categories can be translated in a natural way into our general context.