Two double-spend attack strategies on a proof-of-stake consensus are considered. For each strategy, the probability of its success is obtained, which depends on the network parameters and the number of confirmation blocks. These results can be used to define how many confirmation blocks a vendor should wait after a correspondent transaction before sending goods or services.

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.

The properties of quadratic and twisted supersingular Edwards curves that form pairs of quadratic twist with order over a prime field are considered. A modification of the CSIDH algorithm based on odd degree isogenies of these curves is considered. A simple model for the implementation of the CSIDH algorithm in 3 minimal odd isogeny degrees 3, 5, 7, with the prime field modulus and the order of supersingular curves is constructed. At the precipitation stage, the parameters of isogenic chains of all degrees for these two classes of supersingular Edwards curves are calculated and tabulated. An example of the implementation of the CSIDH algorithm as a non-interactive secret sharing scheme based on the secret and public keys of Alice and Bob is given. A new randomized CSIDH algorithm with a random equiprobable choice of one of the curves of these two classes at each step of the isogeny chain is proposed. The choice of the degree of each isogeny is randomized. The operation of the randomized algorithm by an example is illustrated. This algorithm as a possible alternative to "CSIDH with constant time" is considered. A combination of the two approaches is possible to counter side channel attacks. Estimates of the probability of a successful side-channel attack in a randomized algorithm are given. It is noted that all calculations in the CSIDH algorithm necessary to calculate the shared secret are reduced only to calculating the parameter of the isogenic curve and are performed by field and group operations, in particular, scalar point multiplications and doubling points of the isogeny kernel. In the new algorithm we propose to abandon the calculation of the isogenic function of random point , which significantly speeds up the algorithm.

Among the most significant problems that almost any blockchain faces are the problems of increasing its throughput (i.e., the number of transactions per unit of time) and the problem of a long waiting time before block confirmation. Thus, for example, in the most common BTC blockchain, according to various estimates, throughput is from 3 to 7 tps (transactions per second), and the average block confirmation time (block is considered confirmed if it has at least 6 blocks over it) is 1 h. At the same time, it is impossible to solve these problems directly by increasing the block size or increasing block generation intensity because this leads to essentially a decrease in the security of the blockchain in the first turn against double spend and splitting attacks. Such problems lead to the inconvenience of the practical use of cryptocurrencies to pay for goods and services. Proposed a few years ago, the PoP consensus protocol potentially helps to solve the problem of increasing blockchain throughput, although it was originally intended to ensure the stability of “young” blockchains, with “small” PoW, through the use of a secure blockchain, such as BTC. A blockchain that has provable security is called the security-provided blockchain (SPB), and one that uses SPB to achieve its security is called the security-inherited blockchain. In this paper, we give explicit formulas which describe how the number of confirmation blocks in the security-inherited blockchain, which is sufficient to achieve a given security level of this blockchain to a double spend attack, depends on the parameters of both blockchains. It is essential that we use a realistic model to obtain the results, taking into account the synchronization times of both blockchains. Such a model is much closer to the real situation, but at the same time, it leads to significant analytical difficulties in obtaining results. The obtained formulas are convenient for numerical calculations, the numerous examples of which are also given in this work.