The issue of secure data exchange and performing external transactions between robust distributed ledgers has recently been among the most significant in the sphere of designing and implementing decentralized technologies. Several approaches have been proposed to speed up the process of verifying transactions on adjacent blockchains. The problem of search has not been under research yet. The paper contains security evaluation of data exchange between independent robust distributed ledgers inside multidimensional blockchain. Main principles, basic steps of the protocol and major requirements for it are observed: centralized approach, subset principle and robust SVP. An equivalence of centralized approach and ideal search and verification functionality is proven. The probability of successful verification in case of using fully connected network graph or equivalent approach with fully connected graph between parent and child blockchain is shown. The insecurity of approach with one-to-one links between child and parent ledgers or with a subset principle is proven. A robust search and verification protocol for blocks and transactions based on the features of robust distributed ledgers is presented. The probability of attack on this protocol is mostly defined by the probability of attack on verification and not on search. An approach to protection against an attacker with 50% of nodes in the network is given. It is based on combination of various search and verification techniques.