Key distribution system based on Hadamard designs

The problem of key distribution in a community for providing secure communication between its participants is studied. To solve this problem, key predistribution systems can be used, in which each user receives some key information that can later be used to independently calculate required shared secret keys for conferences they participate in. Such key distribution systems can be based on different structures, such as error-correcting codes and combinatorial designs. The drawback of such systems is the possibility of collusive attacks, when traitors within the system can form a coalition and use their key information to try to calculate shared secret keys of other users. But the secrecy of keys is guaranteed by the system when the number of traitors in the coalition does not exceed a threshold defined by the system structure. In this paper, a key distribution system is based on combinatorial designs and, in particular, on Hadamard 3-design that guarantees the secrecy of communications in the presence of coalitions with less than three users. New notions of combinatorial span and combinatorial rank of a subset of Hadamard code that are required for the study of the resilience of the system to collusive attacks are introduced. The probability of successful collusive attack on an arbitrary conference against the cardinality of coalition is calculated for this system.