Designing error-correcting and combinatorial cryptographic solutions using discrete mathematics

In this paper, we introduce an innovative method of secure sharing of data in distributed systems, which is based on the combination of Error-Correcting Codes (ECCs) and combinatorial cryptographic algorithms, rooted in the realms of Discrete Mathematics. The desired solution utilizes finite fields, combinatorial block designs, and graph-theoretic models to overcome critical obstacles to robust information integrity and confidentiality in adversarial communication environments. Our construction yields stronger ECCs for reliable identification based on pliant combinatorial structures. We provide experimental results showing up to a 20% increase in ECC error correction and 15% reduction in cryptographic overhead, while maintaining the scalability and fault-tolerance. This integrated approach results in a secure, efficient, and mathematically sound communication scheme for distributed systems, and can be applied not only in IoT, cloud networks, but also in secure data transmission in adversarial environments.