Blockchain sharding enhances scalability by dividing the network and ledger into parallel-processing shards, but dynamic sharding introduces security risks, especially under adaptive adversarial attacks. This paper presents a rigorous discrete mathematical framework for secure and encrypted dynamic sharding. It models shard formation, node assignment, and reconfiguration as discrete-time stochastic processes over dynamic graphs, with cryptographic hash functions ensuring secure shard allocation. Security is analyzed using combinatorics and probability theory, while a Markov-chain-based model captures system evolution under adversarial influence. The paper derives bounds on shard compromise probability using binomial distributions and Chernoff bounds. Additionally, a key-evolving encryption mechanism is introduced to ensure correct and forward-secure state transitions. Theoretical results show that shard takeover probability decreases exponentially with shard size, while encrypted state migration preserves consistency. Overall, this work provides a mathematically rigorous foundation with provable security and correctness guarantees for scalable blockchain systems.