Conventional cryptographic systems such as factorization and discrete logarithm problems are threatened by the quick development of quantum computing. This paper offers a thorough study of number-theoretic problems and structures underlying post-quantum cryptographic protocols based on modular arithmetic, algebraic structures, lattices and computational hardness assumptions. The research is based on the mathematical problems of Learning with Errors (LWE) and Short Integer Solution (SIS) which are used for secure quantum-resistant encryption, digital signature and key exchange mechanisms. In secure communication scenarios, a framework of structured protocol is analyzed, which combines lattice-based operations and modular polynomial transformations. Experimental comparisons show that these schemes are not only practically efficient in computation but also have a level of security greater than 95% against the quantum attack, both for the LWE and SIS schemes. The results point towards the scalability, robustness and future applicability of number theoretic post quantum cryptographic systems.