Classical encryption algorithms are in serious danger from quantum computing, so new algorithms that can’t be broken by quantum computers need to be made. This essay looks at discrete mathematical concepts that are important for making these kinds of algorithms. It mainly talks about number theory, lattice theory, and coding theory. We look at both code-based and lattice-based cryptographic methods, which look like good options for security after quantum computing.  There are hardness assumptions that can’t be broken by quantum attacks in lattice problems like the smallest vector problem, and error-correcting codes are the basis of code-based cryptosystems that have been shown to be strong.  We provide a strict framework that makes these quantum-resistant algorithms safer and more efficient by using vector measure spaces, orthogonal sets, and sequences.