Torsion subgroups are important in algebraic structures, and in the current cryptographic systems, especially elliptic curve systems. In this paper the theoretical background of the torsion subgroups is addressed in the framework of the structure theorem of finite abelian groups and extended to the argument of an elliptic curve and algebraic groups. The focus is on the definition of the torsion points and their algebraic properties that carry the interesting implication on the security and efficiency of cryptographic protocols. Torsion subgroups play an important role in elliptic curve cryptography (ECC) since they are applicable to key generation, subgroup confinement and subgroup resistance to small subgroup attacks. Their generalizability to zero-knowledge proofs is also covered, and specifically in the construction of secure and verifiable protocols. The paper contains both, analytical and comparative analysis showing how torsion-based properties can be utilized to provide robustness, scalability and provide a better cryptographic security to the modern distributed and privacy preserving systems.