Applications of finite-field wavelets, paraunitary matrices, and multivariate polynomials in the design of efficient cryptographic algorithms for resource-limited devices and wireless sensor nodes is the main topic of this thesis. In this research, multivariate paraunitary matrices over fields of characteristic two are of special importance. Therefore, the factorization of their bivariate counterpart into the product of fully-parameterized building blocks was studied. Result were a two-level factorization algorithm and new building blocks over the ring of polynomials that allow a complete first-level factorization. One of the contributions in this thesis was a completely new design for self-synchronizing stream ciphers based on wavelets over fields of characteristic two. Since these wavelets can be efficiently designed and implemented using paraunitary matrices, the designed cipher is highly efficient in terms of encryption and decryption complexities. The cryptanalysis of the proposed cipher did not reveal any vulnerabilities to the current state of the art attacks developed for stream ciphers. A completely novel framework for the design of multivariate asymmetric cryptosystems (based on paraunitary matrices) is a main contribution in this thesis. Using algebraic properties of paraunitary matrices, the computational security of systems designed based on this framework was studied. It was proved, for the first time, that breaking any instance of such systems provides a positive answer to an algebraic longstanding (non- computational) open problem. Therefore, the proposed framework certainly is an improvement toward the design of provably secure multivariate cryptosystems. Using this approach, a public-key cryptosystem and a digital signature scheme was proposed. Considering the attractiveness of algebraic techniques, their applications in the design of cryptographic algorithms for wireless sensor networks was investigated. A novel key pre-distribution scheme for data confidentiality in sensor networks was proposed. This scheme outperforms all previous designs in terms of network resiliency against the node capture. Theoretical analysis showed improvement over previous schemes and also robustness in design. In addition to key pre-distribution, a location-aware scheme was proposed that provides authenticity and availability for sensor networks. Main ingredients of this scheme are node collaboration for entity authenticity, hash tree for data authenticity, and random network coding for data availability. This scheme is the first one in its category that provides a practical solution to all the aforementioned security services.