In recent years, the technological landscape has borne witness to the proliferation of embedded systems, low-power devices, and the Internet of Things, all of which have unique security needs. Moreover, the computing space has seen a push toward dedicated cryptoprocessors for the generation, storage, and management of cryptographic keys, for example, in the form of Trusted Platform Modules. There has been a growing interest in the field of lightweight cryptography, demonstrated by the recent introduction of a lightweight cryptography standard with the conclusion of the NIST Lightweight Cryptography Competition (2018-2023). A crucial aspect of designing lightweight cryptographic hardware is the investigation of primitives and structures that lend themselves to compact, low-power hardware implementations, with feedback registers serving as a popular example of a lightweight structure that has cryptographic applications. In particular, many lightweight cryptographic schemes from the literature use nonlinear feedback shift registers (NLFSRs) for the generation of exponential pseudorandom sequences. This dissertation proposes the use of a new class of feedback register (Product Registers) for lightweight cryptography, with emphasis on a derivative class of nonlinear feedback registers called Composite Mersenne Product Registers (CMPRs). Compared to NLFSRs, which are typically constructed using empirical evaluations, CMPRs provide mathematical guarantees for exponential average periodicity, are straightforward to construct for arbitrary state sizes, and admit an exponential number of constructions for a given state size. The use of Product Registers in various cryptographic schemes, including stream ciphers, a message authentication code, and a permutation-based cipher is covered, in addition to hardware efficiency comparisons with lightweight schemes from the literature and applications to critical infrastructure security. This dissertation also furthers the theory of Product Registers by introducing another derivative class of nonlinear feedback register called Hybrid Registers, which enable the direct use of feedback registers from the literature with Product Registers. The overall contribution of this dissertation is in-depth design space exploration of feedback register-based lightweight cryptography.