One of the challenges that wireless sensor networks (WSNs) need to address is achieving security and privacy while keeping low power consumption at sensor nodes. Physically unclonable functions (PUFs) offer a challenge–response functionality that leverages the inherent variations in the manufacturing process of a device, making them an optimal solution for sensor node authentication in WSNs. Thus, identifiability is the fundamental property of any PUF. Consequently, it is necessary to design structures that optimize the PUF in terms of identifiability. This work studies different architectures of oscillators to analyze which ones exhibit the best properties to construct a RO-based PUF. For this purpose, Generalized Galois Ring Oscillators (GenGAROs) are used. A GenGARO is a novel type of oscillator formed by a combination of up to two input logical operations connected in cascade, where one input is the output of the previous operation and the other is the feedback signal. GenGAROs include some previously proposed oscillators as well as many new oscillator designs. Thus, the architecture of GenGAROs is analyzed to implement a GenGARO-PUF on an Artix-FPGA. With this purpose, an exhaustive study of logical operation combinations that optimize PUF performance in terms of identifiability has been conducted. From this, it has been observed that certain logic gates in specific positions within the oscillator contribute to constructing a PUF with good properties, and by applying certain constraints, any oscillator generated with these constraints can be used to construct a PUF with an equal error rate on the order of or below 10−11 using 100-bit responses. As a result, a design methodology for FPGA-based RO-PUFs has been developed, enabling the generation of multiple PUF primitives with high identifiability that other designers could exploit to implement RO-based PUFs with good properties.