Physical unclonable functions (PUFs) are new hardware security primitives proposed for protecting resource-constrained devices. This paper presents two PUFs in voltage-and current-modes. The new designs are based on carbon nanotube field effect transistors (CNTFETs). Sensitivity to strong process variation is considered as a demerit of this emerging nanoscale device. However, this deficiency can be the source of constructing unique PUF instances. The proposed circuits are simulated and tested by Synopsys HSPICE using a standard 32nm CNTFET technology. The properties of randomness, uniqueness, reliability, energy efficiency, and area of the implemented CNTFET-based PUFs are evaluated showing very promising results.