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Four optical combinational circuits based on hybrid plasmonic waveguides utilizing squared-shaped nanoring resonators have been presented, designed, and studied standing on finite element method with COMSOL software package version 5.5. The optical combinational circuits half adder, half subtractor, full adder, and 4-bit binary to gray converter have been established in a layout with dimensions of 850 nm × 400 nm and 1750 nm × 400 nm for the 4-bit converter. The leading guide to building these optical circuits is the constructive and destructive interference founded on the phase shift angles between the input ports and the control port(s). The performance of the four optical combinational circuits has been evaluated based on the transmission value at each output port which has a threshold of 30%, contrast ratio, modulation depth, and insertion loss, with a system for an operating wavelength of 1310 nm. The transmission exceeds 100% in the full adder, in the output carry of the half adder, and in the most significant bit of the 4-bit binary to gray converter with a 202.6% each. Forty-six decibels is the value of the contrast ratio in the most significant bit of the converter and 99.99% as a modulation depth for the sum output of the full adder. [ABSTRACT FROM AUTHOR]