Among biometric recognition systems, a system using brain waves via EEG will hold a special place. The EEG signal, with its nonlinear structure, is unique to the individual and nearly impossible to replicate. In designing such a system, various signal processing and classification methods are considered. In this study, nonlinear features such as Fractal Dimension, Second Order Sample Entropy, Quantities Graph, and Visibility Graph were used, allowing the examination of the EEG signal independently of the amplitude scale. To reduce computational load, the resting state alpha waves, which are prominent features of the EEG, were focused on, and a low number (8) of electrodes were used. The obtained features were analyzed separately for each electrode, aiming to identify the most distinctive feature and electrode. The classification was performed using five different machine-learning methods. The highest accuracy was achieved by the Random Forest algorithm. The most distinctive electrode and features were identified as the Fractal Dimension of the F5 electrode and the Fractal Dimension of the Oz electrode.