AppendixA. Measures for the security level of a biometric system

The performance of a biometric system is primarily defined by its security level, and three measures are frequently employed to evaluate the performance of a system:

False acceptance rate (FAR) is the frequency of unauthorized access instances caused by individuals falsely claiming an identity that is not their own.

False rejection rate (FRR) is the frequency of rejection of individuals who should be accurately verified.

Error equal rate (EER), at which both the FAR and FRR attain identical values, can be employed as a distinctive metric to characterize the security level of a biometric system.

Several preliminary studies formulated probabilistic definitions of these measures and demonstrated a fundamental relationship between measures [28, 29]. In this study, the proposed method distinguished eye-written patterns, not users; therefore, the imposters were incorrect patterns, and the authentics were password patterns. Therefore, the FAR and FRR were defined as

$$FAR\left(\mu \right)= \frac,$$

(2)

$$FRR\left(\mu \right)= \frac,$$

(3)

respectively, where \(FN\) denotes false negative, \(TP\) denotes true positive, \(FP\) denotes false positive, and \(TN\) denotes true negative. The FAR and FRR depend on the threshold \(\mu\), which was determined in the fitting threshold stage in this study.

B. Each class of template constructs one or fewer dimensions

Let there be ten distinguishable classes that are template patterns \(^,^,^,\dots ,^\). For one of pattern \(^\) such that \(^\in ^\), the vector \(}}^}}\) can be calculated from DPW values with every element of template set.

$$}}^}}=\langle _^,_^,_^,...,_^,...,_^\rangle ,\hspace$$

(4)

where \(k\)th component is evaluated by averaging all DPW values with elements of \(k\)th class of template set, where \(k\) is an integer of \(1 \le k \le 10\).

Because the output of DPW is a positive real value, which increases when the calculated two patterns are completely different, there is a real number \(\tau\) that has significantly arithmetic difference between \(\text _^/\tau\) and \(\text_^/\tau\), where \(k\ne i\).

$$}}^}}=-\text\frac}}^}}}\hspace\hspace \hspace$$

(5)

$$}}^}} = \left\langle }\frac^ }}, - \log }\frac^ }}, - \log }\frac^ }},..., - \log }\frac}^ }}} \right\rangle$$

(6)

By subtracting constant \(c\), which is

$$c=\frac__^, \hspace \hspace \hspace\hspace\hspace\hspace$$

(7)

the \(i\) th component of vector \(}}^}}\) is remarkably large, and the other components are relatively close to zero.

Because the space in \(}\) is an \(n\)-dimensional real space, the vector field and space are well defined.

C. Patterns provided to participants in this study

See Figs. 7 and 8.

Fig. 7

Ten template patterns used in this study. Ten template patterns were designated so that the EOG signals of eye-written patterns were different from each other because an arbitrary pattern was used as an input and the distance between each template pattern was computed

Fig. 8

Five test patterns used in this study