# An Improved Fuzzy Vector Signature with Reusability
## 5 Performance Analysis
### Storage or Transmission Costs  
In the authentication phase, helper data \(P_{ID}\) is needed to generate a signing key from user biometric data. There are two options through which the user obtains an \(P_{ID}\). The first way is to store it in their own personal device, and the other is to receive it from the server for each instance of authentication. The first method can reduce the amount of transmission, while carrying a personal storage device. Conversely, the second one does not require a personal device, while the server sends out a huge amount of transmission, and it has the advantage in that the user authentication can work on secure devices that are shared by multiple users. For comparison purposes, we present the size of \(P_{ID}\) as the storage or transmission cost in Table 2.  
Figure 1: Authentication with a fuzzy extractor. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(n\) & \begin{tabular}{c} **Error Tolerance** \\ **Rate (\(t/n\))** \\ \end{tabular} & \begin{tabular}{c} \# of Components \\ **In Each Subset (\(\ell\))** \\ \end{tabular} & \
\begin{tabular}{c} **\# of Subsets** \\ **(\(d\))** \\ \end{tabular} &
\begin{tabular}{c} **Helper Data** \\ **(\(P_{ID}\), **SP**)** \\ \end{tabular} \\ \hline \multirow{3}{*}{[17]} & 512 & & 80 & \(61.6\times 10^{5}\) & 0.83 GB \\  & 1024 & 20\% & 80 & \(61.6\times 10^{5}\) & 0.90 GB \\  & 2048 & & 80 & \(61.6\times 10^{5}\) & 0.96 GB \\ \hline \multirow{3}{*}{[18]} & 512 & & \(16\) (\(\times 5\)) & \(53.6\times 10^{3}\) & 3.76 MB \\  & 1024 & 20\% & \(20\) (\(\times 4\)) & \(26.3\times 10^{3}\) & 2.01 MB \\  & 2048 & & \(27\) (\(\times 3\)) & \(11.6\times 10^{4}\) & 10.16 MB \\ \hline \multirow{3}{*}{[20]} & 512 & & 80 & \(61.6\times 10^{5}\) & 0.05 MB \\  & 1024 & 20\% & 80 & \(61.6\times 10^{5}\) & 0.09 MB \\  & 2048 & & 80 & \(61.6\times 10^{5}\) & 0.19 MB \\ \hline \multirow{3}{*}{Ours} & 512 & & 80 & \(61.6\times 10^{5}\) & 0.03 MB \\  & 1024 & 20\% & 80 & \(61.6\times 10^{5}\) & 0.06 MB \\  & 2048 & & 80 & \(61.6\times 10^{5}\) & 0.13 MB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of storage or transmission costs with helper data. Figure 2: Authentication with our fuzzy vector signature (FVS). Let \(n\) be the dimension of biometric data, \(d\) be the number of subsets, \(\ell\) be the number of elements in each subset, and \(t\) be the maximum number of errors among \(n\) elements. With fuzzy extractors [17; 18], helper data \(P_{ID}\) consists of an information set, a nonce, and an output of a hash function per subset. To begin, the number \(d\) of subsets is obtained via \(d\approx\ln(1/\delta)\cdot e^{\frac{t\ell}{n}}\) in Reference [17], whereas \(d\) is computed as \(d\approx m/2q\) in Reference [18], where \(q=\frac{\tau(\frac{n}{n})}{(\frac{n}{t})}\sum_{t=\tau}^{m}(-1)^{\eta-\tau}\cdot \frac{(\frac{n-\tau}{\tau-t})\times(\frac{n-\tau}{t})}{t}\) for \((\tau,m)\)-threshold scheme [32]. In this case, the information set per subset has \(\ell\) indices which are represented by \(\ell\log n\) bits. When using SHA-256 as the hash function, we set the size of a nonce to be sufficient at \(176(=256-80)\) bits. As a result, the whole size of \(P_{ID}\) is about \(d\cdot(\ell\log n+256+176)\) for \(n=\)512, 1024, and 2048 cases, which, as shown in Table 2, becomes huge when setting \(t/n=0.20\) as an error tolerance rate and \(\ell=80\). On the other hand, with FVS schemes, the number of subsets is determined by \(d\approx\ln(1/\delta)\cdot e^{\frac{t\ell}{n}}\) following [17], but helper data \(P_{ID}\) consists of only a signing parameter SP, regardless of the number of subsets. Indeed, in Figure 3, SP in Reference [20] is \(3n+1\) elements in \(\mathbb{G}_{1}\), whereas SP in ours is \(2n+2\) elements in \(\mathbb{G}_{1}\). When taking the Type-3 pairing [33] at the 100-bit security level, the size of elements in \(\mathbb{G}_{1}\) and \(\mathbb{Z}_{p}\) is 256 bits.