# An Improved Fuzzy Vector Signature with Reusability
## 1 Introduction
### Related Work  
**Reusable Fuzzy Extractor. ** The concept of a fuzzy extractor and secure sketch was first proposed by [PERSON] et al. [6]. Following this, [PERSON] et al. [8] introduced the notion of reusability, meaning that additional keys could be securely generated for any helper data, which is required to regenerate the key, even if the helper data or key pairs were exposed. [PERSON] et al. [14] subsequently proposed a reusable fuzzy extractor based on the Decisional Diffie-Hellman (DDH) problem, which can tolerate a linear size of errors. However, their scheme requires that, for any two distinct readings of the same source, the differences between them should not leak significant information about the source--a requirement that is too strict, as each component of fuzzy (biometric) data is non-uniformly distributed. In response to this, [PERSON] et al. [15] proposed a DDH-based reusable fuzzy extractor to remove the requirement [14] by changing the source distribution. Apart from these studies, [PERSON] et al. [13] also proposed a reusable fuzzy extractor that allowed a linear fraction of errors based on the Learning with Errors (LWE) assumption [22]. However, their schemes [13; 14; 15] used a secure sketch as a method for controlling noise of data. A secure sketch is used for recovering \(w\) from \(w^{\prime}\) if \(w\) and \(w^{\prime}\) are close to within a fixed distance, but it nevertheless causes a small leakage of biometric information. Therefore, when considering reusability scenarios, it is noted that these schemes require an input source to have sufficiently high min-entropy to avoid brute-force attack even after security loss. On the other side of the spectrum are reusable fuzzy extractors that did not suggest using secure sketches. [PERSON] et al. [17] proposed a reusable fuzzy extractor that is constructed using the subset-based sampling technique instead of a secure sketch to control a noisy data source. This scheme relied on the use of a digital locker to generate helper data. However, even if a digital locker was simply instantiated with a hash function, the size of helper data would increase significantly (e.g., about 0.8 GB to achieve an error tolerance rate of 20%). [PERSON] et al. [18] modified this scheme [17] to reduce the size of the helper data, but in turn, the computational cost for **Rep** increased significantly. [PERSON] et al. [12] also improved [17] and suggested a fuzzy extractor that achieved a linear fraction of errors. However, they also used a digital locker and this scheme had an unrealistic requirement that each component of a fuzzy (biometric) source had to have significant min-entropy. [PERSON] et al. [16] modified a non-reusable fuzzy extractor [23] based on the LWE assumption into a reusable one. However, their scheme can tolerate only a logarithmic fraction of errors due to the time-consuming process of reproducing a key even with a small number of errors. **Fuzzy Signature. ** The concept of a fuzzy signature was proposed by [PERSON] et al. [7]. The fuzzy signature does not need helper data, unlike the fuzzy extractor, because it only requires a valid correlation between two input fuzzy (biometric) data relevant to a verification key and a signature, which can be achieved using a linear sketch. However, in Reference [7], a strong assumption was required that input fuzzy (biometric) data should be uniformly distributed, which is then relaxed in Reference [24]. Afterwards, [PERSON] et al. announced that a linear sketch of fuzzy extractors [7; 24] is vulnerable to recovery attacks [25]. In Reference [26], the merge of Reference [7; 24], [PERSON] et al. proposed two instantiations of a fuzzy signature secure against recovery attacks. In 2019, [PERSON] et al. first introduced the notion of reusability in a fuzzy signature [19]. To construct a reusable fuzzy signature, they adopted a reusable fuzzy extractor in Reference [16]. Consequently, the reusability is limited only to the generation of verification keys, ignoring the privacy of signatures. **Fuzzy Vector Signature.