## 2 Related Works  
_The Fuzzy Vault Scheme_  
The fuzzy vault scheme is the most popular key-binding-based bio-cryptographic algorithm. It can help share secret information such as a password, PIN, or short key for protecting biometric templates. The key in a conventional cryptographic algorithm has the same bit sequence every time it is used. However, biometric features have a different bit sequence at every sensor reading, making it almost impossible to generate a conventional cryptographic key from biometric features. Instead of devising a key generation scheme from biometrics, the fuzzy vault scheme is an alternative solution for applying biometric-based security advantages to existing cryptography by binding the secret key with biometric features. The fuzzy vault scheme, proposed by [PERSON] and [PERSON] [13], is a biometric cryptosystem based on the polynomial reconstruction problem and error-correcting codes. It is an improvement on the previous research by [PERSON] and [PERSON] [12]. Since its creation, more practical fuzzy vault schemes have been proposed by many researchers; a representative work is the fingerprint-based fuzzy vault scheme proposed by [PERSON] et al. [14; 15]. [PERSON] and [PERSON]'s fuzzy vault scheme consists of LOCK and UNLOCK algorithms that encode and decode a secret message using biometric features same as Algorithms 1 and 2. The parameters are a finite field \(F\) of cardinality \(q\), a constructed polynomial \(P(x)\), polynomial degree \(k\), a number of genuine points \(t\), a number of chaff points \(r\), the number of all points \(n(k\leq t\leq r<n<q)\), a secret \(S\), biometric features \(f=\alpha_{1},\ldots,\alpha_{t}\) and \(f^{\prime}=\alpha_{1}^{\prime},\ldots,\alpha_{t^{\prime}}^{\prime}\) (\(t\leq t^{\prime}\)), a set of genuine points \(G=(\alpha_{1},p(\alpha_{1})),\ldots,(\alpha_{t},p(\alpha_{t}))\), a set of chaff points \(C=(\alpha_{1},y_{1}),\ldots,(\alpha_{r},y_{r})\), a set of all points \(R=(\alpha_{1},p(\alpha_{1})),\ldots,(\alpha_{t},p(\alpha_{t})),(\alpha_{t+1},y _{t+1})\), \(\ldots,(\alpha_{n},y_{n})\) when \(n=t+r\), and a set of candidate points \(Q=(\alpha_{1}^{\prime},y_{1}^{\prime}),\ldots,(\alpha_{t}^{\prime},y_{t}^{ \prime})\). ```
input :A secret \(S\) and biometric features \(f\) output :A set of all points \(R\)
1 Initialize \(R\) to 0;
2 Construct \(P(x)\);
3 The secret \(S\) is divided into \(s_{0},\ldots,s_{k-1}\) and encoded in coefficients of \(P(x)=s_{0}+s_{1}x^{1}+\ldots+s_{k-1}x^{k-1}\);
4 Generate \(G\);
5\(G\) is the output of the polynomial projection \((\alpha_{1},p(\alpha_{1})),\ldots,(\alpha_{t},p(\alpha_{t}))\) by calculating \(P(x)\) with \(f\), for which each element is an input of \(P(x)\). The cardinality of \(G\) is \(t\), and every element in \(G\) is a genuine point on the polynomial;
6 Generate \(C\) ;
7\(\alpha_{i}\in F_{q}\) and \(y_{i}\in F_{q}\) (\(t+1\leq i\leq n,n=t+r\)) are randomly picked. It is checked whether \(\alpha_{i}\
otin f\) and \(y_{i}\
otin p(\alpha_{1}),\ldots,p(\alpha_{t})\). If not, choose and check a random number repeatedly. Every \((\alpha_{i},y_{i})\) in \(C=(\alpha_{1},y_{1}),\ldots,(\alpha_{r},y_{r})\) is a chaff point that does not lie on \(P(x)\);
8 Return \(R\) as a permutation of \(G\) and \(C\);
```  
**Algorithm 1**Locking Algorithm in Juels and Sudan's Fuzzy Vault Scheme  
After the introduction of the fuzzy vault scheme, [PERSON] [14] proposed a more practical version with fingerprints. Many modified fuzzy vault schemes have since been proposed based on these preliminary versions. Some of these have been analyzed, and their efficiency or security vulnerabilities compared [47; 48; 49]. In these previous works, researchers have encountered the problems of decreasing recognition rates with increasing chaff points, practical implementation of polynomial reconstruction using Reed-Solomon decoding, and correlation attacks that reconstruct a polynomial by exploiting the similarity of biometric features [17; 18; 35; 36; 45; 50].