Fuzzy Vault Security Enhancement Avoid Statistical Biases

Sara Majbour, Morgan Barbier and Jean-Marie Le Bars

Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYC, 14000 Caen, France

Keywords:

Authentication, Biometric, Fuzzy Vault, Statistical Biases.

Abstract:

We assess the fuzzy vault’s security against the exploitation of statistical biases, conducting bias examination

through features on a sample of biometric set. Our comparative analysis quantiﬁes the scheme’s vulnerability

to security-compromising attacks, using three bases of feature templates derived from real biometric databases

of various modalities, showcasing variable quality levels, and quantifying scheme weaknesses. This study

shows a decrease in the scheme’s security under such attacks and signiﬁcantly contributes to understanding

the fuzzy vault’s limitations regarding biases in the stored set. Moreover, we propose the ﬁrst solution without

requiring additional information, preserving the security of the fuzzy vault against such attacks.

1 INTRODUCTION

Biometric authentication systems enhance security

compared to traditional methods like passwords or

keys. They refer to individual traits like ﬁnger-

prints, facial features, retinal scans, and iris patterns

for identiﬁcation (Dargan and Kumar, 2020; Daug-

man, 2004). To authenticate, biometric data is cap-

tured, converted into a digital template, and com-

pared to the enrolled template to conﬁrm the individ-

ual’s identity (Sharma et al., 2015). Since biometric

templates are sensitive, various cryptographic meth-

ods are implemented to secure them (Uludag et al.,

2004). A particularly effective solution in address-

ing the variability of biometric data involves adopting

the cryptographic scheme of the fuzzy vault, devel-

oped by (Juels and Sudan, 2002). This scheme, incor-

porating error correction codes and an unordered set,

enables error-tolerant authentication while preserving

the conﬁdentiality of the data. The standard fuzzy

vault process hides a set by connecting it to a nonce

with Reed-Solomon error correction codes. Authenti-

cation succeeds when a presented set closely matches

the reference.

Despite signiﬁcant progress in studying fuzzy

vault schemes recently, each proposed approach is

tailored to the context examined and the biometrics

modalities used (Uludag et al., 2005; Nandakumar

et al., 2007a; Rathgeb et al., 2023). Numerous stud-

ies have adapted the scheme to biometrics, focusing

primarily on security (Benhammadi and Bey, 2014;

Radha et al., 2010). The original article established

an upper security bound based on the assumption of

uniform distribution. However, deviations from this

assumption in real data distributions have been noted,

which could lower the security level.

It has been recognized that biases exist, but the

correlation between their nature, magnitude, and im-

pact on fuzzy vault security is yet to be deﬁned. Our

study quantiﬁes the impact of the lack of a uniform

distribution on the fuzzy vault security deterioration,

highlighting its overall signiﬁcance by generally ren-

dering the scheme unusable. Additionally, we un-

derscore that the attacker model proposed by Juels

and Sudan, possessing partial knowledge of the in-

formation, lacks realism and relevance. In the lit-

erature, some studies have proposed incorporating a

password as a solution to mitigate bias issues and

enhance the fuzzy vault’s security against statistical

attacks (Benhammadi and Bey, 2014; Radha et al.,

2010). However, introducing passwords fundamen-

tally establishes strong multi-factor authentication,

differing from the original fuzzy vault scheme, and

other vulnerabilities may arise with a multi-factor

scheme. No secure single-factor fuzzy vault solution

has been proposed yet. Our investigation speciﬁcally

focuses on examining the single-factor fuzzy vault

as initially proposed, to lead to a new secure single-

factor system.

Our approach manages feature templates generic

from various biometric modalities. We use three

bases of biometric templates-ﬁngerprint, face, and

electrocardiogram, represented as feature vectors. A

distinctive attribute of the fuzzy vault is its use of

Majbour, S., Barbier, M. and Le Bars, J.

Fuzzy Vault Security Enhancement Avoid Statistical Biases.

DOI: 10.5220/0012715500003767

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 21st International Conference on Security and Cryptography (SECRYPT 2024), pages 40-51

ISBN: 978-989-758-709-2; ISSN: 2184-7711

an unordered set of elements. In our work, the term

biometric set refers to the outcome of transforming a

biometric template of features, into a set of elements

within a ﬁnite ﬁeld, as illustrated Figure 1.

Figure 1: Biometric set construction for fuzzy vault.

Our analysis focuses speciﬁcally on applying the

fuzzy vault scheme as originally proposed, without

incorporating additional information altering the au-

thentication type. For the ﬁrst time, we introduce the

quantiﬁcation of the signiﬁcant advantage of an attack

exploiting biases by features within a sample biomet-

ric set. Our results from different template bases show

that the scheme as presented in the original article is

ineffective and lacks relevance against such attacks.

Taking into consideration these biases, we propose

the ﬁrst solution for applying the fuzzy vault scheme

without passwords, using quantile methods during the

transformation into a biometric set to ensure an equi-

table distribution of elements for each feature. The

aim is to ensure the security of the single-factor fuzzy

vault authentication scheme and neutralize attacks ex-

ploiting biases by features.

The remainder of this article follows this structure:

Section 2 recalls the fuzzy vault concept and exist-

ing work on its application in biometrics. Then, Sec-

tion 3, we explain our generic methodology for any

biometric template base in a feature vector format.

Within Section 4 we outline the conditions of the up-

coming attack and the biometric templates used. Then

Section 5, we present the obtained results. This un-

derstanding of biases allows us to propose Section 6

the ﬁrst solution to secure the fuzzy vault without any

additional information.

2 BACKGROUND

In this section, we aim to provide a more detailed ex-

planation of the fundamental encoding and decoding

process of the fuzzy vault scheme. Following that, we

present an extensive review of prior research on ap-

plying this cryptographic scheme in a biometric do-

main.

2.1 Fuzzy Vault

The fuzzy vault concept is a versatile crypto-

graphic approach that applies to various domains, like

privacy-protected matching, personal entropy sys-

tems, and biometrics. Its strength lies in effectively

handling differences between sets and having the ca-

pability to correct them, which is a standout feature

of this approach. Its primary objective is to authen-

ticate an individual based on a comparison between

an authentication set B, and another enrollment set

A, concealed within a vault using a nonce K . Lin-

ear error correction codes, particularly Reed-Solomon

codes (Juels and Sudan, 2002), are crucial to ensure

reliable and efﬁcient recovery of the nonce in this

cryptographic system, allowing to handle variability

in the enrollment and authentication sets effectively.

2.1.1 Enrollment Stage

The ﬁrst stage, referred to as enrollment, involves

recording user information into the system. This cor-

responds to a reference set V within the fuzzy vault,

known as the vault. To construct it, a Reed-Solomon

error correction code is employed, incorporating a

biometric set A with n elements from a ﬁnite ﬁeld

, along with a random nonce vector K of length k .

where each component corresponds to a coefﬁcient of

the polynomial P ∈ F

[X]. The choice of n exceeding

k introduces redundancy to permit error detection and

correction during decoding. Concurrently, to ensure

conﬁdentiality, additional chaff points are uniformly

integrated, forming the vault V as illustrated Fig-

ure 2b. This stage consists of two steps: encoding

and adding noise.

a) Encoding: the biometric set is encoded with the

nonce K by associating the elements of A with

their polynomial evaluations. Each element x in A

is evaluated by applying the polynomial P, result-

ing in a value P(x) = y. These pairs (x,y) repre-

sent points in the fuzzy vault (Refer to Figure 2a).

b) Noise: at this step, random pairs (x

′

) ∈ F

known as chaff points, are introduced into the set

V . They are speciﬁcally chosen to not correspond

to genuine points of A and do not follow the pat-

tern of polynomial evaluations: ∀x

′

/∈ A, y

′

̸=

P(x

′

) (see Figure 2b). Adding these chaff points

aims to make it challenging for attackers to dis-

cern genuine points from chaff points within set

V .

2.1.2 Authentication Stage

The subsequent stage, denoted as authentication,

serves to verify the user’s identity and grant access to

Fuzzy Vault Security Enhancement Avoid Statistical Biases

Elements of A

(a) Encoding.

Elements of A

Chaff points

(b) Noise.

Figure 2: Enrollment stage of the fuzzy vault scheme.

appropriate resources. Authentication evaluation de-

pends on the similarity between enrollment set A and

authentication set B, both of size n, irrespective of the

order of their elements.

The retrieval of the random nonce K used during

enrollment is contingent upon the similarity of these

sets. This condition is ensured by a decoding algo-

rithm associated with the codes used during the en-

rollment process, having the ability to correct up to e

errors, referred to as the decoding radius. Therefore,

to accurately recover the nonce K , we check if the

number of discrepancies between the two sets A and

B is bounded by e.

The authentication stage can be divided into three

steps: extraction, decoding, and veriﬁcation.

a) Extraction: from the reference set V , we extract

pairs (x,y) where x belongs to the authentication

set B; thus, the set Q of size less than or equal to n

is composed of the extracted pairs (see Figure 3a).

b) Decoding: in this phase, a Reed-Solomon decod-

ing algorithm is employed, receiving the set Q

and the length k as inputs. The algorithm produces

a secret K

′

if there’s adequate matching, facilitat-

ing potential error correction (see Figure 3b). Al-

ternatively, it may return a null value if no polyno-

mial aligns with the decoding of the received set

Q .

c) Validation: authentication is successful, enabling

user authentication, only if the candidate nonce

′

is identical to the nonce K used during en-

rollment. This condition corresponds to the sufﬁ-

cient sharing of elements between two sets A and

B (see Figure 3c).

2.2 Related Work

Original fuzzy vault scheme proposed by Juels and

Sudan, is generic and can be applied to different pur-

poses. When employed in a biometric context, it re-

quires speciﬁc adjustments, primarily driven by con-

siderations related to security, the choice of modality,

the error correction codes parameters choice, and the

Elements of V

Elements of Q

(a) Extraction.

Elements of V

Elements of Q

(b) Decoding.

Elements of V

Elements of Q

Elements of A

Figure 3: Authentication stage of fuzzy vault scheme.

concept of a set which is a fundamental characteris-

tic of the fuzzy vault. This is why numerous different

works have been done, to use it in a practical world.

Highlighting the inability to directly implement this

scheme.

The initial difﬁculty in implementing the fuzzy

vault lies in transforming a biometric template into

a biometric set, a process inﬂuenced by the bio-

metric modality and extraction algorithm. Notably,

the minutiae template, is extensively explored in the

fuzzy vault implementations (Uludag et al., 2005;

Nandakumar et al., 2007a; Poon and Miri, 2012; You

and Wang, 2018). Each minutia is characterized by

three values (x, y,θ). The set is achieved through the

concatenation of bits derived from x,y, or x, y and θ.

Minutiae template is a rare case where the concept of

a set is inherently present. However, alignment issues

complicate this process (Merkle et al., 2010; Nagar

et al., 2008). Many studies have investigated the ap-

plication of the fuzzy vault with biometric templates

represented as feature vectors. The type of template

we are interested in our study for different modali-

ties. For example, in study (Lee et al., 2008), a se-

cure fuzzy vault system employs local iris features

and clustering techniques for accuracy, ensuring prox-

imity between sets A and B in both biometric data

and error-correcting codes. Another instance is seen

in (Rathgeb et al., 2016), which introduces a multi-

instance iris biometric cryptosystem, requiring a par-

titioning method for subsets based on each feature set.

The secondary difﬁculty pertains to the parameter

selection of an error correction code and its contex-

tual relevance. Various methodologies are employed

in reconstructing the secret K from Reed-Solomon

SECRYPT 2024 - 21st International Conference on Security and Cryptography

codes. Some researchers, faced with issues, opt for

using a secret structured by a code such as Cyclic

Redundancy Check (CRC), involving polynomial re-

construction through exhaustive search (Nandakumar

et al., 2007a; You and Wang, 2018; V.S.Meenakshi

and Padmavathi, 2010), aiming to avoid False Rejects.

However, considerations regarding execution time are

discussed (Khalil-Hani et al., 2013). Different stud-

ies have highlighted concerns about the efﬁciency and

this approach’s security, suggesting the use of decod-

ing algorithms for Reed-Solomon codes (Velciu et al.,

2015).

The scheme’s security is a major concern, eval-

uated in various contexts and against different at-

tack scenarios (Nandakumar et al., 2007b; Benham-

madi and Bey, 2014; Radha et al., 2010). Studies

focus on improving biometric data integrity, espe-

cially by assessing the False Acceptance Rate (FAR).

Another approach involves evaluating min-entropy in

protected models, offering insights into leaked infor-

mation and the maximum probability of uncovering

the secret (Merkle et al., 2010; Dodis et al., 2008). In

their theoretical framework, Juels and Sudan do not

explicitly consider biases in security analysis. They

rely on a uniform distribution, establishing a security

upper bound, and acknowledging the lack of justiﬁca-

tion for assuming uniformity. Other research supports

this assertion by noting that non-uniformity in stored

data can jeopardize the scheme’s security (Nandaku-

mar et al., 2007a; Merkle et al., 2010; Nagar et al.,

2008). To overcome this limitation, some studies pro-

pose adding a password (Reddy and Babu, 2008; Nan-

dakumar et al., 2007b; Benhammadi and Bey, 2014),

known as a Hard fuzzy vault. The user is required

to capture their biometric data and enter a password.

Currently, no proposal for a secure fuzzy vault against

such statistical attacks without additional information

is known.

3 BIOMETRIC SET AND BIAS

ASSESSMENT

This section details vault construction and its bias re-

sistance, particularly against database attacks. We

discuss the set creation function designed for feature

models, explore theoretical measures for bias assess-

ment in biometric sets, and review attacker models to

evaluate fuzzy vault security.

3.1 Construction Biometric Set

The fuzzy vault scheme is characterized by its use of

unordered sets. Our methodology employs a transfor-

mation approach tailored for feature templates from

diverse biometric modalities, conceptually similar to

the method in (Rathgeb et al., 2016) which encodes

already features as unordered sets. However, our ap-

proach differs by taking into account individual vari-

ability and speciﬁc parameters, thereby enhancing

authentication reliability and ensuring accuracy and

compatibility with various biometric systems.

Feature values in biometric templates vary across

captures from the same user. To ensure reliable identi-

ﬁcation and appropriate access, we use min-max nor-

malization to address this variability. This technique,

widely acknowledged for enhancing system reliabil-

ity through precise and consistent data representation,

is crucial for maintaining consistency in each fea-

ture’s distribution relative to the initial data and effec-

tively managing data variability (Zheng and Casari,

2018). Employing this approach also facilitates the

security assessment of the fuzzy vault system by ac-

counting for statistical biases in feature distributions

within biometric samples.

This methodology enables the grouping of multi-

ple feature values within a similar range. The normal-

ization process converts the real values of each fea-

ture i into values ranging from 0 to 1 (⋆), using the

minimum min

and maximum max

values associated

with each feature. Subsequently, the value is encoded

using m

bits, representing the exponent of a prime

number in our system. The selection of m

is based

on the template base, aiming to choose the value that

best distinguishes users. This selection results from

experiments to determine the optimal number of bits

to use, which may vary from one base to another.

Given a template containing n features, the nor-

malized value f

of each features f

, encoded in m

bits, is obtained by applying the following formula:

= ⌊

( f

−min

)

max

−min

× 2

⌋.

(⋆)

To construct a biometric set, we introduce the

function S that maps elements encoded with m

bits

into a biometric set within a ﬁnite ﬁeld F

, where

each element corresponds to speciﬁc features. Let S

represent function (⋆⋆), which takes as input the m

bits of the index, encoded with the binary length of the

biometric template size, and the m

bits of the feature

value obtained from (⋆). Through the concatenation

of these input elements, this function produces an el-

ement within the ﬁnite ﬁeld F

, where p = m

+ m

S : {0,1}

× {0, 1}

→ F

(i, f

) 7→ e = i| f

. (⋆⋆)

Fuzzy Vault Security Enhancement Avoid Statistical Biases

3.2 Bias Quantiﬁcation

Our analysis speciﬁcally focuses on statistical biases

in biometric sets by features, aiming to understand

how they affect the fuzzy vault system’s security and

quantitatively evaluate their effects. Using a training

sample TrainingS of the biometric template, we gen-

erate the biometric set from this sample using the spe-

ciﬁc transformation function (⋆⋆). We propose quan-

tifying these biases in two distinct scenarios.

A. Scenario 1: in this context, we disregard features

and calculate the frequencies of elements in the ﬁ-

nite ﬁeld F

within biometric sets of TrainingS.

This calculation is established by scaling the repe-

titions of each element e ∈ F

(denoted as rep(e))

by the product of sample size |TrainingS | and 2

representing the total number of elements (Rice,

2006):

freq(e) =

rep(e)

|TrainingS| × 2

B. Scenario 2: in this context, the signiﬁcance of the

order of features is considered. We calculate the

frequencies of elements f

for each feature, from

biometric sets of TrainingS . We compute the fre-

quencies of the 2

elements for each feature i,

using the following formula:

freq( f

) =

rep( f

)

|TrainingS|

To assess the overall distribution of these elements

in each scenario, we employ Shannon entropy, a met-

ric designed to measure the distribution of occur-

rences and its proximity to a uniform distribution.

Shannon entropy (Cover and Thomas, 2006), de-

noted H

(X,D), quantiﬁes the uncertainty of a ran-

dom variable X with outcomes in a ﬁnite ﬁeld and

their associated probabilities D = (p

,. .., p

), where

= Pr(X = e

). It is calculated using the base

2 logarithm to express information in bits and is

deﬁned as: H

(X,D) = −

∑

i=1

log

). Subse-

quently, for a uniform distribution D

, where all out-

comes are equally probable, the entropy simpliﬁes to

(X,D

) = log

(n), serving as a benchmark to com-

pare D against an ideal scenario of equal likelihood.

Finally, we propose assessing this outcome

through the measure of statistical bias, denoted as

M (D). This measure is deﬁned as the ratio of Shan-

non entropy of the calculated distribution to the en-

tropy of the uniform distribution, formulated as fol-

lows:

M (D) =

(X,D)

(X,D

)

The measure M generates a numerical value

lower than 1. Any deviation from M compared to

1 indicates that the distribution is far from being uni-

form. This observation facilitates the quantiﬁcation of

the effect of statistical biases, thereby aiding in evalu-

ating their inﬂuence on the security of the fuzzy vault.

3.3 Authentication Attacker Models

We evaluate the impact of statistical biases on secu-

rity by introducing an attacker model that uses knowl-

edge of biometric set distribution. This model con-

trasts with Juels and Sudan’s, which is based on par-

tial knowledge and is less clear. Additionally, we pro-

pose a model based on vault knowledge alone. To

compare these models and understand how vault size

and attacker knowledge inﬂuence security, we assess

authentication efﬁcacy across various vault sizes and

attack scenarios.

A- Distribution-Knowledge Attacker (DKA:) in this

model, the attacker, having gained access to the

vault, also possesses knowledge of the distribu-

tion of elements from the biometric set within the

ﬁnite ﬁeld. To generate n elements for an authen-

tication set B, the attacker draws from the vault

according to this distribution. When the vault’s

size is smaller than the size of elements with prob-

ability in the sample, it becomes vital to concen-

trate solely on the present elements in the vault.

To address this, we use a smoothing technique (Si-

monoff, 2012) to establish a new probability dis-

tribution. This method assigns a minimal, yet

nonzero, probability to elements not found in r.

This probability is contingent on the lowest prob-

ability within the sample, signifying that absent

occurrences are improbable but not impossible.

Using this updated distribution, the attacker ran-

domly chooses elements from r to compose n ele-

ments within the authentication set.

B- Partial Knowledge Attacker (PKA:) this model,

proposed in (Juels and Sudan, 2002), formalizes

the security framework by incorporating the con-

cept of partial knowledge attributed to the ad-

versary. Authors assume that the enrollment set

is chosen according to a potentially non-uniform

distribution, but they do not quantify security with

biases, they assume a scenario where an attacker

possesses knowledge of a part of the set. How-

ever, these are two distinct concepts. To illus-

trate, consider attempting to guess a password.

Knowing the distribution of passwords provides

a signiﬁcant advantage, but it doesn’t reveal spe-

ciﬁc letters in a given password from that distribu-

tion. Conversely, knowing speciﬁc letters offers

SECRYPT 2024 - 21st International Conference on Security and Cryptography

an additional advantage. Thus, these two forms of

knowledge appear complementary.

To identify values of the partial set α that hold

signiﬁcance for the attacker and serve as a real-

istic prerequisite, we analyze this attack scenario

to evaluate the vault’s security across different

knowledge levels. Our method entails being in-

formed about different values of α elements from

A. To complete the remaining n − α elements, we

use a uniform selection from the remaining ele-

ments of V .

C- Uniform Attacker (UA:) we intend to compare

the advantage of the previous model with that ob-

served when no prior information is available. In

this context, the attacker possesses only knowl-

edge about the vault itself. To generate the au-

thentication set B, the attacker uniformly selects

n elements from the r elements contained in the

vault. This implies that each element in the vault

has an equal probability of being chosen to be part

of the authentication set B.

Studying and comparing the attacker’s advantage

in these three models provides valuable insights into

assessing the impact on the security of the fuzzy vault.

This analysis aids in identifying the most sensitive or

vulnerable information within each model, facilitat-

ing adjustments in security measures and countermea-

sures accordingly.

4 FUZZY VAULT

CONSTRUCTION

In this section, we introduce the three biometric tem-

plate bases derived from different modalities, each

exhibiting unique quality levels. Quality Quality as-

sessment of each base relies on False Rejection Rates

(FRR) and False Acceptance Rates (FAR). When

these rates are equal, they represent the Equal Error

Rate (EER). Minimizing these rates enhances the sys-

tem’s authentication performance. Next, we present

the fuzzy vault parameters obtained for each base

used.

4.1 Biometric Templates Bases

Our approach is dedicated to biometric template bases

categorized by features, We use three bases derived

from different modalities (see Table 1), each ex-

hibiting varying levels of quality in terms of EER.

These include the FVC ﬁngerprint base, the PTB

base of electrocardiograms used in (Gernot and

Lacharme, 2022), and the higher-quality LFW base,

used in (Dong et al., 2019). Each base contains T

biometric templates for each of the N distinct individ-

uals. An extraction algorithm applied to the image

yields a feature vector of size n.

4.2 Fuzzy Vault Parameters

This section provides a concise overview of the es-

sential parameters for constructing the reference set

V , stored in the fuzzy vault. The speciﬁcations are

tailored to the speciﬁcities of each biometric template

base, as presented Table 2.

A. Biometric set size n: our set construction func-

tion establishes a mapping between each binary

feature sequence of m

bits and an element of the

ﬁnite ﬁeld F

associated with the biometric set.

The set size corresponds to the number of features

in the template speciﬁc to the biometric template

base, denoted as n.

B. Secret length k: the secret length k is directly

linked to e and inﬂuences the authentication algo-

rithm based on Peterson-Berlekamp-Massey. This

algorithm can correct up to e = ⌊

n−k

⌋ errors. The

minimal intersection I between the sets A and B

is crucial, where I = n − e = ⌊

n+k

⌋.

To select k, we calculate the FAR and FRR rates

for various threshold values I. The objective

is to achieve a balance between security (mini-

mized FAR) and user-friendly interaction (con-

trolled FRR). The optimal threshold I is deter-

mined as the intersection between the discrete

curves of FAR and FRR, corresponding to the

(EER) with the biometric set. If achieving this

equality proves impossible, a preferred threshold

is selected to minimize FAR. Depending on each

base, different thresholds are obtained. To main-

tain these levels of authentication as indicated Ta-

ble 3, we select k with consideration to variable

I, taking into account error corrections with the

decoding algorithm, k = 2I − n.

C. Vault size r: the construction of the set V re-

lies on the addition of chaff points, dependent on

the function S and parameters (m

) speciﬁc to

each biometric template base, ensuring their in-

discernibility from the biometric set’s elements.

The choice of variable m

is made to allow bet-

ter differentiation between biometric sets of dif-

ferent users and to manage variability among dif-

ferent templates from the same user. This decision

is also correlated with the FAR and FRR rates.

Thus, following tests on our bases, the appropri-

ate choice is m

= 2. As for m

, it represents

the smallest possible value satisfying the condi-

Fuzzy Vault Security Enhancement Avoid Statistical Biases

Table 1: Biometric templates bases.

FVC PTB LFW

Modality Fingerprints Electrocardiogram Face

Image

database

FVC2002 (Maio et al.,

2002)

LFW (Bousseljot et al., 1995; Gold-

berger et al., 2000)

(Huang et al., 2008)

Extraction

algorithm

Gabor ﬁlters (Belguechi

et al., 2016)

ECG wave delineation (Martinez

et al., 2004; Makowski et al., 2021)

deep network Insight-

Face (Dong et al., 2019)

N 100 158 158

T 8 7 10

n 512 990 512

tion n ≤ 2

, with n being the number of features

in the vector. Our construction of set V involves

choosing sizes ranging from n × 2 to n × 2

Table 2: Fuzzy vault parameters for each template base.

Fuzzy vault parameters

Biometric template base (m

) n k r

FVC (2,9) 512 65 [1024,2048]

PTB (2,10) 990 273 [1980,3960]

LFW (2,9) 512 9 [1024,2048]

The objective of the fuzzy vault is to enable com-

parison for authentication. Our encoding method ef-

fectively protects data with minimal impact on au-

thentication system performance, as evidenced by the

obtained rates Table 3. The degradation in the authen-

tication rate remains limited, with an observed impact

of about 7%. For example, in the FVC base, an EER

of 17% is obtained. However, in the PTB base, our

method leads to an improvement in authentication and

a reduction of approximately 3.8% in the EER. For

the LFW base, no EER is obtained, and therefore, we

choose a threshold of k = 9 for which FAR rates are

minimal and do not impact security.

Table 3: Authentication rates.

biometric template biometric set

Biometric template base EER FAR FRR EER

FVC 10% 17% 17% 17%

PTB 10.8% 7% 7% 7%

LFW 0.2% 9.5 × 10

−4

% 4% -

5 CONDUCTING

AUTHENTICATION ATTACKS

Based on the framework from Section 4, this section

evaluates the security and practicality of the fuzzy

vault. We detail quantitative results on its resilience

against statistical biases exploited through biometric

set features. Multiple vaults are constructed for each

base, followed by authentication tests against various

attacker models previously introduced.

During each testing phase, a sample TrainingS,

comprising 60% of individuals from the base, is used

to evaluate biases in the biometric set, with one tem-

plate per person. The remaining 40% sample, referred

to as TestS, is reserved for authentication testing. This

distribution is consistently maintained across all three

template bases. The initial stage in the fuzzy vault

process is enrollment. Considering sample TestS with

|TestS| = N ∗ 40%, we construct the associated vault

by incorporating the previously presented parameters,

generating vaults with sizes r ranging from 2n to

n×2

. During authentication stage, we create 50 au-

thentication sets B for each vault. This authentication

process is conducted using the three attacker models,

depending on the speciﬁc scenario being investigated.

The validity of authentication is conditioned by

the number of common elements between the authen-

tication set B and enrolment set A. We deﬁne two

sets to be close if their intersection is at least equal

to ⌊

n+k

⌋, and then the decoding algorithm guarantees

the correction of other errors. To calculate the authen-

tication rate for each vault, we determine the ratio of

valid sets B to the total number of constructed sets,

which is |TestS|×50. The success rates computed for

each attacker model will be compared to analyze the

impact of the acquired knowledge, as visualized in the

graphs below.

5.1 Results

To exploit biases, we focus on two attack scenarios.

The ﬁrst scenario relies on a global frequency analy-

sis, where the assembly of the biometric set is per-

formed without considering the order of elements.

Conversely, the second scenario takes into account

the speciﬁc distribution of each feature, thus adopt-

ing a more in-depth approach based on feature-based

construction. Starting from each of the three tem-

plate bases, we initially assess the biases inherent in

the associated biometric sets and perform quantitative

measurements M . Subsequently, we conduct authen-

tication by leveraging the quantiﬁed biases, as well as

considering two other attacker models. The outcomes

of the authentication rates are then presented in the

form of curve graphs for each reference vault set of

size r.

SECRYPT 2024 - 21st International Conference on Security and Cryptography

Every color in graph denotes the authentication

rate associated with a speciﬁc attacker model, the red

represents an attacker with knowledge of the distri-

bution (DKA), while the black signiﬁes a uniform at-

tacker (UA). The green, purple, and blue shades rep-

resent attackers with partial knowledge, each having

different levels of information (PKAα%).

A. Scenario 1: in this context, the order of features is

disregarded. From a sample of the biometric set,

the frequency of each element e ∈ F

is calcu-

lated, thereby determining the measure M .

The speciﬁc results for the FVC, PTB, and LFW

bases are 0.92 and 0.93, 0.94 respectively Table 4,

indicating that they deviate signiﬁcantly from a

uniform distribution. Notably, there is a range of

information concentration levels observed among

the three bases, reﬂecting their divergence from

uniformity. This observed diversity is associated

with the quality of the base. A less pronounced

dispersion is noted in the FVC database, which

exhibits an EER rate of 17%, while the results for

the LFW database, with a FAR of 4% and an FRR

close to 0%, show greater dispersion.

Table 4: M (D) of biometric set.

template base FVC PTB LFW

M (D) 0.92 0.93 0.94

Overall, the measurements indicate pronounced

diversity within the biometric set, with signiﬁcant

deviations from the uniform distribution. Speciﬁ-

cally, pronounced disparities are observed in bi-

ases among biometric sets from different bio-

metric template bases, particularly with the FVC

showing a more marked deviation with a measure-

ment of 0.92 compared to the uniform distribu-

tion, in contrast to the set from the base LFW

with a measurement of 0.94. Through subse-

quent tests, we will seek to determine whether the

slight differences observed among these measures

will have a signiﬁcant or negligible impact on the

vault’s security.

The results from Figure 4 indicate that, across all

bases, the advantages of the three applied attacker

models gradually decrease as the size of the vault

r increases. With the FVC base, it is observed

that knowledge of the set distribution, as depicted

by the red curve, provides a signiﬁcant advantage

to the attacker compared to the other two bases

where this advantage decreases for sizes starting

from 3 ∗ n.

For the partial knowledge attacker model, we pos-

sess, in each case, knowledge of proportions of

25%, 40%, and 50% of the enrollment biometric

set. The results obtained reveal that for all bases,

knowledge of the distribution is always more im-

portant than knowledge of 25% of the informa-

tion (represented in green), which still constitutes

a signiﬁcant bound. Even in the case of the FVC

with multiple vault size r Figure 4a, the advantage

obtained using the distribution is more signiﬁcant

than knowledge of 40%, and close to 50%. On

the PTB base, we achieve the best rate with the

red curve, surpassing the advantage gained with

the knowledge of 50% of the enrollment set Fig-

ure 4b. However, it is important to note that

these knowledge percentages are unrealistic and

do not represent a plausible attacker model, un-

like knowledge of the distribution, which seems

more practical.

The ﬁfth black curve illustrates the advantage of

a database-side attacker without any additional

information, attempting to build elements of the

set by drawing uniformly. It is observed that

the vault’s security remains intact, with the at-

tacker frequently unable to gain any advantage.

This supports the highest security threshold of the

fuzzy vault under uniform conditions, consistent

with assertions made by prior researchers. Com-

paring (DKA) and (UA) models, the red curve’s

notable advantage over the black indicates that ex-

ploiting set biases can weaken the fuzzy vault’s

security.

In connection with the base quality and measure

M , it is observed that the difference of 0.1 be-

tween the calculated measures Table 4 signiﬁ-

cantly impacts the vault’s security against the at-

tacker model considering the distribution. This

is evident when comparing the degradation of the

red curve across the three bases.

B. Scenario 2:

In this context, the relevance of knowledge

through distribution becomes more pronounced,

allowing for the provision of speciﬁc information

regarding the distribution of each feature. We cal-

culate the M

metric for each feature i among the

n in the template to assess the irregularity or dis-

persion of the feature values distribution.

The results from measuring feature biases reveal

pronounced differences across the various bases.

The majority of features for all three bases showed

a M value ranging between 0.6 and 0.8, suggest-

ing signiﬁcant biases when compared to a uniform

distribution. A few features, however, scored

above 0.9, indicating less bias. Notably, in the

FVC and PTB bases, some features recorded en-

tropy measures below 0.5. A few feature groups

even demonstrated distributions that were quite

Fuzzy Vault Security Enhancement Avoid Statistical Biases

(a) FVC. (b) PTB. (c) LFW.

Figure 4: Authentication for scenario 1.

structured or condensed, with measures below

0.3. These ﬁndings will be used to assess their

impact on the security of the vault.

In establishing the construction of the authenti-

cation set based on features, using each of the

considered models. For the partial knowledge

attacker model, we assume knowledge of 25%,

30%, and 35% proportions of the enrollment set.

The results presented Figure 5 highlight a signif-

icant deterioration in the vault’s security through

this feature-based approach. The three bases cor-

roborate this observation. In general, authentica-

tion rates in this scenario are signiﬁcantly higher,

attributable to additional speciﬁc information as-

sociated with each feature. With FVC Figure 5a,

success rates approach 40% for most vault sizes

and reach around 10% for sizes approaching the

maximum value of r. In this scenario, compared

to Figure 4b and 4c, we observe that more chaff

points are needed to diminish the signiﬁcance of

the distribution advantage, as illustrated in Fig-

ure 5b and 5c.

Furthermore, with these bases of varying quality,

a correlation is observed between the base qual-

ity and knowledge of the distribution: the higher

the base quality, the less signiﬁcant the impact of

biases. As depicted Figure 5c, knowing the distri-

bution is more advantageous than knowing 25%

of the biometric set with the LFW base, in con-

trast to the PTB and FVC bases. In these cases,

knowledge of the distribution proves to be more

advantageous than knowing 35% of the set, with

the distribution conferring a more substantial ad-

vantage over all other models.

5.2 Discussion

We have presented the results of our attack scenarios

applied to three distinct template bases, characterized

by different modalities and qualities. We assessed the

resilience of the fuzzy vault against these three differ-

ent attacker models.

Our initial observation reveals that the authenti-

cation rates obtained using the distribution model are

substantial, afﬁrming the scheme’s inability to with-

stand such attacks, this casts doubt on its feasibility

in such conditions and raises concerns about the fuzzy

vault’s security, especially in the case of feature-based

construction, where we quantify biases related to fea-

tures.

In comparison with the model suggested by Juels

and Sudan, we illustrate that this model requires sub-

stantial knowledge to achieve a security level compa-

rable to distribution knowledge, necessitating at least

30% of the enrollment set. This signiﬁcant require-

ment is less apparent in the context of individual sen-

sitive data.

In the scenario involving global data, the advan-

tage is less pronounced than with feature-based con-

struction. However, we observe a convergence of suc-

cess rates with a non-realistic information knowledge

model. This underscores the critical importance of

the model used and its direct impact on security. It

calls into question the scheme’s ability to withstand

this speciﬁc model, highlighting a potential vulnera-

bility.

6 SECURE SINGLE-FACTOR

FUZZY VAULT

The results of the previous section indicate that

the original fuzzy vault cannot be considered se-

cure, particularly when exploiting biases by features.

This quantiﬁcation, along with the suggested attacker

model, helps us understand the dispersion of elements

in the set. Differently from previous studies, we

present the ﬁrst solution while maintaining the orig-

inal proposition of the fuzzy vault scheme, without

modifying its inherent nature.

For our proposed solution, the goal is to avoid

signiﬁcant concentration within value ranges for each

feature, thereby reducing biases in biometric sets by

ensuring a balanced distribution for each feature. We

SECRYPT 2024 - 21st International Conference on Security and Cryptography

(a) FVC. (b) PTB. (c) LFW.

Figure 5: Authentication for scenario 2.

rely on a quantile method, dividing the data for each

feature into intervals of equal probability, where each

integer appears with the same frequency.

6.1 Fuzzy Vault Parameters

Using the quantile method to mitigate biases in each

feature decreases dominant feature values, conse-

quently lowering the system’s authentication level.

Therefore, we reassess our selection of the value m

for the quantile method. Setting m

= 1 uses binary

encoding for each feature based on its median, which

minimizes data dispersion and aids in managing tem-

plate value variability to achieve an EER with the bio-

metric set. However, this parameter decreases the

number of image elements of the function (⋆⋆), re-

sulting in fewer chaff points for vault V . Assuming

the attacker knows all parameters, this compromises

the vault’s security. Despite reduced authentication

from increased data dispersion, for security reasons,

we choose m

= 2, providing four possible values cor-

responding to employing the quartile method. This di-

vides the data into four equal parts, each representing

25%. Subsequently, an integer between 0 and 2

− 1

is associated with each interval, aiming to generate a

sufﬁciently large ﬁnite ﬁeld with function (⋆⋆). With

this choice of m

, we obtain the same values for n and

r as those presented Table 2.

The correlation between the secret length k and

authentication rates indicates that longer lengths are

associated with reduced error correction. Employ-

ing the quartile method reveals variations in FAR and

FRR rates. Regardless of speciﬁc k values, an in-

crease in length is associated with higher FRR and

lower FAR. During vault construction, emphasis is

placed on selecting the smallest suitable value for k,

tailored to the biometric template base. The follow-

ing Table 5 presents the FAR and FRR results ob-

tained for each base along with the corresponding k.

Overall, the three bases do not exhibit an EER,

displaying FAR rates below 2% for all bases, thereby

enhancing the fuzzy vault’s security. However, the

FRR rate has shown a signiﬁcant increase for two

Table 5: Authentication rates.

biometric template biometric set

Biometric template base EER k FAR FRR

FVC 10% 5 1.3% 56%

PTB 10.8% 9 2% 6%

LFW 0.2% 4 0% 75%

databases, FVC and LFW, making usability less

straightforward, requiring the user to authenticate re-

peatedly with a more precise representation of the

biometric data until successful, while with PTB bases,

no degradation is observed Table 5. Using our method

removes biases from features, but considering the

FRR results for some bases, one approach to reducing

them could involve retaining certain biases, balancing

system usability and security against statistical biases

in features.

6.2 Results

Given that scenario 2, characterized by a feature-

based attack, poses the most formidable threat com-

promising the integrity of the fuzzy vault and render-

ing it intricate to use. In this same scenario, we aim to

quantify the biases of each feature obtained through

the application of the quartile method. Subsequently,

we intend to assess the advantages resulting from the

exploitation of these biases and determine if this com-

promises the security of the fuzzy vault scheme once

again.

The objective of our quartile method is to obtain

features with more balanced values closer to unifor-

mity. Reassessing the biases of features using the

measure M , as expected, reveals a predominance of

features with a M measurement exceeding 0.9, con-

trasting with previous results where the majority were

between 0.6 and 0.8.

To quantify the effectiveness of our proposed so-

lution, we replicate the attack on the three bases using

identical attacker models similar to those in scenario

2. The results are visually presented Figure 6.

The advantage of an attacker exploiting biases de-

creases signiﬁcantly. Based on the results from the

three bases, in line with our method used to remove

Fuzzy Vault Security Enhancement Avoid Statistical Biases

(a) FVC. (b) PTB. (c) LFW.

Figure 6: Authentication rate with quantile method.

biases by features, we achieve an advantage of the

red curve similar to that of the black curve, obtained

through uniform sampling. Speciﬁcally, with vault

sizes much smaller than n ∗ 3, it is observed that the

vault becomes more secure without the need to add

numerous chaff points. These conclusions also hold

for Scenario 1, where there is no knowledge of bi-

ases speciﬁc to the features. Thus, the quartile method

eliminates the effectiveness of bias attacks while pre-

serving the nature of the proposed original scheme.

To conclude this study, we have proposed a ﬁrst

method based on eliminating biases from each feature

during the transformation of a template into a biomet-

ric set, This approach employs the quantile method

to ensure a fair presentation of each feature’s values.

Unlike methods to enhance the vault’s security by

adding passwords, our single-factor-based proposal

maintains an equivalent level of security in the case

of uniformity without requiring additional informa-

tion. We have presented a solution to mitigate biases

in each feature and mitigate attacks speciﬁcally aimed

at these features. Nonetheless, the potential for an at-

tack exploiting the correlation between features per-

sists. This scenario could render biases exploitable,

thus compromising the vault’s security.

7 CONCLUSION

In this study, we have quantiﬁed the advantage of ex-

ploiting statistical biases in the biometric sets of the

fuzzy vault, compromising its security and rendering

its use by the initial proposal unfeasible. In response

to these vulnerabilities, we have introduced a ﬁrst

method for a secure single-factor fuzzy vault authenti-

cation, aligned with the initial proposition. Quantiles

were employed to achieve a balanced distribution of

the values for each feature, eliminating the need for

additional information and avoiding the use of multi-

factor authentication.

We have obtained preliminary results support-

ing the effectiveness of the single-factor fuzzy vault,

which is not sensitive to feature biases, highlighting

a signiﬁcant correlation between its security and the

construction function of the biometric sets. Currently,

it is still possible to deduce the corresponding feature

of the biometric template from an element of the bio-

metric set. One method to prevent this is to add a

secret on the server side to protect the parameters of

the construction function, thereby enhancing security

while maintaining the single-factor scheme and elim-

inating attacks based on features, thus avoiding a cor-

relation attack between features. This approach also

helps avoid constraints related to parameter choices,

improving system performance.

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