# Cryptobiometrics for the Generation of Cancellable Symmetric and Asymmetric Ciphers with Perfect Secrecy

^{*}

## Abstract

**:**

## 1. Introduction. The Objectives of Security, Cryptography, and Biometrics

## 2. Cryptobiometrics and Ways of Obtaining Keys

**Variability**: Biometric data varies, whereas cryptography calls for exact and invariable data, as only one change in one bit of the key invalidates its identification and utility.**Irreversibility**: Cryptobiometric keys are not pure biometric data, which usually remain permanent and immutable in each human being, and are never to be obtained in a direct manner, which would oblige the need to revoke them (thus not permitting their later use) in the case of being compromised. Instead, they are generated as the result of some type of secure and irreversible mathematical transformation using that pure data.**Cancellability**: Biometric data are inexhaustible as, from them, many different keys can be generated as required. Thus, from a given biometric feature (iris, fingerprint, face, odor, gait, and so on), one is able to extract as many keys (of different lengths) as desired, according to the different applications.**Irrevocability**: In spite of the possibility of canceling cryptobiometric keys, our genuine biometric data can still be used, as they cannot be revoked.**Unlinkability**: Even if an attacker knows many keys, be it proceeding from the same or from a different pattern, they cannot obtain the original biometric datum.**Reliability**: Biometrics seeks an acceptable reliability average value, depending on the application and situation, between false rejection rate and false acceptance rate, while looking for the most possible reduced values. However, we always speak of the maximum probability of acceptance, rejection, or identification.**Biometric bit-length**: At present, symmetric cryptographic keys of order 10${}^{2}$–10${}^{3}$ and asymmetric keys of order 10${}^{3}$–10${}^{4}$ are required. However, not all biometric features (e.g., the iris is better than the fingerprint, and the latter better than typing patterns) enable us to extract such a number of bits, due to extraction difficulties, environmental variability, errors, and so on; or, they may (for reasons intrinsic to the very biometric feature) be highly variable in different samples and circumstances. It is, therefore, necessary to find better mathematical techniques which are capable of extracting a sufficient number of representative bits.

**Key-release**: The key is totally detached with respect to the biometric pattern. It is not really regarded as a cryptobiometrics system; the comparison between the identified or verified external biometric capture and the pattern saved releases a cryptographic key that has no relation whatsoever to the biometric datum. It is of no interest and is cited here only as a proposal previous to the two following methods.**Key-binding**: Departing from a key that has no relation with the biometric datum, a complete monolithic cohesion is realized a posteriori between the biometric pattern and the key.The development of this model started with Soutar, Roberge, Stojanov, Gilroy, and Kumar [14,15]. The procedure, originating from the ideas patented by Tomko, Soutar, and Schmidt [16,17] and by Tomko and Stoianov [18], involves the use of correlation functions in the phase of processing, followed by carrying out binding with the key, generating (ad hoc) a lookup table from the key’s bits and a series of points in the pattern image.Juels and Wattemberg [19] introduced the so-called “fuzzy commitment,” where the key corresponds to a codeword generated after a process of correction of errors, and it later applies the binding with the value obtained from the biometric datum. The differential of bits with a new entry is corrected using error-correcting codes. In this model, the scheme of Hao, Anderson, and Daugman [5] was applied to the iris; some important improvements were added, such as the concatenation of diverse error-correcting codes (Hadamard and Reed–Solomon). Apart from that, using the iris, similar fuzzy commitment schemes have been made in other modalities (e.g., fingerprint, face, and so on) [20,21,22].Goh and Ngo [23], in turn, applied quantification transformations (projections) in facial biometry to generate random information streams.Juels and Sudan [24] developed the “fuzzy vault” over the model “fuzzy commitment,” the practical application of which has been shown by Clancy, Kiyavash, and Lin [25]. It generates a polynomial from the key, with no relation with the biometric data, where the characteristics of the subject are placed in particular points of the polynomial, followed by the insertion of masking data. This scheme has been later applied in numerous biometric categories: fingerprint, iris, palms of hands, face, and so on [26,27,28,29], improving its design [30].On the other hand, Dodis, Ostrovsky, Reyzin, and Smith [31] theoretically developed the so-called “fuzzy extractor/secure sketch,” with which they were able to extract quasi-random information from a biometric input, which was shown to be tolerant to variations and errors.Linnartz and Tuyls [32] proposed the “shielding functions” model in a theoretical manner, where the key and the biometric data were operated upon by these functions, thereby generating support data (or “helper data”) that could be later used to generate the key again, in the case of authenticated input data.On the other hand, Van der Veen, Kevenaar, Schrijen, Akkermans, and Zuo [33] followed the “fuzzy commitment” model together with “helper data,” already introduced theoretically by Tuyls and Goseling [34] and later by Verbitskiy and Denteneer [35].“Password hardening,” proposed by Monrose, Reiter, Li, and Wetzel, adds biometric information to a previous password [38,39,40,41]; the Teoh, Ngo, and Goh BioHashing scheme [23,42,43,44,45] also falls within the key-binding classification.More recently, the work of Iida and Kiya has focused on error-correcting codes and fuzzy commitment schemes for JPEG image [46]. The different methodology of Malarvizhi, Selvarani, and Raj uses fuzzy logic and adaptive genetic algorithms [47]. It is also worth mentioning the proposal of Liew, Shaw, Li and Yang, who made use of Bernoulli mapping and chaotic encryption [48]; and the use of face and fingerprint biometrics along with watermarking and hyper-chaotic maps by Abdul, Nafea, and Ghouzali [49].Another approach is that of Priya, Kathigaikumar, and Mangai, who used random bits mixed in an AES cipher [50]. Asymmetric encryption and irrevocability were used by Barman, Samanta, and Chattopadhyay [51]. The fuzzy extractor with McEliece encryption, which is resistant to quantum attacks, was proposed by Kuznetsov, Kiyan, Uvarova, Serhiienko, and Smirnov [52]. In the work of Chang, Garg, Hasan, and Mishra, a cancelable multi-biometric authentication fuzzy extractor has been proposed, where a novel bit-wise encryption scheme irreversibly transforms a biometric template to a protected one using a secret key generated from another biometric template [53].We also note the works of Damasevicius, Maskeliunas, Kazanavicius, and Wozniak, who used data from electroencephalography and Bose–Chaudhuri–Hocquenghem error-correcting codes [54]. Olanrewaju, Oyebiyi, Misra, Maskeliunas, and Damasevicius [55] used the same type of correction codes with Principal Component Analysis and fast Chebyshev transform hashing for ear biometrics. Some other works have discarded the use of error-correcting codes, such as that of Chai, Goi, Tay, and Jin, where an alignment-free and cancelable iris key binding scheme was constructed through the use of a non-invertible transform [56]. Finally, another work used another type of biometrics which is not very common: key binding with finger vein [57].**Key-generation**: The key is extracted from the biometric pattern.The first proposal in this area was the patent of Bodo [58], where the cryptographic key was obtained fully from the biometric data, even though it posed cancellability and security problems related to the theft of data specific to the subject.An improvement was proposed by Davida, Frankel, Matt, and Peralta [59,60], which generated the key from the hash function over the biometric data after the correction of errors.Other approaches and proposals have been designed using the quantification and intervals of biometric features, such as those of Vielhauer, Steinmetz, and Mayerhofer [61] or Feng and Wah [62].Drahanský, in turn, obtained the key from quantification and the use of graphs [63].Other authors have utilized the combined use of quantification and “fuzzy extractors/secure sketches” as key generators [64,65,66].More recently, the proposal of Aness and Chen used discriminative binary feature learning and quantization [67], and Yuliana, Wirawan, and Suwadi [68] combined pre-processing with multi-level quantization. Furthermore, Chen, Wo, Xie, Wu, and Han improved quantization techniques against leakage and similarity-based attacks [69].With regards to cancellability, a crucial aspect of the works of Ratha, Connell, Zuo, and Bolle [70,71], and later, of Savvides, Kumar, and Khosla [72], introduced biometric cancellability, proposing systems that protect the original biometric data. Along with the above, the closest in time was the study of Trivedi, Thounaojam, and Pal [73], or those who used symmetric cryptography, such as Barman, Samanta, and Chattopadhyay [74]. A secured feature matrix from the template and AES cipher was used by Gaddam and Lal [75]. We also note the novel approach using random slopes of Kaur and Khanna [76], and the technique for achieving cancellability through geometric triangulation and steganography of Neethu and Akbar [77]. The work of Punithavathi and Subbiah [78] introduced partial DCT-based cancellability for IoT applications. In key-binding, cancellability and revocability are naturally simpler; however, in key-generation, the situation is not as easy or simple.

## 3. Voice Biometry

#### 3.1. Introduction

#### 3.1.1. Speech Recognition

#### 3.1.2. Voice Recognition Techniques

- (1)
**Template Matching:**A maximum accuracy or maximum likelihood is sought between the samples previously stored as a voice template and the new voice sample input. This is called the speaker-dependent model.- (2)
**Feature Analysis:**This is also called the speaker-independent model, as it searches for characteristics within human discourse, and from them, it searches for similarities among the input speakers compared to the stored data in the system.

#### 3.2. Methodology

#### 3.2.1. Spectral Analysis of Frequencies

#### 3.2.2. Periodogram of the Signal

#### 3.2.3. Welch Method

- Dividing the signal (overlapping segments).
- Windowing and FFT.
- Averaging.

- Dividing the signal (overlapping segments). We divide the signal into overlapping segments. We consider ${2}^{9}=512$ overlapped samples herein.
- Windowing and FFT. We use an efficient algorithm for computing the DFT, known as fast Fourier transform (FFT); a Hamming window with an FFT size of ${2}^{10}=1024$ and normalization constant $U=0.3970$, obtained from the window $w\left[n\right]$; and the size of the FFT.$$w\left[n\right]=\left\{\begin{array}{cc}0.54-0.46cos\left(\frac{2\pi n}{M}\right)& 0\le n\le \phantom{\rule{3.33333pt}{0ex}}M\\ 0& otherwise\end{array}\right.$$
- Averaging. The average and normalized values are calculated from the vectorized values of the overlapped fragments of the windowed and FFT-processed signal.

## 4. Proposed Model

#### 4.1. Schemes of Ciphers

**RSA Cipher**

**Elgamal Cipher**

**Elliptic Curve Cipher**

**Paillier Cipher**

**Diffie–Hellman Key-Exchange**

**AES Cipher**

#### 4.2. Protocols

**Asymmetric ciphers**

- User A generates their own template $\overrightarrow{T}$, which we can consider to be a binary vector of any length, ${\{0,1\}}^{\ast}$.
- A also generates a random value $\overrightarrow{{R}_{H}}$, a binary vector ${\{0,1\}}^{n}$.
- The binary values $\overrightarrow{T}$ and $\overrightarrow{{R}_{H}}$ are adjusted to the right, where options $n<,\phantom{\rule{0.277778em}{0ex}}=,\phantom{\rule{0.277778em}{0ex}}>\ast $ can be given. Then, the following is calculated:$$H2\left[H1\left(\overrightarrow{T}\oplus \overrightarrow{{R}_{H}}\right)\left|\right|H1\left(\overrightarrow{T}\oplus CL{S}_{1}\left(\overrightarrow{{R}_{H}}\right)\right)\left|\right|\dots \left|\right|H1\left(\overrightarrow{T}\oplus CL{S}_{max(\ast ,n)-1}\left(\overrightarrow{{R}_{H}}\right)\right)\right]=\overrightarrow{{T}_{e}},$$
- Over-randomization:
- (a)
- User A generates a random number $\overrightarrow{R{T}_{e}}$ of length ${l}_{e}>h$ (normally, the output of a hash function is lower in length than the order of our elements, for security reasons).
- (b)
- A generates a set ${R}_{L}$ of h different values from the set $[1,{l}_{e}]$.
- (c)
- Calculate the perfect substitution transposition cipher $PST(\overrightarrow{{T}_{e}},\overrightarrow{R{T}_{e}},{R}_{L})=\overrightarrow{{T}_{e}^{\prime}}$ of length ${l}_{e}$.

- Depending on the element e that we are considering, we can have the following cases:
- (a)
- $\overrightarrow{{T}_{e}^{\prime}}$ generates a prime number: Here, user A carries out $GenPrime\left(\overrightarrow{{T}_{e}^{\prime}}\right)=p$ prime, where $GenPrime$ is a procedure to generate a prime number—applying the usual methods of generation through primality tests—from $\overrightarrow{{T}_{e}^{\prime}}$: using its value (if it is already prime), the next closest prime, or a strong prime.
- (b)
- $\overrightarrow{{T}_{e}^{\prime}}$ generates an element of G, generally ${\mathbb{Z}}_{p}$, ${\mathbb{Z}}_{q}$, ${\mathbb{Z}}_{r}$, ${\mathbb{Z}}_{{r}^{2}}$: In this case, we do not have to make any changes in e; in any case, calculate its modular value in G.
- (c)
- $\overrightarrow{{T}_{e}^{\prime}}$ generates a point of an elliptic curve: Here, user A carries out $GenPointEC(\overrightarrow{{T}_{e}^{\prime}},u,v,p)=({J}_{x},{p}^{\prime})$, with $GenPointEC$ being a procedure to generate ${J}_{x}$, the x coordinate of a point J located on the elliptic curve, such that ${J}_{y}^{2}={J}_{x}^{3}+u{J}_{x}+v\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}p$.

**Symmetric ciphers**

- User A generates their own template $\overrightarrow{{T}_{A}}$, which we can consider to be a binary vector of any length, ${\{0,1\}}^{\ast}$.
- User B generates their own template $\overrightarrow{{T}_{B}}$, again a binary vector of any length, ${\{0,1\}}^{\ast}$.
- A also generates a random value $\overrightarrow{{R}_{HA}}$, a binary vector ${\{0,1\}}^{n}$.
- B also generates a random value $\overrightarrow{{R}_{HB}}$, a binary vector ${\{0,1\}}^{n}$.
- The binary values $\overrightarrow{{T}_{A}}$ and $\overrightarrow{{R}_{HA}}$ are adjusted to the right, where options $n<,\phantom{\rule{0.277778em}{0ex}}=,\phantom{\rule{0.277778em}{0ex}}>\ast $ can be given. Then, the following is calculated:$$H2\left[H1\left(\overrightarrow{{T}_{A}}\oplus \overrightarrow{{R}_{HA}}\right)\left|\right|H1\left(\overrightarrow{{T}_{A}}\oplus CL{S}_{1}\left(\overrightarrow{{R}_{HA}}\right)\right)\left|\right|\dots \left|\right|H1\left(\overrightarrow{{T}_{A}}\oplus CL{S}_{max(\ast ,n)-1}\left(\overrightarrow{{R}_{HA}}\right)\right)\right]=\overrightarrow{{T}_{KA}},$$
- The binary values $\overrightarrow{{T}_{B}}$ and $\overrightarrow{{R}_{HB}}$ are adjusted to the right, where options $n<,\phantom{\rule{0.277778em}{0ex}}=,\phantom{\rule{0.277778em}{0ex}}>\ast $ can be given. Then, the following is calculated$$H2\left[H1\left(\overrightarrow{{T}_{B}}\oplus \overrightarrow{{R}_{HB}}\right)\left|\right|H1\left(\overrightarrow{{T}_{B}}\oplus CL{S}_{1}\left(\overrightarrow{{R}_{HB}}\right)\right)\left|\right|\dots \left|\right|H1\left(\overrightarrow{{T}_{B}}\oplus CL{S}_{max(\ast ,n)-1}\left(\overrightarrow{{R}_{HB}}\right)\right)\right]=\overrightarrow{{T}_{KB}},$$
- Over-randomization:
- (a)
- User A generates a random value $\overrightarrow{R{T}_{KA}}$ with the length of the symmetric key t; generally, $t<h$ (normally, the output of a hash function is higher in length than the order of our element K).
- (b)
- User B generates a random value $\overrightarrow{R{T}_{KB}}$ with the length of the symmetric key t; generally $t<h$ (normally, the output of a hash function is higher in length than the order of our element K).
- (c)
- A generates a set ${R}_{LA}$ of t different values from the set $[1,h]$.
- (d)
- B generates a set ${R}_{LB}$ of t different values from the set $[1,h]$.
- (e)
- User A applies the perfect substitution transposition cipher $PST(\overrightarrow{{T}_{KA}},\overrightarrow{R{T}_{KA}},{R}_{LA})=\overrightarrow{{K}_{A}^{\prime}}$ of length t.
- (f)
- User B applies the perfect substitution transposition cipher $PST(\overrightarrow{{T}_{KB}},\overrightarrow{R{T}_{KB}},{R}_{LB})=\overrightarrow{{K}_{B}^{\prime}}$ of length t.

- User A sends B the vector $\overrightarrow{{K}_{A}^{\prime}}$, using the asymmetric cryptography of a cryptobiometrically generated key.
- User B sends A the vector $\overrightarrow{{K}_{B}^{\prime}}$, using the asymmetric cryptography of a cryptobiometrically generated key.
- Both users apply $\overrightarrow{{K}_{A}^{\prime}}\oplus \overrightarrow{{K}_{B}^{\prime}}=\overrightarrow{K}$.

#### 4.3. Main Elements of the Protocols

**The Biometric Pattern (Template)**

**Hash Functions and Randomization**

**Over-Randomization with Perfect Ciphers**

**Generation of a point in the elliptic curve**

**Strong Primes**

#### 4.4. Security Analysis

**Irreversibility**: Given an output of our system, an eventual attacker cannot reconstruct $\overrightarrow{T}$ (similar with $\overrightarrow{{T}_{A}}$ or $\overrightarrow{{T}_{B}}$), the genuine biometric data.As we examine below, this aspect is achieved by the security properties of the first sub-system of hash functions and the subsequent sub-system of over-randomization with perfect encryption.**Cancellability**: From an input $\overrightarrow{T}$, $\overrightarrow{{T}_{A}}$, or $\overrightarrow{{T}_{B}}$, we can generate as many outputs as we want.This property is achieved, in the scenario of asymmetric ciphers, through all those initial moments in which the parameters of the cryptographic scheme must be generated, on which all the encrypted communications subsequently take place, by the random values $\overrightarrow{{R}_{H}}$, $\overrightarrow{R{T}_{e}}$, and ${R}_{L}$. The binary length of $\overrightarrow{{R}_{H}}$ is n, and ${l}_{e}$ is the binary length of $\overrightarrow{R{T}_{e}}$. On the other hand, with ${R}_{L}$, we have the h-element variations of ${l}_{e}$ elements (with repetition not allowed), $\frac{{l}_{e}!}{({l}_{e}-h)!}$, as possible options. Thus, the number of possible options is given by ${2}^{(n+{l}_{e})}\frac{{l}_{e}!}{({l}_{e}-h)!}$.In the scenario of symmetric ciphers, cancellability is achieved by the random values $\overrightarrow{{R}_{HA}}$, $\overrightarrow{R{T}_{KA}}$, and ${R}_{LA}$ (similarly $\overrightarrow{{R}_{HB}}$, $\overrightarrow{R{T}_{KB}}$, and ${R}_{LB}$). The binary length of $\overrightarrow{{R}_{HA}}$ is n, and t is the binary length of $\overrightarrow{R{T}_{KA}}$.On the other hand, with ${R}_{LA}$ we have, as possible options, the h-element variations of t elements with repetition not allowed $\frac{t!}{(t-h)!}$. Thus, the number of possible options for the user A is given by ${2}^{(n+t)}\frac{t!}{(t-h)!}$.**Irrevocability**: The elements obtained for the schemes of ciphers can be changed, when necessary, and the biometric data of the initial template, $\overrightarrow{T}$, $\overrightarrow{{T}_{A}}$, or $\overrightarrow{{T}_{B}}$, can be used together with new random values, masking the template, which can be used permanently and irrevocably.**Unlinkability**: For a single biometric sample $\overrightarrow{T}$, $\overrightarrow{{T}_{A}}$, or $\overrightarrow{{T}_{B}}$, we should be able to generate different outputs in a way such that it is not feasible to determine whether those outputs belong to a single subject or not.The proof of this is that, although the template is the same, as it originates from biological and/or behavioral aspects of a subject, the random variables of the system, as well as the properties of the hash functions (one-way or pre-image resistance, resistance to second pre-image, and collision resistance), and the perfect secret property of the over-randomization sub-system, by which the probability a posteriori that the original text is x if the ciphered text is y, is identical to the probability a priori that the original text is x.

**Analysis of the hash structure**

- Pre-image resistance: this property means that H is a one-way function, and so, for a randomly chosen $x\in {\{0,1\}}^{\ast}$, it is hard to find, given $y=H\left(x\right)$, an ${x}^{\prime}\in {\{0,1\}}^{\ast}$ such that $H\left({x}^{\prime}\right)=y$.
- Second pre-image resistance: given $x\in {\{0,1\}}^{\ast}$, it is hard to find ${x}^{\prime}\ne x$ such that $H\left({x}^{\prime}\right)=H\left(x\right)$.
- Collision resistance: it is hard to find a pair $x,\phantom{\rule{0.277778em}{0ex}}{x}^{\prime}\in {\{0,1\}}^{\ast}$, $x\ne {x}^{\prime}$, such that $H\left({x}^{\prime}\right)=H\left(x\right)$.

**Analysis of the over-randomization structure**

- -
- The probability distribution on K is uniform.
- -
- For every $m\in M$ and every $c\in C$, there exists a unique $k\in K$ such that $En{c}_{k}\left(m\right)=c$.

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Experiments

#### Generation of Fundamental Elements of Cryptographic Schemes

**Elliptic Curve**

- (a)
- $p=A{p}_{1}+1$ (with ${p}_{1}$ a high prime and A any integer);
- (b)
- ${p}_{1}=B{p}_{2}+1$ (with ${p}_{2}$ a high prime and B any integer);
- (c)
- $p=C{p}_{3}-1$ (with ${p}_{3}$ a high prime and C any integer).

**AES**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**

**Bottom:**Waveform of the sounds of the word “biometrics,” where you can see the consistency in time of each sound that makes it up.

**Top:**Spectrogram of the time-dependent Fourier transform of the waveform.

**Figure 2.**Scheme of steps for processing a signal from its original form to its value discretized by the discrete Fourier transform (DFT).

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Jara-Vera, V.; Sánchez-Ávila, C. Cryptobiometrics for the Generation of Cancellable Symmetric and Asymmetric Ciphers with Perfect Secrecy. *Mathematics* **2020**, *8*, 1536.
https://doi.org/10.3390/math8091536

**AMA Style**

Jara-Vera V, Sánchez-Ávila C. Cryptobiometrics for the Generation of Cancellable Symmetric and Asymmetric Ciphers with Perfect Secrecy. *Mathematics*. 2020; 8(9):1536.
https://doi.org/10.3390/math8091536

**Chicago/Turabian Style**

Jara-Vera, Vicente, and Carmen Sánchez-Ávila. 2020. "Cryptobiometrics for the Generation of Cancellable Symmetric and Asymmetric Ciphers with Perfect Secrecy" *Mathematics* 8, no. 9: 1536.
https://doi.org/10.3390/math8091536