#
Biometric Identification Systems with Noisy Enrollment for Gaussian Sources and Channels^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. System Model and Converted System

#### 2.1. Notation and System Model

#### 2.2. Converted System

**Remark**

**1.**

## 3. Problem Formulation and Main Results

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**2.**

**Theorem**

**1.**

**Corollary**

**1.**

**Remark**

**3.**

## 4. Behaviors of the Capacity Region

#### 4.1. Optimal Asymptotic Rates and Zero-Rate Slopes

#### 4.2. Examples

- Ex.1:
- (a) ${\rho}_{1}^{2}=3/4,{\rho}_{2}^{2}=2/3$, (b) ${\rho}_{1}^{2}=7/8,{\rho}_{2}^{2}=2/3$, (c) ${\rho}_{1}^{2}=15/16,{\rho}_{2}^{2}=2/3$,
- Ex.2:
- (a) ${\rho}_{1}^{2}=3/4,{\rho}_{2}^{2}=2/3$, (b) ${\rho}_{1}^{2}=9/10,{\rho}_{2}^{2}=7/8$, (c) ${\rho}_{1}^{2}=15/16,{\rho}_{2}^{2}=11/12$,
- Ex.3:
- (a) ${\rho}_{1}^{2}=3/4,{\rho}_{2}^{2}=2/3$, (b) ${\rho}_{1}^{2}=3/4,{\rho}_{2}^{2}=8/9$, (c) ${\rho}_{1}^{2}=3/4,{\rho}_{2}^{2}=14/15$.

## 5. Overviews of the Proof of Theorem 1

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Equation (19)

#### Appendix A.1. Weakly Typical Sets and Modified Typical Sets

**Lemma**

**A1.**

- 1
- For $\forall S\subseteq \{{X}_{1},{X}_{2},\cdots ,{X}_{k}\}$ and large enough n,$$\begin{array}{cc}\hfill Pr\left\{{\mathcal{A}}_{\u03f5}^{\left(n\right)}\left(S\right)\right\}& \ge 1-\u03f5.\hfill \end{array}$$
- 2
- For $\forall S,V\subseteq \{{X}_{1},{X}_{2},\cdots ,{X}_{k}\}$ $(S\cap V=\xd8)$, we have that$$\begin{array}{c}\hfill \mathrm{Vol}\left({\mathcal{A}}_{\u03f5}^{\left(n\right)}\left(V\right|{s}^{n})\right)\le {e}^{n\left(h\right(V\left|S\right)+2\u03f5)},\end{array}$$where $\mathrm{Vol}(\xb7)$ denotes the volume of a set.
- 3
- Fix $k=2$. If $({\tilde{X}}_{1}^{n},{\tilde{X}}_{2}^{n})$ are independent sequences with the same marginals as ${f}_{{X}_{1}^{n}{X}_{2}^{n}}({x}_{1}^{n},{x}_{2}^{n})$, then$$\begin{array}{c}\hfill Pr\{({\tilde{X}}_{1}^{n},{\tilde{X}}_{2}^{n})\in {\mathcal{A}}_{\u03f5}^{\left(n\right)}\left({X}_{1}{X}_{2}\right)\}\le {e}^{-n(I({X}_{1};{X}_{2})-2\u03f5)}.\end{array}$$Moreover, for n large enough,$$\begin{array}{c}\hfill Pr\{({\tilde{X}}_{1}^{n},{\tilde{X}}_{2}^{n})\in {\mathcal{A}}_{\u03f5}^{\left(n\right)}\left({X}_{1}{X}_{2}\right)\}\ge (1-\u03f5){e}^{-n(I({X}_{1};{X}_{2})+2\u03f5)}.\end{array}$$

**Proof.**

**Lemma**

**A2.**

- Property 1.
- If $({y}^{n},{u}^{n})\in {\mathcal{B}}_{\u03f5}^{\left(n\right)}\left(YU\right)$ then also $({y}^{n},{u}^{n})\in {\mathcal{A}}_{\u03f5}^{\left(n\right)}\left(YU\right)$.
- Property 2.
- Assume that $({U}^{n},{Y}^{n},{X}^{n})\sim {f}_{{U}^{n}{X}^{n}{Y}^{n}}={\prod}_{t=1}^{n}{f}_{{X}_{t}{Y}_{t}}{f}_{{U}_{t}|{Y}_{t}}$. Then, for $\u03f5\in (0,1)$ and n large enough, ${\sum}_{({y}^{n},{u}^{n})\in {\mathcal{B}}_{\u03f5}^{\left(n\right)}\left(YU\right)}{P}_{{Y}^{n}{U}^{n}}({y}^{n},{u}^{n})\ge 1-\u03f5$.

**Proof.**

#### Appendix A.2. Achievability Part

- $\mathcal{E}$
_{1} - : {$({Y}_{i}^{n},{U}^{n}(s,j))\notin {\mathcal{B}}_{\delta}^{\left(n\right)}\left(YU\right)$ for all $s\in \mathcal{S}$ and $j\in \mathcal{J}$},

- $\mathcal{E}$
_{2} - : {$({Z}^{n},{U}_{i}^{n})\notin {\mathcal{A}}_{\delta}^{\left(n\right)}\left(ZU\right)$},
- $\mathcal{E}$
_{3} - : {$({Z}^{n},{U}^{n}({s}^{\prime},J\left(i\right)))\in {\mathcal{A}}_{\delta}^{\left(n\right)}\left(ZU\right)$ for some ${s}^{\prime}\in \mathcal{S},\phantom{\rule{3.33333pt}{0ex}}{s}^{\prime}\ne S\left(i\right)$},
- $\mathcal{E}$
_{4} - : {$({Z}^{n},{U}^{n}(s,J\left({i}^{\prime}\right)))\in {\mathcal{A}}_{\delta}^{\left(n\right)}\left(ZU\right)$ for some ${i}^{\prime}\in \mathcal{I},\phantom{\rule{3.33333pt}{0ex}}{i}^{\prime}\ne i$, and $s\in \mathcal{S}$}.

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

- (c)
- follows as $\left(S\right(i),J(i\left)\right)$ determines ${U}_{i}^{n}$,
- (d)
- follows because conditioning reduces entropy,
- (e)
- follows as $h\left({Y}_{i}^{n}\right|{U}_{i}^{n},T=0)\le h\left({Y}_{i}^{n}\right)=\frac{n}{2}log\left(2\pi e\right)$, and we define ${\u03f5}_{n}=\frac{1}{n}+\frac{\delta}{2}log\left(2\pi e\right)$,
- (f)
- follows by applying Jensen’s inequality to the concave function $\varphi \left(t\right)=-tlogt$,
- (g)
- is due to (A17) in Lemma A3,
- (h)
- is due to (A3) in Lemma A1.

- (i)
- follows as $J\left(i\right)$ and $S\left(i\right)$ are functions of ${Y}_{i}^{n}$ for given codebook ${\mathcal{C}}_{n}$,
- (j)
- follows since $({Y}_{i}^{n},{X}_{i}^{n})$ are independent of ${\mathcal{C}}_{n}$, and the Markov chain $S\left(i\right)-({X}_{i}^{n},J\left(i\right))-{Z}^{n}$ holds,
- (k)
- follows because conditioning reduces entropy and $S\left(i\right)-(J\left(i\right),{Z}^{n})-\mathit{J}\setminus J\left(i\right)$ is applied,
- (l)
- follows by applying Fano’s inequality since $S\left(i\right)$ can be reliably reconstructed from $(\mathit{J},{Z}^{n})$ for given codebook ${\mathcal{C}}_{n}$, and ${\delta}_{n}^{\prime}\downarrow 0$ as $\delta \downarrow 0$ and $n\to \infty $,
- (m)
- is due to (A19).

#### Appendix A.3. Converse Part

- (a)
- holds since $(\widehat{W},\widehat{S\left(W\right)})$ is a function of $({Z}^{n},\mathit{J})$,
- (b)
- follows because conditioning reduces entropy, and only $J\left(W\right)$ is possibly dependent on ${Z}^{n},S\left(W\right)$,
- (c)
- is due to Fano’s inequality with ${\delta}_{n}=\frac{1}{n}(1+\delta log{M}_{I}{M}_{S})$,
- (d)
- follows since (A28) is applied, and W is independent of other RVs,
- (e)
- follows because conditioning reduces entropy.

- (f)
- holds as W is independent of ${Y}_{W}^{n}$,
- (g)
- holds as W is independent of other RVs and $S\left(W\right)$ is a function of ${Y}_{W}^{n}$,
- (h)
- follows because $h\left({Y}_{W}^{n}\right)=h\left({Z}^{n}\right)=\frac{n}{2}log\left(2\pi e\right)$, and by combining (A30) and (A31), we obtain that $H\left(S\left(W\right)\right|W)\le h\left({Z}^{n}\right)-h\left({Z}^{n}\right|J\left(W\right),S\left(W\right))-n({R}_{I}-({\delta}_{n}+2\delta ))$.

- (i)
- holds as W is independent of ${X}_{W}^{n}$,
- (j)
- follows because $h\left({X}_{W}^{n}\right)=h\left({Z}^{n}\right)$, and the same reason of (h) in (A33) is used.

## Appendix B. Proof Sketch of Equation (20)

#### Appendix B.1. Achievability Part

#### Appendix B.2. Converse Part

- (a)
- follows since $J\left(W\right)$ is a function of $({Y}_{W}^{n},S\left(W\right))$,
- (b)
- follows as W is independent of other RVs and $S\left(W\right)$ is chosen independently of ${Y}_{W}^{n}$,
- (c)
- follows because (A40) is applied.

## Appendix C. Convexity of the Regions ${\mathcal{R}}_{G}$ and ${\mathcal{R}}_{C}$

- (a)
- follows as $logx\phantom{\rule{3.33333pt}{0ex}}(x>0)$ is a concave function,
- (b)
- holds as we define ${\eta}^{\prime}=\lambda {\eta}_{1}+(1-\lambda ){\eta}_{2}$.

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**Figure 3.**(

**a**,

**b**) are the explanations of the optimal values of identification, secret-key, storage, and privacy-leakage rates in the regions ${\mathcal{R}}_{G}$ and ${\mathcal{R}}_{C}$, respectively, for a fixed $\alpha $.

**Figure 4.**Projections of the capacity region ${\mathcal{R}}_{G}$ onto (

**a**) ${R}_{J}{R}_{S}{R}_{I}$-space and (

**b**) ${R}_{J}{R}_{I}$-plane.

**Figure 5.**The projections of the capacity region ${\mathcal{R}}_{G}$ onto two dimension figures for Exs. 2 and 3. (

**a**) is the boundary of the capacity region ${\mathcal{R}}_{G}$ onto ${R}_{J}{R}_{S}$-plane for Ex. 2. (

**b**) is the boundary of the capacity region ${\mathcal{R}}_{G}$ onto ${R}_{J}{R}_{L}$-plane for Ex. 2. (

**c**) is the boundary of the capacity region ${\mathcal{R}}_{G}$ onto ${R}_{J}{R}_{S}$-plane for Ex. 3. (

**d**) is the boundary of the capacity region ${\mathcal{R}}_{G}$ into ${R}_{J}{R}_{L}$-plane for Ex. 3.

Cases | The Optimal Secret-Key Rate | Privacy-Leakage Rate | ||||
---|---|---|---|---|---|---|

(a) | (b) | (c) | (a) | (b) | (c) | |

Ex. 1 | $0.35$ | $0.44$ | $0.49$ | $0.35$ | $0.6$ | $0.90$ |

Ex. 2 | $0.35$ | $0.77$ | $0.98$ | $0.35$ | $0.38$ | $0.41$ |

Ex. 3 | $0.35$ | $0.55$ | $0.6$ | $0.35$ | 0.14 | 0.09 |

Cases | The Slope of Secret-Key Rate | The Slope of Privacy-Leakage Rate | ||||
---|---|---|---|---|---|---|

(a) | (b) | (c) | (a) | (b) | (c) | |

Ex. 1 | $1.0$ | $1.40$ | $1.67$ | $0.5$ | $0.7$ | $0.83$ |

Ex. 2 | $1.0$ | $3.71$ | $6.11$ | $0.5$ | $0.53$ | $0.56$ |

Ex. 3 | $1.0$ | $2.0$ | $2.33$ | $0.5$ | $0.25$ | $0.17$ |

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**MDPI and ACS Style**

Yachongka, V.; Yagi, H.; Oohama, Y.
Biometric Identification Systems with Noisy Enrollment for Gaussian Sources and Channels. *Entropy* **2021**, *23*, 1049.
https://doi.org/10.3390/e23081049

**AMA Style**

Yachongka V, Yagi H, Oohama Y.
Biometric Identification Systems with Noisy Enrollment for Gaussian Sources and Channels. *Entropy*. 2021; 23(8):1049.
https://doi.org/10.3390/e23081049

**Chicago/Turabian Style**

Yachongka, Vamoua, Hideki Yagi, and Yasutada Oohama.
2021. "Biometric Identification Systems with Noisy Enrollment for Gaussian Sources and Channels" *Entropy* 23, no. 8: 1049.
https://doi.org/10.3390/e23081049