---
kind: concept
slug: unlinkability
title: Unlinkability
summary: >-
  How hard it is to tell that two stored, protected records belong to the same person when they were enrolled under different keys. Measured as a score-wise value D(s) and a system-wide value D_sys in [0, 1], built from mated and non-mated comparison scores and a prior ratio omega. Shown for biometric template protection.
tags: [privacy, biometrics, evaluation]
papers: ["2311.04633"]
related: [biometric-template-protection, homomorphic-encryption]
status: draft
updated_at: 2026-11-06
---

## Overview

Unlinkability asks whether two protected records of one person, stored under
different keys in two applications, can be matched to each other
[arxiv:2311.04633 §1, §3]. The record is a protected biometric template, and
the same biometric sample is enrolled in both applications through an encoder
that uses a different secret key for each [arxiv:2311.04633 §3].

The paper's contribution is a number, not a yes or no. It defines a local value
D(s) for each comparison score s and a global value D_sys, both in [0, 1]. Both
come from two score distributions: pairs of records from the same person
(mated) and pairs from different people (non-mated)
[arxiv:2311.04633 §4.1, §4.2]. A value of 0 means the records cannot be linked
at any score. A value of 1 means they can be linked with near certainty
[arxiv:2311.04633 §4.2]. The same definition is applied to schemes built on
iris, signature, face and fingervein data [arxiv:2311.04633 §6.1]. The
comparison to earlier analyses is covered in [Contradictions](#contradictions).

Card: [General Framework to Evaluate Unlinkability in Biometric Template
Protection Systems](/papers/2311.04633).

## Variants

These are the parts of the method that you choose, not competing methods.

### Score-wise measure D(s)

The measure starts from the likelihood ratio LR(s), the mated density divided
by the non-mated density at score s [arxiv:2311.04633 eq. 4]. Multiplying LR(s)
by a prior ratio omega gives the posterior odds that the pair is mated, and the
difference between the two posterior probabilities is D(s)
[arxiv:2311.04633 eq. 3, eq. 6]. The final form is piecewise. D(s) is 0 while
the odds are at most 1. Above 1, D(s) = 2x / (1 + x) − 1, where x is the odds
[arxiv:2311.04633 eq. 13]. It is continuous at the boundary, never decreases
as the score grows, and is defined at every score [arxiv:2311.04633 app. A.1].
Use it to see which score ranges link the records, not only a single figure
[arxiv:2311.04633 §4.1].

### System-wide measure D_sys

D_sys is the mated-weighted integral of D(s), normalised by the mated
probability [arxiv:2311.04633 eq. 16, eq. 17, eq. 18]. It lies in [0, 1]
[arxiv:2311.04633 app. A.2]. It is 1 when the two distributions do not overlap
and 0 when they fully overlap [arxiv:2311.04633 §4.2]. If some scores favour
non-mated pairs, the distributions must favour mated pairs elsewhere, so
D_sys stays above 0 in that case [arxiv:2311.04633 §4.3]. This is the number to
report and compare across schemes.

### Prior ratio omega

omega is the mated prior divided by the non-mated prior
[arxiv:2311.04633 eq. 6]. When the priors are known, use them. When they are
not, set omega = 1, which the paper calls the worst case for unlinkability
[arxiv:2311.04633 §4.1]. The paper sweeps omega from 0.0001 to 1 and finds that
a larger omega gives a larger D(s) at the same score [arxiv:2311.04633 §6.4].

With N enrolled subjects, the two prior probabilities are 1/N and (N − 1)/N
[arxiv:2311.04633 eq. 14, eq. 15]. Taking their ratio gives omega = 1/(N − 1).
This follows from those two equations and is not stated in the paper, which
says only that omega is at most 1 and equals 1 at N = 2 [arxiv:2311.04633 §4.1].

## When to use which

| Situation | Use | Why |
|---|---|---|
| One number to compare schemes | D_sys | Single value in [0, 1] from the full score distributions [arxiv:2311.04633 §4.2] |
| You need to know which scores link | D(s) plot with both distributions | The local value shows the score ranges that favour mated pairs [arxiv:2311.04633 §4.1] [arxiv:2311.04633 §5] |
| Priors are known | omega from the priors | The paper's formula uses the prior ratio directly [arxiv:2311.04633 §4.1] |
| Priors are unknown | omega = 1 | The paper's stated worst case for unlinkability [arxiv:2311.04633 §4.1] |
| You have only binary accept curves (DET or ROC) | Do not use them alone for unlinkability | The paper's fingervein case: DET analyses call the system unlinkable, the measure gives 0.92 [arxiv:2311.04633 §6.5]. The DET reading is the authors' |
| Scores are discrete | Map them to a continuous distance first | The method needs continuous, normalisable scores. The paper does not say how to do the mapping [arxiv:2311.04633 §5] |
| You claim a scheme is unlinkable | Report D_sys with the linkage functions and attacker model used | D_sys = 0 holds only for the functions tested [arxiv:2311.04633 §5] [arxiv:2311.04633 §6.2] |

## Implementation sketch

The sketch follows the paper's equations and the evaluation protocol. The
density estimator is a choice the paper leaves open, so `estimate_density` is
a placeholder. The paper computes the likelihood ratio point-wise
[arxiv:2311.04633 §6.1].

```pseudocode
# Local and global unlinkability from two score samples [§4] [§5]
# Input:  mated, nonmated (1-D score samples), omega, grid of scores
# Output: D(s) on the grid, D_sys in [0, 1]

def unlinkability(mated, nonmated, omega, grid):
    # Estimate p(s | H_m) and p(s | H_nm) on the grid [§5 steps 2-3]
    p_m = estimate_density(mated, grid)
    p_nm = estimate_density(nonmated, grid)

    # Likelihood ratio [eq. 4]
    lr = p_m / p_nm

    # Posterior odds, scaled by the prior ratio [eq. 6]
    x = lr * omega

    # Local measure: 0 up to odds of 1, then 2x/(1+x) - 1 [eq. 13]
    d_local = where(x > 1, 2 * x / (1 + x) - 1, 0.0)

    # Where the non-mated density is zero, the limit is 1 [eq. 13] [§6.3]
    d_local[p_nm == 0] = 1.0

    # Global measure: mated-weighted integral, normalised [eqs. 16-18]
    d_sys = integrate(d_local * p_m, grid) / integrate(p_m, grid)
    return d_local, d_sys
```

Run the sketch once per linkage function, then report the largest D_sys over
the functions [arxiv:2311.04633 §5].

## Contradictions

- **Fingervein.** The paper's measure gives D_sys = 0.92 for block re-mapping.
  The DET and RTMR analyses it cites call the same system unlinkable, with the
  EER rising from 1.91 % to 4.48 % [arxiv:2311.04633 §6.5]. The two readings
  are not reconciled in the paper.
- **Face, Hamming weight difference.** The reported value is 0.07, and the text
  then says it rises to 0.08 [arxiv:2311.04633 §6.3] [arxiv:2311.04633 fig. 5].
  The paper gives both numbers and does not explain the difference.

## Pitfalls

- **omega = 1 is a worst case, not a neutral default.** A larger omega gives a
  larger D(s) for the same score [arxiv:2311.04633 §6.4]. Report which omega
  was used.
- **Empty non-mated bins.** Where the non-mated density is zero, D(s) is set to
  1 [arxiv:2311.04633 eq. 13]. With a finite sample, empty bins can arise from
  sampling alone. The paper's strong face-attack claims rest on this rule
  [arxiv:2311.04633 §6.3].
- **Low-probability scores.** For one face score, some values below 0.94 favour
  mated pairs, yet D_sys is 0.07. The paper says these scores have a mated
  density below 0.005, so they carry little weight in D_sys
  [arxiv:2311.04633 §6.2].
- **The worst linkage function sets the value.** A system is only as unlinkable
  as its most linkable function. The paper reports the maximum over functions
  [arxiv:2311.04633 §5] [arxiv:2311.04633 §6.3]. A single D_sys = 0 does not
  cover attacks that were not tried.
- **Sampling design matters.** Mated pairs must come from different keys.
  Non-mated pairs follow the paper's pairing of one sample with the first sample
  of other subjects [arxiv:2311.04633 §5] [arxiv:2311.04633 §6.1]. Report the
  pairing.
- **Bin counts and bandwidths are not given.** The paper does not state the
  histogram or kernel settings used for the densities, so results from different
  implementations may differ [arxiv:2311.04633 §6.1].
- **Results are the authors' own.** The numbers on this page are
  author-reported. The authors' code is under a custom licence from their
  institution and is not open source, so this page does not use its code.
  The script was not run for this page.

## Lineage

- Builds on the view of unlinkability as a continuous property, from a fuzzy
  scheme analysis by Buhan et al. [arxiv:2311.04633 §2].
- Adapts the mated and non-mated analysis of Ferrara et al. and reduces it from
  three distributions to two [arxiv:2311.04633 §3].
- Compared against DET- and ROC-based unlinkability analyses by Kelkboom et al.
  and Piciucco et al. [arxiv:2311.04633 §2] [arxiv:2311.04633 §6.5].
- Superseded by: not known from the content in this wiki.

## Papers

- [General Framework to Evaluate Unlinkability in Biometric Template Protection Systems](/papers/2311.04633): defines D(s) and D_sys, sets out the evaluation protocol, and reports values for iris, signature, face and fingervein schemes.