# Random Hash Code Generation for Cancelable Fingerprint Templates using
Vector Permutation and Shift-order Process

Sani M. Abdullahi and Shuifa Sun
College of Computer and Information Technology
China Three Gorges University, Yichang 443002, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Cancelable biometric techniques have been used to prevent the compromise of
biometric data by generating and using their corresponding cancelable
templates for user authentication. However, the non-invertible distance
preserving transformation methods employed in various schemes are often
vulnerable to information leakage since matching is performed in the
transformed domain. In this paper, we proposed a non-invertible distance
preserving scheme based on vector permutation and shift-order process. First,
the dimension of feature vectors is reduced using kernelized principle
component analysis (KPCA) prior to randomly permuting the extracted vector
features. A shift-order process is then applied to the generated features in
order to achieve non-invertibility and combat similarity-based attacks. The
generated hash codes are resilient to different security and privacy attacks
whilst fulfilling the major revocability and unlinkability requirements.
Experimental evaluation conducted on 6 datasets of FVC2002 and FVC2004 reveals
a high-performance accuracy of the proposed scheme better than other existing
state-of-the-art schemes.

##  1 Introduction

Fingerprint authentication systems have witnessed a progressive deployment in
various areas of our daily endeavors, stretching from a simple task of
verifying ourselves on our smartphones to a broader area of identity
management such as visa application and cross boarder checking procedures.
Such huge reliant on fingerprints amongst other modalities has to do with its
permanence, uniqueness, proven stability, assurance, individuality and cost-
effectiveness[13][24][25][14]. However, the stored fingerprint templates are
vulnerable to attacks, and once compromised, there would be a permanent loss
of vital data since it is irrevocable and irreplaceable. As such, the
devastating consequence that comes with the loss is immeasurable. To overcome
these drawbacks, cancelable templates are used to efficiently preserve the
security and privacy concerns of fingerprint biometric data by permitting
compromised templates to be replaced by a new one. They are built on the idea
that they can be revoked and a new instance of the corresponding templates can
be re-generated if the original data is tampered with. This is often achieved
through non-invertible transformation, i.e., an intentional (but crafted)
distortion of the biometric data by means of transformation function and user-
predefined parameters. The transformation function can be applied either in a
form of blending mechanism (salting) or a many-to-one transformation (non-
invertibility). The essential properties of cancelable biometric schemes are
as follows:
a) Diversity: Different cancelable templates should be generated from same
biometric data. In other words, same biometric data should not correlate in
different applications.
b) Non-invertibility: It should be impossible to reconstruct the original
biometric data from its corresponding transformed version even if the
transformed templates and the transformation methods are known.
c) Revocability: It should be possible to revoke a compromised template and
replace it with a new one based on the original data.
d) Performance: The accuracy performance should not be degraded after the
template transformation. In other words, the incorporation of security
mechanism should not reduce the systems’ recognition accuracy.

Generally, the accuracy performance of a cancelable biometric system is known
to be directly impacted by the feature extraction and representation of the
biometric data. Minutiae descriptors provides us with the efficient means of
representing their discriminative features for effective construction of
fingerprint templates. An example of such descriptor is the state-of-the-art
minutiae cylinder code (MCC)[3] which is known for its superior matching
performance. However, this descriptor exhibits some security and privacy
shortcomings as they are prone to inversion. On the other hand, the
incorporation of security mechanism such as in the protected version of MCC,
known as 2PMCC[8]. The method is able to avert the inversion but still prone
to revocability. Besides non-invertibility and revocability, the accuracy
performance is often degraded after transformation due to modification of the
minutiae descriptor, coupled with information leakage in the matching
procedure which raise concerns over the systems’ full non-invertibility.

##  2 Review works

In the section, several works that focus on the non-invertibility of
fingerprint templates including the recent distance preserving drawbacks in
most of these approaches are discussed. Readers can as well refer to these
papers [27][22][25] for a more in-depth survey on cancelable biometrics. The
earlier works of [8][21][29], and [17] are all designed with notion of
achieving non-invertibility. However, even though it is not fully achieved due
to the fact that the distance between features is not attained, their accuracy
is also degraded considering that the security mechanisms are directly applied
on the descriptors or incorporated in the mapping approach, except for [8].
Even so, it also suffers accuracy degradation depending on the parameter
tuning. Wang et al [36] designed a cancelable fingerprint templates using
zoned minutiae pairs. Local minutiae structures are leveraged to construct the
minutiae pairs with the incorporation of partial discrete Fourier transform in
order to generate discriminating features for matching. The scheme however,
only shows good accuracy performance on DB1 of FVC2002, while achieving a
relatively low accuracy in other datasets. In another different work [35], the
authors used a partial Hadamard transform approach for the design of
cancelable templates. The stochastic distance between binary vectors is
theoretically proven to be preserved after transformation. Even though the
achieved accuracy is better than the previous scheme, it still lacks a
profound distance preservation analysis.

Recently, Trivedi et al. [33] proposed a non-invertible approach based on
information extracted from the Delaunay triangulation of the minutiae points.
A random binary sting is utilized with a user specific key for cancelability.
However, the scheme is evaluated on a single dataset only, which of course is
insufficient for effective validation of the accuracy performance and other
cancelable properties of the scheme. Another novel design of cancelable
templates using randomized non-negative least squares is performed by Kho et
al. [19]. The authors formulated a unique PR-NNLS descriptor for non-
invertible transformation but instead of applying it directly on the minutiae
descriptor, they chose to apply it on the PLS descriptor dictionary in order
to retain the accuracy. However, its full non-invertibility still cannot be
ascertained because there is no analysis conducted on the distance
preservation. In a rather similar approach by Shahzad et al [32], authors
explored the concept of window-shift-XOR and discrete wavelet transform to
avert ARM. Some similarity-based attacks are evaluated on their dual
protection mechanism to validate the non-invertibility of the scheme but
without an analysis.

The new prevailing problem applicable to most cancelable biometric systems is
their inability to achieve full non-invertibility due to the fact that the
transformation techniques often used are not preserving the distance space
between original features and their transformed versions. However, the best
possible ways of overcoming these issues are 1) to avoid matching in the
transformed domain, instead perform matching in an encrypted domain (which of
course requires the integration of an encryption mechanism); 2) perform
matching in an entirely different dataset that does not require a lot of
attempts for same user verification; 3) develop schemes that take full
cognizance of the non-invertibility of the system, e.g., by incorporating
additional protection mechanisms to secure the parameters [23]. Gomez et al.
[10] gave a comprehensive survey on the inversion of biometric templates with
possible ways of surmounting the weakness associated with the related attacks.
One of such attacks is launched through perceptron learning [7]. A deep secure
quantization approach[5], genetic algorithm [28], and constraint optimization
approach [34] are some of the recent works aimed at preventing such attacks.
Another survey by Dong et al.[6] analyzes the distance preserving properties
in cancelable biometrics. They also digressed into the security risks in
relation to pre-image attacks, and proposed a framework for countering such
attacks. Their previous works on random projection [15] [2], index-of-max
hashing [16], and kernalized hashing [20] are however, susceptible to
similarity-based attacks. A cryptanalysis conducted by Ghamman et al. [9] also
proves the vulnerability of [16].

##  3 Proposed scheme

First, the minutiae feature descriptor, popularly known as Minutiae Cylinder
Code (MCC)[3] is employed to extract fingerprint vector features, and a fixed
length representation technique using Kernel Principle Component Analysis
(KPCA)[18] to convert the extracted feature descriptors into fixed-length
vectors is applied. We choose to use MCC feature descriptor due to its
invariance property and its capability of local minutiae matching, which leads
to high performance accuracy during authentication. The accuracy is also
retained after feature transformation using KPCA, which has also proven better
performance and accuracy preservation than other descriptors [12] [1].

###  3.1 Feature Extraction with MCC

This is done by capturing the directional and spatial relationship between the
reference minutiae ${m_{r}=\left\\{p_{r},q_{r},\theta_{r}\right\\}}$. For
$p_{r}$ and $q_{r}$ denotes the minutiae point location, while $\theta_{r}$
represents the minutiae orientation. Their corresponding aligned neighbor
minutiae
${m^{t}=\left\\{p_{i}^{t},{q_{i}{t},\theta^{\prime}_{i}{t}}|i=1,...,N-1\right\\}}$
is within the selected pre-defined radius R. The extraction technique mainly
relies on a robust discretization of the neighborhood of each minutia into a
3D cell-based structure known as the cylinder, as each cylinder is discretized
into a number of cells. Hence, a numerical value is computed from each cell
based on the likelihood of finding minutiae that are closer to the cell with
respect to location and direction. MCC is a prominent fingerprint recognition
technique based on minutiae template. Due to its cylinder invariance, fixed-
length binary string, and bit-oriented coding, it provides us with the perfect
suitability for achieving non-invertibility, cancelability and unlinkability
of our proposed template protection technique.
Generate the feature vector: In order to reduce the dimension of the binary
strings derived from MCC, KPCA is employed for such purpose. To achieve this,
MCC descriptor variable size is converted into a fixed-length vector
representation. Given ${\Phi=\left\\{{{\Phi^{l}}(i)|i=1,...{N_{l}}}\right\\}}$
as the set of MCC training samples, where ${N_{l}}$ is the total number of
${\Phi^{l}}$, the kernel matrix of the fixed features
${F_{k}}\in\mathbb{Q}{{}^{{v_{l}}\times{v_{l}}}}$ is computed with its
resultant function as given in Eq. 1 below.

$\begin{array}[]{l}{F_{f}}\left({i,j}\right)=f\left({{\Phi^{i}}\left(i\right),{\Phi^{i}}\left(j\right)}\right)=\\\
\exp\left({-1{{\left({1-{D_{{s_{MCC}}}}\left({i,j}\right)}\right)}^{2}}/2{\sigma^{2}}}\right)\end{array}$
(1)

where ${D_{{s_{MCC}}}}$ denotes the MCC dissimilarity measure, given
${\sigma^{2}}$ as the speed factor. The eigenvectors of KPCA projection matrix
${P_{r}}\in\mathbb{Q}{{}^{{v_{l}}\times d}}$ is derived as in [31], where $d$
represents the output dimension of a considerate desired projection. Then, a
query instance of the MCC given as ${\Phi^{q}}$ is used to match with all the
training samples ${\Phi^{l}}(i)$, such that $1\leq i\leq{N_{l}}$. Afterwards,
${N_{l}}$ is concatenated with ${v_{i}}$ as the matching scores
${v_{i}}:\leftarrow sim\left({{\Phi^{q}},{\Phi^{l}}(i)}\right)$, in order to
generate a vector $v\in\mathbb{Q}{{}^{{v_{l}}}}$. $v$ can then be transformed
using the initial kernel function to have
${F(v)=exp\left({-1{{\left({1-v}\right)}^{2}}/2{\sigma^{2}}}\right)}$. Hence,
the fingerprint vector is generated through
$\omega=v^{\prime}T\in\mathbb{Q}{{}^{d}}$, where ${v^{\prime}=F(v)}$, for
$v^{\prime}\Rightarrow v\in\mathbb{Q}{{}^{{N_{l}}}}$, and
$T\in\mathbb{Q}{{}^{{N_{l}}\times d}}$.

###  3.2 Feature vector permutation

The generated fixed-length vectors from KPCA are used as the input to our
protection mechanism. For each of the derived vectors
$v\in\mathbb{Q}{{}^{{N_{l}}}}$ and $T\in\mathbb{Q}{{}^{{N_{l}}\times d}}$, we
utilized an instance of LSH [4] from the function $h(.)$ to generate ${h_{i}}$
together with a permutation set $\left[{{P_{\theta}}}\right]$. Lets assume the
parent fingerprint feature vector $\omega=v^{\prime}T\in\mathbb{Q}{{}^{d}}$
and an $n$ independent LSH family hash function denoted by
${h_{i}}(\omega)\in[1,b]|i=1,...,n$, where each hash function constitutes a
product function. Therefore, for each ${h_{i}}(\omega)$, $p$ independent
random permutation seeds are generated to make up the permutation set
$\left[{{P_{\theta}}}\right]$. Consequently, $\omega$ is permuted so as to get
the permuted vectors ${\bar{\omega}_{k}}={\mathbb{Q}^{d}}|k=1,...,p$. Next, a
hash sample ${h_{({{\bar{\omega}}_{k}}i)}}$ is formed by performing $n$ times
permutation of randomly independent enrolled vectors. Hence,
${h_{({{\bar{\omega}}_{k}}i)}}$ yields $n$ number of permuted feature vector
${h_{({{\bar{\omega}}_{k}}i)}}\in\mathbb{Q}{{}^{{v_{l}}}}|i=1,...,n.$
Sampling bit location: The hash generation procedure can be denoted by the
function $\mathbb{F}$ which constitutes of the randomly sampled bit location
$\xi$ of each permuted hash vector ${h_{({{\bar{\omega}}_{k}}i)}}$. Thus, each
hash sample can be denoted by Eq. 2 below.

${\rm{\mathbb{F}}}\left(\xi\right):{\rm{\mathbb{Q}}}^{v_{l}}\to
h_{(\bar{\omega}_{k}i)}\left({\rm{\mathbb{Q}}}_{i}\right)|\left\\{i=1,2,...,n\right\\}$
(2)

We extract the bits ${\theta_{bits}}$ corresponding to each marked position
$b$ of $\xi_{b}\left(\theta_{(h_{i,n}^{{}^{\prime}})_{bits}}\right)$ from the
entire hash components $H_{C_{(i,n)}}$, where $h_{i}^{{}^{\prime}}$ denotes
the permuted hash vector deduced from ${h_{({{\bar{\omega}}_{k}}i)}}$.
Subsequently, the bit position of each vector element
$i=\left\\{{1,2,...,n}\right\\}$ from the vector component ${C_{(i,n)}}$ is
retained. Then its equivalent is concatenated within the hash component
${H_{{C_{(i,n)}}}}$. Thus, we have
$\xi_{b}\left(\theta_{(h_{i,n}^{{}^{\prime}})_{bits}}\right)||H_{C_{(i,n)}}$.

###  3.3 Non-invertible hash component

Even though the randomized permutation of the enrolled vectors in the previous
step can provide us with sufficient security, we still consider storing
${C_{(i,n)}}$ in the plain as insecure because an attacker can make use of the
vector components to effectively infer to the feature vectors in each of the
components available in the marked position. To avoid such scenario from
occurring, a security parameter ${S_{k}}$ is introduced such that
${S_{k}}:{L_{ow}}\leq{S_{k}}\leq{H_{igh}}$ for all permuted vector components.
${L_{ow}}$ and ${H_{igh}}$ represent the predefined lowest and highest values
of the given threshold, respectively. This threshold is also tied to the
accuracy preservation of the scheme. Hence, eliminating or limiting the trade-
off impact between accuracy and non-invertibility. Consequently, ${S_{k}}$ can
be incorporated into each vector component with respect to their bit location
using Eq. 3 below.

$S_{k}\left[{\rm{\mathbb{Q}}}_{C_{(i,n)}}^{{}^{\prime}}\right]=\xi_{b}\left(\theta_{(h_{i,n}^{{}^{\prime}})_{bits}}\right)||H_{C_{(i,n)}}$
(3)

Next, we perform a random search via a shift-max order process on the entire
hash component ${H_{{C_{(i,n)}}}}$ by taking the index of the maximum value
from the 1st two elements (in which one of them must be the highest in the
entire hash component). We term this as the shift-max value order scenario.
Each component ${H_{{C_{(i,n)}}}}$ consists of $m$ number of elements with a
fixed window size $k$. Thus, ${H_{{C_{(i,n)}}}}$ can be represented as
${H_{{C_{(i,n)}}}}=\left[{H_{{1_{(i,n)}}}^{C},H_{{2_{(i,n)}}}^{C},...,H_{{m_{(i,n)}}}^{C}}\right]$
and $m=l/k$, where $l$ denotes the length of the vector component. In the
first order shifting, each $H_{{1_{(i,n)}}}^{C}$ is shifted in a sequentially
cyclical order in each of the hash component ${H_{{C_{(i,n)}}}}$, until the
last element $H_{{m_{(i,n)}}}^{C}$. The entire shifting procedure can be
represented by the matrix in Eq. 4.

$\displaystyle\tiny\begin{array}[]{l}{M_{shift}}=\left[{\begin{array}[]{*{20}{c}}{H_{{1_{(i,n)}}}^{C}}\\\
{H_{{2_{(i,n)}}}^{C}}\\\ \vdots\\\
{H_{{m_{(i,n)}}}^{C}}\end{array}}\right]=\\\
{\left[{\begin{array}[]{*{20}{c}}{H_{{1_{(i,n)}}}^{C}\left(a\right)}\\\
{H_{{2_{(i,n)}}}^{C}\left(a\right)}\\\ \vdots\\\
{H_{{m_{(i,n)}}}^{C}(a)}\end{array}{\rm{}}\begin{array}[]{*{20}{c}}{H_{{1_{(i,n)}}}^{C}\left(b\right){\rm{}}\cdots}\\\
{H_{{2_{(i,n)}}}^{C}\left(b\right){\rm{}}\cdots}\\\ \vdots\\\
{H_{{m_{(i,n)}}}^{C}\left(b\right){\rm{}}\cdots}\end{array}\begin{array}[]{*{20}{c}}{H_{{1_{(i,n)}}}^{C}\left(k\right)}\\\
{H_{{2_{(i,n)}}}^{C}\left(k\right)}\\\ \vdots\\\
{H_{{m_{(i,n)}}}^{C}\left(k\right)}\end{array}}\right]_{m\times
k}}\end{array}$ (22)

where
$\left[{H_{{1_{(i,n)}}}^{C}\left(a\right){\rm{,}}H_{{1_{(i,n)}}}^{C}\left(b\right){\rm{}}\cdots{\rm{}}H_{{1_{(i,n)}}}^{C}\left(k\right)}\right]$
to
$\left[{H_{{m_{(i,n)}}}^{C}\left(a\right){\rm{,}}H_{{m_{(i,n)}}}^{C}\left(b\right){\rm{}}\cdots{\rm{}}H_{{m_{(i,n)}}}^{C}\left(k\right)}\right]$
are the first and last components constituting the entire elements of a given
$k-size$ window, while $m\times k$ denote the size of the entire ${M_{shift}}$
matrix.

After selecting the highest value from the 1st two elements
$\left[{H_{{1_{(1,n)}}}^{C}\left(a\right),{\rm{}}H_{{1_{(1,n)}}}^{C}\left(b\right)}\right]$
of the 1st component $\left[{H_{{1_{(i,n)}}}^{C}}\right]$, it is then passed
on to the second component $\left[{H_{{2_{(i,n)}}}^{C}}\right]$, until the
entire component is searched whilst preserving the bits position of the
selected elements (i.e., shifting bits corresponding to each element in the
vector components according to the matrix order). From the second component,
the highest value of $h_{i}^{{}^{\prime}}$ from each component is thus passed
to the hash bin ${H_{bin}}$ for final hash construction. As such, the
randomized permutation, security parameter ${S_{k}}$, coupled with the shift-
max value procedure, all contribute to the non-invertibility of the templates.
Fig. 1 illustrate the entire hash generation procedure at both enrollment and
querying stages.

Figure 1: Illustration of feature vector enrolling and querying for hash code
generation

Final hash construction: In order to construct the final hash code, we take
the index of the highest value from the bin such that for each
$h_{i}^{{}^{\prime}}\in H_{bin}$, we have;

$\xi_{b}\left(\theta_{(h_{i,n}^{{}^{\prime}})}\right)=\max_{j=1,...,n}\left\langle
h_{i}^{{}^{\prime}},y_{i}^{{}^{\prime}}\right\rangle|i=1,...,n$ (23)

for $h_{i}^{{}^{\prime}}\in H_{bin}$ and $y_{i}^{{}^{\prime}}\in H_{bin}$,
where $H_{bin}$ denotes the hash bin, and $y_{i}^{{}^{\prime}}$ denote the
index of the highest value in the bin. The highest value is then recorded in
the first $k-size$ window and its index value is stored. Therefore, we will
have
$\left[\theta_{h_{(i,n)}^{{}^{\prime}}}\right]\in\left[1,k\right]{\rm\;for\;}i=1,...,n$.
We repeat stage 2 and 3 on different hash components using different
permutation sets, and construct the final hash code
$h_{{C_{(i,n)}}}^{{}^{\prime\prime}}|i=1,...,n$. Note that every single hash
component ${H_{{C_{(i,n)}}}}$ constitutes of the hash code
$h_{C}^{{}^{\prime\prime}}$ that emanates from the permuted feature vector
$h_{(\bar{\omega}_{k}i)}\in{\rm{\mathbb{Q}}}^{v_{l}}$. Hence,

$h_{C}^{{}^{\prime\prime}}=h_{(\bar{\omega}_{k}i)}\in{\rm{\mathbb{Q}}}^{v_{l}}[1,b]|i=1,...,n$
(24)

Similarity score: A similarity score is used to evaluate the comparison
between the reference (enrolled) and probe (queried) hash codes of the
proposed scheme. Since LSH instance is used to ascertain the similarity of
permuted vector features within the inherent set of ordered elements, whereby
their hash is directly dependent on their relative distances, we therefore
cannot opt for traditional similarity measures that does not support such
ordering. Instead, the global similarity is estimated between the enrolled and
queried hash codes by computing the number of elements within each component
that matches with the two sets. Each set here correspond to the correlation
order of each index of the maximum value over $k-size$ window. In order to
enhance security, matching between the enrolled and queried vectors is still
conducted in the transformed domain, but by taking into cognizance the
similarity between feature vectors with respect to the shifting process, which
might be argued of being prone to similarity-based attacks (Our future work
will give an in-depth analysis on this).
For each hash code $h_{{C_{(i,n)}}}^{{}^{\prime\prime}}$ and
$h_{{C_{(j,n)}}}^{{}^{\prime\prime}}$, let
$h_{{C_{(i,1)}}}^{{}^{\prime\prime}},h_{{C_{(i,2)}}}^{{}^{\prime\prime}},...,h_{{C_{(i,n)}}}^{{}^{\prime\prime}}$
and
$h_{{C_{(j,1)}}}^{{}^{\prime\prime}},h_{{C_{(j,2)}}}^{{}^{\prime\prime}},...,h_{{C_{(j,n)}}}^{{}^{\prime\prime}}$
be the enrolled and queried instances of the non-invertible hash components,
respectively. Thus, the two sets of hash components from
$h_{{C_{(i,n)}}}^{{}^{\prime\prime}}$ and
$h_{{C_{(j,n)}}}^{{}^{\prime\prime}}$ can be given as:

$E\left[{h_{{C_{(i,n)}}}^{{}^{\prime\prime}}}\right]=h_{{C_{(i,1)}}}^{{}^{\prime\prime}},h_{{C_{(i,2)}}}^{{}^{\prime\prime}},...,h_{{C_{(i,n)}}}^{{}^{\prime\prime}}|i=1,...,n$
(25)
$Q\left[{h_{{C_{(j,n)}}}^{{}^{\prime\prime}}}\right]=h_{{C_{(j,1)}}}^{{}^{\prime\prime}},h_{{C_{(j,2)}}}^{{}^{\prime\prime}},...,h_{{C_{(j,n)}}}^{{}^{\prime\prime}}|j=1,...,n$
(26)

Suppose the collision probability of the two hash codes is
$CP\left\\{{E\left[{h_{{C_{(i,n)}}}^{{}^{\prime\prime}}}\right]}\right.,\left.{Q\left[{h_{{C_{(j,n)}}}^{{}^{\prime\prime}}}\right]}\right\\}$.
Therefore,

$\begin{array}[]{l}{\mathbb{C}_{P}}=\left[{E\left[{h_{{C_{(i,n)}}}^{{}^{\prime\prime}}}\right]=\left.{Q\left[{h_{{C_{(j,n)}}}^{{}^{\prime\prime}}}\right]}\right]}\right.\Rightarrow\\\
CP\left\\{{E\left[{h_{{C_{(i,n)}}}^{{}^{\prime\prime}}}\right]}\right.,\left.{Q\left[{h_{{C_{(j,n)}}}^{{}^{\prime\prime}}}\right]}\right\\}{\rm{for}}i=1,...,n\end{array}$
(27)

Hence, the similarity between
$E\left[{h_{{C_{(i,n)}}}^{{}^{\prime\prime}}}\right]$ and
$Q\left[h_{C_{(j,n)}}^{{}^{\prime\prime}}\right]$ indicates a high
probability, while their dissimilarity indicates a less collision probability.
Consequently, a normalized similarity correlation score is derived by dividing
the number of matches between the two components with their set cardinality.
The similarity correlation between the two hash codes can be given as:

$S_{C}=\frac{\sum_{b=1}^{n}\left|E\left[h_{C_{(i,n)}}^{{}^{\prime\prime}}\right]^{r}=Q\left[h_{C_{(j,n)}}^{{}^{\prime\prime}}\right]^{r}\right|/k}{V_{l}^{{}^{\prime}}}$
(28)

where $V_{l}^{{}^{\prime}}=V_{l}/k$ for each ${V_{l}}$ denoting a single
vector element from the vector component, $r$ is the correlation order of each
index, and $b$ is the index of maximum value over $k-size$ window. This $b$
can be selected via a $k-1$ entries that are less than $b$ in both the
enrolled and queried hash codes, which helps determine the exact similarity
between two hash codes and consequently assist with the selection of the
maximum element within a hash component.

##  4 Experiments

In this section, we performed an extensive evaluation of the proposed template
protection scheme over all six public fingerprint datasets which includes,
FVC2002 (DB1, DB2, and DB3), as well as FVC2004 (DB1, DB2, and DB3). Each
dataset contains 800 fingerprints of varying quality collected from 100 users
with 8 impressions per user. Verifinger SDK is utilized for the extraction of
ISO-compliant minutiae information which was further feed to MCC. Four major
parameters including security threshold ${S_{key}}$, window size $k$, Hashvecs
${H_{{C_{(i,n)}}}}$, and number of elements in each vector $Nu{m_{perm}}$ are
assigned specific values with respect to analyzing the accuracy of the
proposed scheme. Also, the security and privacy analysis including non-
invertibility, cancelability/revocability, and unlinkability, were all carried
out in order to evaluate the efficacy of the proposed scheme.

Different performance measures are used for the evaluation. These are the
False Match Rate (FMR), False Non-Match Rate (FNMR), and Equal Error Rate
(EER). FMR results from the decision threshold of a biometric system to
misinterpret the probability of two different impressions from two fingers as
an impression of one finger, while FNMR results from the decision of the
system to misinterpret the probability of two different impressions from one
finger as an impression of two fingers. The common value between FMR and FNMR
is the ERR, which determines the accuracy of the scheme. (i.e., the error rate
when FMR = FNMR). Likewise, the Genuine Match Rate (GMR) is akin to the FMR
when GMR + FMR = 1. We employed the original FVC protocol for all evaluations
and analysis.

Figure 2: Effect of security threshold and window size w.r.t. FMR and FMNR | FVC2002DB1 | FVC2002DB2 | FVC2002DB3 | FVC2004DB1 | FVC2004DB2 | FVC2004DB3
---|---|---|---|---|---|---
${S_{key}}$ | GMR | FMR | GMR | FMR | GMR | FMR | GMR | FMR | GMR | FMR | GMR | FMR
2 | 99.61 | 0.38 | 98.80 | 1.15 | 96.00 | 3.94 | 97.60 | 2.42 | 92.70 | 7.41 | 93.40 | 6.95
6 | 99.00 | 0.95 | 97.40 | 2.85 | 91.60 | 8.10 | 94.80 | 5.52 | 88.60 | 11.01 | 89.00 | 11.41
10 | 97.50 | 2.18 | 96.10 | 3.64 | 90.40 | 8.59 | 93.60 | 7.01 | 87.60 | 13.25 | 85.80 | 13.09
14 | 97.00 | 2.59 | 94.80 | 4.57 | 88.70 | 10.61 | 92.50 | 6.55 | 86.30 | 13.41 | 84.70 | 14.26
18 | 96.30 | 3.05 | 94.70 | 5.11 | 89.20 | 11.47 | 93.20 | 7.62 | 86.60 | 13.88 | 84.20 | 14.55
22 | 96.10 | 3.64 | 94.30 | 5.03 | 88.00 | 12.83 | 91.90 | 7.17 | 85.70 | 14.34 | 85.20 | 16.51
26 | 96.10 | 3.33 | 94.90 | 5.31 | 88.60 | 12.73 | 92.30 | 8.61 | 86.40 | 15.64 | 84.30 | 14.61
30 | 96.20 | 4.36 | 94.10 | 5.37 | 87.40 | 12.79 | 91.80 | 7.70 | 84.90 | 14.91 | 83.90 | 15.13
Table 1: Performance accuracy of GMR against FMR w.r.t. the security
threshold.

###  4.1 Effect of security threshold ${S_{key}}$ and window size $k$

In this analysis, the number of hash vector ${H_{{C_{(i,n)}}}}$ in each
component ${{C_{(i,n)}}}$ (component size) is set at 10, while the number of
vector elements is set at 500. The affected parameters i.e., window size $k$
and security threshold ${S_{key}}$ are set within the specified range of $k$ =
75, 100, 125, 150, and ${S_{key}}$ = 2, 6, 10, 14, 18, 22, 26, 30,
respectively. Fig. 2. shows the overall plots analysis against FMR and FNMR
with the initials $m$ and $n$ as depicted on their distributions $(K,S)$,
respectively. All six datasets of FVC2002 and FVC2004 are utilized for the
evaluation. It can be seen that lower ${S_{key}}$ and higher $k$ generally
exhibits better accuracy as compared to higher ${S_{key}}$ and lower $k$. This
proves the sensitivity of the generated hash codes w.r.t the defined security
threshold. Also, DB1 & DB2 of FVC2002, and DB1 of FVC2004 depicts a better
performance than the other datasets.

Techniques |  | FVC2002
---
DB1
| FVC2002
---
DB2
| FVC2002
---
DB3
| FVC2004
---
DB1
| FVC2004
---
DB2
| FVC2004
---
DB3
[HTML]EFEFEFWith Template Protection Mechanism
$2{\rm{PMC}}{{\rm{C}}_{128,{\rm{128}}}}$ [8] | 1.88 | 0.99 | 1.24 | - | 4.84 | -
$2{\rm{PMC}}{{\rm{C}}_{64,{\rm{64}}}}$ [8] | 3.33 | 1.76 | 7.78 | - | 6.62 | -
$2{\rm{PMC}}{{\rm{C}}_{16,{\rm{16}}}}$ [8] | 12.39 | 10.16 | 19.33 | - | 17.70 | -
Bloom filter with fingerprint [26] | 2.3 | 1.8 | 6.6 | 13.4 | 8.1 | 9.7
PR-NNLS $(\gamma={N_{c}}+150)$ [19] | 2.28 | 1.25 | 6.4 | - | 7 | -
PR-NNLS $(\gamma={N_{c}}+20)$ [19] | 2.48 | 1.51 | 7.03 | - | 7.44 | -
IoM Hashing [16] | 0.43 | 2.10 | 6.60 | 4.51 | 8.02 | 8.46
Fractal Coding Hashing [30] | 0.36 | 0.54 | 2.395 | 2.348 | 5.925 | 2.365
Proposed (Protected) | 0.38 | 1.16 | 3.88 | 2.42 | 7.12 | 6.34
[HTML]EFEFEFWithout Template Protection Mechanism
MCC (Original) | 0.60 | 0.59 | 3.91 | 3.79 | 5.22 | 3.82
Fixed-length Vector (Transformed) | 0.20 | 0.19 | 2.30 | 4.70 | 3.13 | 2.80
TKLSH-DQ [18] | 3.76 | 4.94 | 11.58 | 17.11 | 12.19 | 14.50
TKPCA-DQ [18] | 4.03 | 5.04 | 11.86 | 17.39 | 10.95 | 13.49
Table 2: Accuracy performance comparison between the proposed and other state-
of-the-art schemes Figure 3: Effect of security threshold and window size
w.r.t. FMR and FMNR

###  4.2 Performance Accuracy and Comparison with other state-of-the-arts

In addition to the evaluations with respect to the effect of different
parameters, we also analyze the matching performance of the scheme under lost-
key scenario using the FVC protocol, in relation to the GMR and FMR. All six
datasets are used for the experiment by keeping the window size $k$ at 100,
$Nu{m_{perm}}$ at 500. The security threshold ${S_{key}}$ is however adjusted
from 2 to 30 according to the previous experimentation while keeping the hash
vector at 5. A detailed description of the evaluation is given in Table 1
below. It can easily be judged from the Table that the best performance in all
datasets is achieved when the security threshold is set at 2. To further
analyze the performance accuracy of the scheme, we compare our approach with
other well-known template protection schemes. By setting the security
threshold ${S_{key}}$ = 2 and Hashvec ${H_{{C_{(i,n)}}}}$ = 10, we record the
EER of the proposed method under original FVC protocol (with window size $k$ =
150 for all FVC2002 datasets, $k$ = 100 for DB1 and DB2 of FVC2004, and $k$ =
125 for DB3 of FVC 2004). Table 2 shows the performance comparison between the
proposed method and other state-of-the-art existing cancelable templates
schemes with protection and their corresponding transformed version before the
incorporation of protection mechanism.
It can be seen from Table 2 that the proposed scheme outperforms all other
techniques except for fractal coding-based hashing. However, comparing 2PMCC
at its best $\left({2{\rm{PMC}}{{\rm{C}}_{128,{\rm{128}}}}}\right)$, moderate
$\left({2{\rm{PMC}}{{\rm{C}}_{64,{\rm{64}}}}}\right)$, and worst
$\left({2{\rm{PMC}}{{\rm{C}}_{16,{\rm{16}}}}}\right)$ parameters, we can
notice that our scheme still outperforms
$2{\rm{PMC}}{{\rm{C}}_{128,{\rm{128}}}}$ (best) on FVC2002 DB2, and
$2{\rm{PMC}}{{\rm{C}}_{64,{\rm{64}}}}$ (moderate), and
$2{\rm{PMC}}{{\rm{C}}_{16,{\rm{16}}}}$ (worst) with a huge margin. For the
best $(\gamma={N_{c}}+150)$ and worst $(\gamma={N_{c}}+20)$ performances of
PR-NNLS based scheme, it shows a slightly better performance on DB2 of FVC2004
at $(\gamma={N_{c}}+150)$. As for techniques without incorporation of template
protection, the transformed fixed length vector outperforms the original MCC
with a huge margin as compared to the proposed protected version (due to the
slight manifestation of trade off between accuracy and security even with a
moderate Hashvec ${H_{{C_{(i,n)}}}}$ value. Meanwhile, the protected version
also outperforms the original MCC. The EER of all 3 approaches from same
features is depicted in Fig. 3, with MCC as unprotected, fixed-length vector
as transformed, and the proposed scheme as protected. It can be seen that the
protected scheme generally outperforms both unprotected and transformed
version in BD1 of FVC2004, whilst being outperformed by its counterparts in
DB2 and DB3 of same database. This certainly has to do with the low
identifiable correlation within different elements in the components of these
datasets, which in same vain emanates from the distorted or low-quality images
in the datasets. In FVC2002 however, the protected version performs better
than its unprotected counterpart in DB1 and DB2. But is slightly outperformed
by the transformed version, which of course has to do with the incorporation
of security mechanisms. Let’s also not forget that the rigorous search for a
maximum value between the 1st two elements of each components has a role to
play in reducing the accuracy performance of the protected templates. Overall,
$2{\rm{PMC}}{{\rm{C}}_{16,{\rm{16}}}}$ shows the worst performance on schemes
incorporated with a template protection mechanism, TKPCA-DQ depicts the worse
performance on schemes without a template protection mechanism, while the
fixed length (transformed) version performs best among the three techniques
from same features. Also, it is worth mentioning that due to the randomness
(caused by random vector permutation) and iterative reshuffling (caused by
shifting procedure) of the mechanism, a difference of 0.1 to 0.5 is noticed at
each specific feature reconstruction of the protected template when a loop
iteration without pre-allocation is allowed. This certainly confirms the
sensitivity of the hash code and the entire security mechanism of the scheme.

###  4.3 Revocability

Revocability is an important property of cancelable biometrics which pertains
to the security of a biometric system. When the original stored biometric
template is compromised, it is expected by the cancelable system to re-
generate uncorrelated templates from same fingerprint. Thus, further ensuring
that the reissued templates from same fingerprint are not correlated despite
being generated from the same biometric data. To test the revocability of the
proposed scheme, transformed templates are generated from first impression of
each finger using different randomly generated keys. Same process is
repeatedly applied to DB1 of FVC2002 by matching the pseudo-imposter templates
against its imposter and genuine counterparts. The mean for both pseudo-
impostor and impostor distributions are set at 0.35. Fig. 4 shows the
distributions of pseudo-impostor, impostor, and genuine scores of the dataset.
We can see from the figure that the distributions of pseudo-impostor and
impostor overlap, while being completely unrelated to the genuine
distributions. This signifies that the reissued version of the fingerprint
templates are uncorrelated despite coming from same source.

Figure 4: Revocability analysis

###  4.4 Unlinkability

Unlinkability is another important property of cancelable biometrics which
relates to the privacy of the biometric data. To measure the unlinkability of
a cancelable biometric system, it is required that the protected template
should not cross-match with the same corresponding unprotected fingerprint on
different platforms and/or applications. Two major indicators are used for the
evaluation of unlinkability. These are mated and non-mated score
distributions. According to the framework [11], mated and non-mated sample
score distributions measure the system’s unlinkability at both local and
global level. Calculating the mated sample scores is mainly performed by
comparing the templates that are extracted from fingers with same impression
using different keys, while non-mated scores are computed by comparing
templates that are extracted from fingers with different impressions using
different keys. Unlinkability analysis of the scheme is conducted on DB1 of
FVC2002 dataset. The same mean value used in revocability is also maintained
here. The score distributions between mated and non-mated samples can be seen
in Fig. 5, which revealed a significant overlap. Thus, indicating the
unlinkability of the scheme.

Figure 5: Unlinkability analysis

##  5 Conclusions and Future work

We have introduced a novel approach for the construction of cancelable and
non-invertible fingerprint hash codes using vector permutation and a shift-
order process. Particularly, the security mechanism of the scheme against non-
invertibility and similarity-based attacks is improved through the integration
of shift-order process. Performance accuracy of the scheme outperforms other
state-of-the-art recognition systems that comes with, and without the
incorporation of security mechanisms. The scheme also satisfy other properties
of cancelable biometrics.

Future work will cover security analysis of the scheme against similarity-
based attacks, non-invertiblity, and other important security threats. We will
also prove that the scheme is able to alleviate the similarity preservation
drawback between hash spaces. Consequently, we shall further build upon the
shift-ordering procedure by incorporating homomorphic encryption to avert all
shortcomings associated with matching in the transformed domain. Instead,
matching will be performed in the encrypted domain.

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# Entropy on modules over the group ring of a sofic group

Bingbing Liang  B.L., Max Planck Institute for Mathematics, Vivatsgasse 7,
53111, Bonn, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: September 5, 2017)

###### Abstract.

We partially generalize Peters’ formula [14] on modules over the group ring
${\mathbb{F}}\Gamma$ for a given finite field ${\mathbb{F}}$ and a sofic group
$\Gamma$. It is also discussed that how the values of entropy are related to
the zero divisor conjecture.

###### Key words and phrases:

Sofic group, algebraic entropy, topological entropy, zero-divisor conjecture

###### 2010 Mathematics Subject Classification:

Primary 37B99, 16D10, 13C12.

##  1\. Introduction

Let $\Gamma$ be a countable discrete group. An algebraic action of $\Gamma$ is
the induced (continuous) action $\Gamma\curvearrowright\widehat{\mathcal{M}}$
from some (left) ${\mathbb{Z}\Gamma}$-module ${\mathcal{M}}$. Here
$\widehat{{\mathcal{M}}}$ is the Pontryagin dual of the discrete abelian group
${\mathcal{M}}$. One of main concerns is about the interplay between the
dynamical information of $\Gamma\curvearrowright\widehat{\mathcal{M}}$ and the
algebraic information of ${\mathcal{M}}$. Weiss considered the case that
${\mathcal{M}}$ is a torsion abelian group [18]. For general
${\mathbb{Z}\Gamma}$-module ${\mathcal{M}}$, Peters showed that the
topological entropy of $\Gamma\curvearrowright\widehat{\mathcal{M}}$ coincides
with the algebraic entropy of ${\mathcal{M}}$ when $\Gamma$ is amenable [14].
This general result does not only establish the beautiful connection but also
provides the great flexibility to study the entropy of algebraic actions [5,
13].

Towards more general group actions, Bowen and Kerr-Li developed an entropy
theory when $\Gamma$ can be approximated by finite groups [2, 10]. The groups
admitting such approximations are the so-called sofic groups, which in
particular includes amenable groups and residually groups [8, 19].

On the other hand, given a unital ring $R$ and a sofic group $\Gamma$, in [12,
Definition 3.1], started with a length function ${\rm L}$ on left $R$-modules
([12, Definition 2.1]), Li and the author introduced a mean length function on
$R\Gamma$-modules ${\mathcal{M}}$ which are locally $L$-finite in the sense
that each finitely generated $R$-submodule of ${\mathcal{M}}$ has finite ${\rm
L}$-length. In particular, for $R={\mathbb{Z}}$ and ${\rm
L}(\cdot)=\log|\cdot|$, the corresponding mean length function can be treated
as an algebraic version of entropy, which we may call sofic algebraic entropy.
Then a natural question to ask is whether one can generalize Peters’ formula
to the setting of sofic group actions.

In this paper, we give a partial generalization of Peters’ formula to the case
of sofic group actions. The main inputs of the proof are an approximation
formula of topological entropy given in [7, Lemma 4.8] and some techniques
appeared in [12]. Given a finite field ${\mathbb{F}}$ any
${\mathbb{F}}\Gamma$-module can be treated as a ${\mathbb{Z}\Gamma}$-module.
Fix a sofic approximation $\Sigma$ for $\Gamma$ and a free ultrafilter
$\omega$ over ${\mathbb{N}}$. For any left ${\mathbb{F}}\Gamma$-modules
${\mathcal{M}}_{1}\subseteq{\mathcal{M}}_{2}$, we have the algebraic entropy
of ${\mathcal{M}}_{1}$ relative to ${\mathcal{M}}_{2}$, denoted by
$h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2})$ (Definition 3.1). For
any ${\mathbb{F}}\Gamma$-module ${\mathcal{M}}$, the sofic algebraic entropy
$h_{\Sigma,\omega}({\mathcal{M}})$ of ${\mathcal{M}}$ is then defined as
$h_{\Sigma,\omega}({\mathcal{M}}|{\mathcal{M}})$. We remark that the main
motivation to introduce this relative version of invariants and technical
consideration of ultrafilters is that one can establish a modified addition
formula under this approach [12, Theorem 1.1].

Correspondingly, we have the relative version of topological entropy in
dynamical systems. Let $\Gamma$ act on compact metrizable spaces $X$ and $Y$
continuously. For any factor map from $X$ to $Y$, we have the topological
entropy $h_{\Sigma,\omega}(Y|X)$ of $\Gamma\curvearrowright Y$ relative to
$\Gamma\curvearrowright X$ [12, Definition 9.3]. When $X=Y$ and the factor map
is the identity map, the relative topological entropy coincide with the
original definition of sofic topological entropy, which we denote by
$h_{\Sigma,\omega}(X)$. Similar approaches also independently appears in the
works of other experts. Hayes gave a formula for the relative version of sofic
measure entropy in terms of a given compact model in [9]. A similar notion for
Rokhlin entropy, called outer Rokhlin entropy, was developed by Seward in
[16].

######  Theorem 1.1.

Let $\Gamma$ be a sofic group and ${\mathbb{F}}$ be a finite field. Then for
any finitely generated ${\mathbb{F}}\Gamma$-module ${\mathcal{M}}$, we have

$h_{\Sigma,\omega}(\widehat{\mathcal{M}})=h_{\Sigma,\omega}({\mathcal{M}}).$

Let ${\mathcal{M}}_{1}\subseteq{\mathcal{M}}_{2}$ be countable
${\mathbb{F}}\Gamma$-modules. Then
$h_{\Sigma,\omega}(\widehat{\mathcal{M}}_{1}|\widehat{\mathcal{M}}_{2})\leq
h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2}).$

This paper is organized as follows. We recall some background knowledge in
Section 2. The algebraic entropy is introduced in Section 3 and some basic
properties are discussed there. In particular, we show that the values of
algebraic entropies is related to the zero divisor conjecture. We prove the
main result in Section 4.

Throughout this paper, $\Gamma$ will be a countable discrete group. For any
set $S$, we denote by ${\mathcal{F}}(S)$ the set of all nonempty finite
subsets of $S$. All modules are assumed to be left modules unless specified.
For any $d\in{\mathbb{N}}$, we write $[d]$ for the set $\\{1,\cdots,d\\}$ and
${\rm Sym}(d)$ for the permutation group of $[d]$.

Acknowledgements. We thank the helpful comments from Yongle Jiang and Hanfeng
Li.

##  2\. Preliminaries

###  2.1. Group rings and algebraic actions

Let $R$ be a unital ring. The group ring of $\Gamma$ with coefficients in $R$,
denoted by $R\Gamma$, consists of all finitely supported functions:
$f:\Gamma\to R$. Conventionally, we shall write $f$ as
$\sum_{s\in\Gamma}f_{s}s$, where $f_{s}\in R$ for all $s\in\Gamma$ and
$f_{s}=0$ for all except finitely many $s\in\Gamma$. The algebraic operations
on $R\Gamma$ are defined by

$\sum_{s\in\Gamma}f_{s}s+\sum_{s\in\Gamma}g_{s}s=\sum_{s\in\Gamma}(f_{s}+g_{s})s,\mbox{
and
}\big{(}\sum_{s\in\Gamma}f_{s}s\big{)}\big{(}\sum_{t\in\Gamma}g_{t}t\big{)}=\sum_{s,t\in\Gamma}f_{s}g_{t}(st).$

We similarly have the product if one of $f$ and $g$ sits in $R^{\Gamma}$.

For any countable ${\mathbb{Z}\Gamma}$-module ${\mathcal{M}}$, treated as a
discrete abelian group, its Pontryagin dual $\widehat{\mathcal{M}}$ consisting
of all continuous group homomorphisms
${\mathcal{M}}\to{\mathbb{R}}/{\mathbb{Z}}$, coincides with ${\rm
Hom}_{{\mathbb{Z}}}({\mathcal{M}},{\mathbb{R}}/{\mathbb{Z}})$. By Pontryagin
duality, $\widehat{\mathcal{M}}$ is a compact metrizable space under compact-
open topology. Furthermore, the ${\mathbb{Z}\Gamma}$-module structure of
${\mathcal{M}}$ naturally induces an adjoint action
$\Gamma\curvearrowright\widehat{\mathcal{M}}$ by continuous automorphisms. To
be precise,

$\langle s\chi,u\rangle:=\langle\chi,s^{-1}x\rangle$

for all $\chi\in\widehat{\mathcal{M}},u\in{\mathcal{M}}$, and $s\in\Gamma$.
Conversely, by Pontryagin duality, each action of $\Gamma$ on a compact
metrizable abelian group arise this way and thus we call such a dynamical
system an algebraic action [15].

###  2.2. Amenable and sofic groups

The group $\Gamma$ is called amenable if for any $K\in{\mathcal{F}}(\Gamma)$
and any $\delta>0$ there is an $F\in{\mathcal{F}}(\Gamma)$ with $|KF\setminus
F|<\delta|F|$.

A sequence of maps $\Sigma=\\{\sigma_{i}:\Gamma\to{\rm{\rm
Sym}}(d_{i})\\}_{i\in{\mathbb{N}}}$ is called a sofic approximation for
$\Gamma$ if it satisfies:

  1. (1)

$\lim_{i\to\infty}|\\{v\in[d_{i}]:\sigma_{i,s}\sigma_{i,t}(v)=\sigma_{i,st}(v)\\}|/d_{i}=1$
for all $s,t\in\Gamma$,

  2. (2)

$\lim_{i\to\infty}|\\{v\in[d_{i}]:\sigma_{i,s}(v)\neq\sigma_{i,t}(v)\\}|/d_{i}=1$
for all distinct $s,t\in\Gamma$,

  3. (3)

$\lim_{i\to\infty}d_{i}=+\infty$.

The group $\Gamma$ is called a sofic group if it admits a sofic approximation.

Any amenable group is sofic since one can use a sequence of asymptotically-
invariant subsets of the amenable group, i.e. Følner sequence, to construct a
sofic approximation. Residually finite groups are also sofic since a sequence
of exhausting finite-index subgroups naturally induces a sofic approximation
in which each approximating map is actually a group homomorphism. We refer the
reader to [3, 4] for more information on sofic groups.

Throughout the rest of this paper, $\Gamma$ will be a countable sofic group,
$\Sigma$ will be a sofic approximation for $\Gamma$, and $\omega$ will be a
free ultrafilter over ${\mathbb{N}}$.

###  2.3. Sofic topological entropy

For any continuous pseudometric $\rho_{X}$ on a compact space $X$ and any
$\varepsilon>0$, a subset $Z$ of $X$ is called
($\rho_{X},\varepsilon$)-separated if $\rho_{X}(z_{1},z_{2})>\varepsilon$ for
all distinct $z_{1},z_{2}\in Z$. Set
$N_{\varepsilon}(X,\rho_{X}):=\max_{Z}|Z|$ for $Z$ ranging over all
$(\rho_{X},\varepsilon)$-separated subsets of $X$.

Let $\Gamma$ act continuously on a compact metrizable space $X$.

######  Definition 2.1.

Let $\rho_{X}$ be a continuous pseudometric on $X$. For any
$d\in{\mathbb{N}}$, define continuous pseudometrics $\rho_{X,2}$ and
$\rho_{X,\infty}$ on $X^{d}$ by

$\rho_{X,2}(\varphi,\psi)=\left(\frac{1}{d}\sum_{v\in[d]}\rho_{X}(\varphi_{v},\psi_{v})^{2}\right)^{1/2},{\rm
and\
}\rho_{X,\infty}(\varphi,\psi)=\max_{v\in[d]}\rho_{X}(\varphi_{v},\psi_{v}).$

Let $\sigma$ be a map from $\Gamma$ to ${\rm Sym}(d)$,
$F\in{\mathcal{F}}(\Gamma)$, and $\delta>0$. The set of approximately
equivariant maps ${\rm Map}(\rho,F,\delta,\sigma)$ is defined to be the set of
all maps $\varphi:[d]\to X$ such that
$\rho_{X,2}(s\varphi,\varphi\circ\sigma(s))\leq\delta$.

A continuous pseudometric $\rho_{X}$ on $X$ is called dynamically generating
if $\rho_{X}$ can distinguish all distinct elements of $X$ under the action of
$\Gamma$, i.e. for all distinct $x,x^{\prime}\in X$, there exists some
$s\in\Gamma$ such that $\rho_{X}(sx,sx^{\prime})>0$.

Now let $\Gamma$ act on another compact metrizable space $Y$ and $\pi:X\to Y$
be a factor map. Denote by ${\rm Map}(\pi,\rho,F,\delta,\sigma)$ the set of
all $\pi\circ\varphi$ for $\varphi$ ranging in ${\rm
Map}(\rho,F,\delta,\sigma)$.

######  Definition 2.2.

Let $\rho_{X}$ and $\rho_{Y}$ be two dynamically generating continuous
pseudometrics of $X$ and $Y$ respectively. Let $F\in{\mathcal{F}}(\Gamma)$ and
$\delta,\varepsilon>0$. Define

$h^{\varepsilon}_{\Sigma,\omega,\infty}(\rho_{Y},F,\delta|\rho_{X})=\lim_{i\to\omega}\frac{1}{d_{i}}\log
N_{\varepsilon}({\rm Map}(\pi,\rho_{X},F,\delta,\sigma_{i}),\rho_{Y,\infty}).$

If the set of $i\in{\mathbb{N}}$ with ${\rm
Map}(\rho_{X},F,\delta,\sigma_{i})=\emptyset$ is in $\omega$, we set
$h^{\varepsilon}_{\Sigma,\omega,\infty}(\rho_{Y},F,\delta|\rho_{X})=-\infty$.
We define the sofic topological entropy of $\Gamma\curvearrowright Y$ relative
to $\Gamma\curvearrowright X$ as

$h_{\Sigma,\omega}(Y|X):=\sup_{\varepsilon>0}\inf_{F\in{\mathcal{F}}(\Gamma)}\inf_{\delta>0}h^{\varepsilon}_{\Sigma,\omega,\infty}(\rho_{Y},F,\delta|\rho_{X}).$

The sofic topological entropy of $\Gamma\curvearrowright X$ is defined as
$h_{\Sigma,\omega}(X):=h_{\Sigma,\omega}(X|X)$ for $\pi:X\to X$ being the
identity map.

######  Remark 2.3.

It was shown that the definition of the relative sofic topological entropy
does not depend on the choices of the dynamically generating continuous
pseudometrics $\rho_{X}$ and $\rho_{Y}$ [12, Lemma 8.2, Lemma 9.5]. One has
$h_{\Sigma,\omega}(Y|X)\leq\min(h_{\Sigma,\omega}(X),h_{\Sigma,\omega}(Y))$.

##  3\. Algebraic entropy

When we specialize the length function as $\log|\cdot|$ on
${\mathbb{Z}}$-mdoules, the corresponding sofic mean length function in [12,
Definition 3.1] brings us the notion for the algebraic analogue of sofic
topological entropy. For reader’s convenience, we recall the complete
definition.

For any (abelian) group $G$, denote by ${\mathscr{F}}(G)$ the collection of
finitely generated subgroups of $G$. Let ${\mathcal{M}}$ be a
${\mathbb{Z}\Gamma}$-module and
${\mathscr{A}},{\mathscr{B}}\in{\mathscr{F}}({\mathcal{M}})$. For
$F\in{\mathcal{F}}(\Gamma)$ and a map $\sigma:\Gamma\to{\rm Sym}(d)$ for some
$d\in{\mathbb{N}}$, denote by ${\mathscr{M}}({\mathscr{B}},F,\sigma)$ the
abelian subgroup of
${\mathcal{M}}^{d}\cong{\mathbb{Z}}^{d}\otimes_{\mathbb{Z}}{\mathcal{M}}$
generated by the elements $\delta_{v}\otimes b-\delta_{\sigma(s)v}\otimes sb$
for all $v\in[d],b\in{\mathscr{B}}$, and $s\in F$, where $\delta_{v}$ denotes
the element of ${\mathbb{Z}}^{d}$ taking value 1 at the coordinate $v$ and $0$
everywhere else. Put ${\mathscr{M}}({\mathscr{A}},{\mathscr{B}},F,\sigma)$ as
the image of ${\mathscr{A}}^{d}$ in
${\mathcal{M}}^{d}/{\mathscr{M}}({\mathscr{B}},F,\sigma)$ under the quotient
map
${\mathcal{M}}^{d}\to{\mathcal{M}}^{d}/{\mathscr{M}}({\mathscr{B}},F,\sigma)$.
We remark that any ${\mathbb{Z}\Gamma}$-module ${\mathcal{M}}$ is locally
$\log|\cdot|$-finite if and only if ${\mathcal{M}}$ is torsion as an abelian
group.

######  Definition 3.1.

Let ${\mathcal{M}}_{1}\subseteq{\mathcal{M}}_{2}$ be
${\mathbb{Z}\Gamma}$-modules such that ${\mathcal{M}}_{1}$ is torsion. For any
${\mathscr{A}}\in{\mathscr{F}}({\mathcal{M}}_{1}),{\mathscr{B}}\in{\mathscr{F}}({\mathcal{M}}_{2})$
and $F\in{\mathcal{F}}(\Gamma)$, set

$h_{\Sigma,\omega}({\mathscr{A}}|{\mathscr{B}},F):=\lim_{i\to\omega}\frac{\log|{\mathscr{M}}({\mathscr{A}},{\mathscr{B}},F,\sigma_{i})|}{d_{i}}.$

We define the algebraic entropy of ${\mathcal{M}}_{1}$ relative to
${\mathcal{M}}_{2}$ as

$h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2}):=\sup_{{\mathscr{A}}\in{\mathscr{F}}({\mathcal{M}}_{1})}\inf_{F\in{\mathcal{F}}(\Gamma)}\inf_{{\mathscr{B}}\in{\mathscr{F}}({\mathcal{M}}_{2})}h_{\Sigma,\omega}({\mathscr{A}}|{\mathscr{B}},F).$

The sofic algebraic entropy of ${\mathcal{M}}_{1}$ is then defined as

$h_{\Sigma,\omega}({\mathcal{M}}_{1}):=h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{1}).$

Apply [12, Theorem 1.1] to the length function $\log|\cdot|$ on
${\mathbb{Z}}$-modules, we have the modified addition formula for sofic
algebraic entropy.

######  Theorem 3.2.

Let ${\mathcal{M}}_{1}\subseteq{\mathcal{M}}_{2}$ be
${\mathbb{Z}\Gamma}$-modules such that ${\mathcal{M}}_{2}$ is torsion as an
abelian group. Then

$h_{\Sigma,\omega}({\mathcal{M}}_{2})=h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2})+h_{\Sigma,\omega}({\mathcal{M}}_{2}/{\mathcal{M}}_{1}).$

By the similar argument as in [12, Section 3], we have the following
proposition, which collects basic properties of the sofic algebraic entropy.

######  Proposition 3.3.

Let ${\mathcal{M}}_{1}\subseteq{\mathcal{M}}_{2}$ be
${\mathbb{Z}\Gamma}$-modules such that ${\mathcal{M}}_{2}$ is torsion as an
abelian groups. Then the following are true.

  1. (1)

If ${\mathscr{N}}_{1}\subseteq{\mathscr{N}}_{2}$ are two torsion abelian
groups, then
$h_{\Sigma,\omega}({\mathbb{Z}\Gamma}\otimes_{\mathbb{Z}}{\mathscr{N}}_{1}|{\mathbb{Z}\Gamma}\otimes_{\mathbb{N}}{\mathscr{N}}_{2})=\log|{\mathscr{N}}_{1}|$;

  2. (2)

If $\\{{\mathcal{M}}_{j}^{\prime}\\}_{j\in{\mathcal{J}}}$ is an increasing net
of ${\mathbb{Z}}\Gamma$-submodules of ${\mathcal{M}}_{1}$, then
$h_{\Sigma,\omega}({\mathcal{M}}_{j}^{\prime}|{\mathcal{M}}_{2})\nearrow
h_{\Sigma,\omega}(\bigcup_{j\in{\mathcal{J}}}{\mathcal{M}}_{j}^{\prime}|{\mathcal{M}}_{2})$.
Morerover, if $h_{\Sigma,\omega}({\mathcal{M}}_{2})<\infty$, then
$h_{\Sigma,\omega}({\mathcal{M}}_{2}/{\mathcal{M}}_{j}^{\prime})\searrow
h_{\Sigma,\omega}({\mathcal{M}}_{2}/\bigcup_{j\in{\mathcal{J}}}{\mathcal{M}}_{j}^{\prime})$;

  3. (3)

If ${\mathcal{M}}_{1}$ is a finitely generated ${\mathbb{Z}\Gamma}$-modules,
and $\\{{\mathcal{M}}_{j}^{\prime}\\}_{j\in{\mathbb{N}}}$ is an increasing
sequence of ${\mathbb{Z}\Gamma}$-submodules of ${\mathcal{M}}_{2}$ containing
${\mathcal{M}}_{1}$ with union ${\mathcal{M}}_{2}$, then
$h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{j}^{\prime})\searrow
h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2})$.

Using the similar argument in the proof of [12, Proposition 3.5 (2)],
analogues to [6, Lemma 4.1], we have

######  Lemma 3.4.

Let $R$ be a finite abelian group. Then for any ${\mathbb{Z}\Gamma}$-module
${\mathcal{M}}\subseteq(R\Gamma)^{k}$ for some $k\in{\mathbb{N}}$, we have
${\mathcal{M}}=0$ if and only if
$h_{\Sigma,\omega}({\mathcal{M}}|(R\Gamma)^{k})=0$. In fact, for any
$f\in({\mathbb{Z}\Gamma})^{k}$,

$h_{\Sigma,\omega}(R\Gamma f|(R\Gamma)^{k})\geq\frac{\log|\\{rf:r\in
R\\}|}{(2|{\rm supp}(f)|+1)^{2}}.$

######  Proposition 3.5.

Let $R$ be a finite abelian group. Assume for any nonzero $f$ of $R\Gamma$, we
have $h_{\Sigma}(R\Gamma/R\Gamma f)=0$. Then $R\Gamma$ contains no nontrivial
zero divisors.

###### Proof.

By Theorem 3.2 we have $0=h_{\Sigma,\omega}(R\Gamma/R\Gamma
f)=h_{\Sigma,\omega}(R\Gamma)-h_{\Sigma,\omega}(R\Gamma f|R\Gamma)$. Since
$h_{\Sigma,\omega}(R\Gamma f|R\Gamma)\leq h_{\Sigma,\omega}(R\Gamma f)\leq
h_{\Sigma,\omega}(R\Gamma)$, we have $h_{\Sigma,\omega}(R\Gamma
f)=h_{\Sigma,\omega}(R\Gamma)$.

On the other hand, let $R(f):R\Gamma\to R\Gamma$ be the $R\Gamma$-module
homomorphism sending $x$ to $xf$. Then we have $h_{\Sigma,\omega}(R\Gamma
f)=h_{\Sigma,\omega}(R\Gamma)-h_{\Sigma,\omega}({\rm ker\ }R(f)|R\Gamma)$. It
forces that $h_{\Sigma,\omega}({\rm ker\ }R(f)|R\Gamma)=0$. By Lemma 3.4,
${\rm ker\ }R(f)=0$. That means $R\Gamma$ has no nontrivial zero divisor. ∎

We remark the similar approaches also appear in [6, Lemma 4.4], [17, Theorem
7.14], and [12, Theorem 4.5].

######  Definition 3.6.

When $\Gamma$ is amenable, for any ${\mathbb{Z}\Gamma}$-module ${\mathcal{M}}$
such that ${\mathcal{M}}$ is torsion as an abelian group, the algebraic
entropy of ${\mathcal{M}}$ is defined as

$h({\mathcal{M}})=\sup_{{\mathscr{A}}\in{\mathscr{F}}({\mathcal{M}})}\inf_{F\in{\mathcal{F}}(\Gamma)}\frac{\log|\sum_{s\in
F}s^{-1}{\mathscr{A}}|}{|F|}.$

From [12, Theorem 5.1], we know the sofic algebraic entropy is a
generalization of algebraic entropy.

######  Theorem 3.7.

Let $\Gamma$ be an amenable group. For any ${\mathbb{Z}\Gamma}$-modules
${\mathcal{M}}_{1}\subseteq{\mathcal{M}}_{2}$ such that ${\mathcal{M}}_{2}$ is
torsion as abelian group, then
$h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2})=h({\mathcal{M}}_{1})$.

######  Remark 3.8.

Let $R$ be a finite abelian group and $f\in RH\subseteq R\Gamma$ be nonzero
for some amenable subgroup $H$ of $\Gamma$. If $RH$ has no nontrivial zero
divisor, we have $RH$ is isomorphic to $RHf$ as $RH$-modules. By [12,
Corollary 6.2] and Theorem 3.7, we have

$h_{\Sigma,\omega}(R\Gamma/R\Gamma
f)=h_{\Sigma|_{H},\omega}(RH/RHf)=h(RH/RHf)=h(RH)-h(RHf)=0.$

By Bartholdi’s characterization of nonamenability [1, Theorem 1.1], for any
field ${\mathbb{F}}$ there exists an injective ${\mathbb{F}}\Gamma$-module
homomorphism $\varphi:({\mathbb{F}}\Gamma)^{n}\to({\mathbb{F}}\Gamma)^{n-1}$
for some $n\in{\mathbb{N}}$. In particular, when ${\mathbb{F}}$ is finite, one
has $h_{\Sigma,\omega}(\operatorname{im\
}\varphi|({\mathbb{F}}\Gamma)^{n-1})<h_{\Sigma,\omega}(\operatorname{im\
}\varphi)$. Combining Theorem 3.7, we have

######  Corollary 3.9.

$\Gamma$ is amenable if and only if
$h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2})=h_{\Sigma,\omega}({\mathcal{M}}_{1})$
holds for any ${\mathbb{Z}\Gamma}$-modules
${\mathcal{M}}_{1}\subseteq{\mathcal{M}}_{2}$ such that ${\mathcal{M}}_{2}$ is
torsion as an abelian group.

##  4\. Peters’ formula

In this section, we prove Theorem 1.1, which follows from Proposition 4.1 and
Proposition 4.3.

Let ${\mathbb{F}}$ be a finite field. For a finitely generated free
${\mathbb{F}}\Gamma$-module $({\mathbb{F}}\Gamma)^{1\times n}$, if we identify
$\widehat{({\mathbb{F}}\Gamma)^{1\times n}}$ with
$({\mathbb{F}}^{\Gamma})^{1\times n}=({\mathbb{F}}^{1\times n})^{\Gamma}$
naturally via the pairing

$({\mathbb{F}}\Gamma)^{1\times n}\times({\mathbb{F}}^{\Gamma})^{1\times
n}\to{\mathbb{F}},\ (u,x)\mapsto(xu^{*})_{e_{\Gamma}},$

the induced natural action of $\Gamma$ on
$\widehat{({\mathbb{F}}\Gamma)^{1\times n}}$ is then given by left shift. Here
$u^{*}=\sum_{s\in\Gamma}u_{s}s^{-1}$. For a finitely presented
${\mathbb{F}}\Gamma$-module ${\mathcal{M}}$, write it as
${\mathcal{M}}=({\mathbb{F}}\Gamma)^{n}/({\mathbb{F}}\Gamma)^{m}f$ for some
$m,n\in{\mathbb{N}}$ and $f\in M_{m,n}({\mathbb{F}}\Gamma)$, we can further
identify $\widehat{\mathcal{M}}$ as ${\rm ker\ }R(f^{\ast})$ for the linear
map $R(f^{\ast}):({\mathbb{F}}^{\Gamma})^{1\times
n}\to({\mathbb{F}}^{\Gamma})^{1\times m}$ sending $x$ to $xf^{\ast}$.

######  Proposition 4.1.

Let ${\mathbb{F}}$ be a finite field. Then for any finitely generated
${\mathbb{F}}\Gamma$-module ${\mathcal{M}}$, we have
$h_{\Sigma,\omega}(\widehat{\mathcal{M}})=h_{\Sigma,\omega}({\mathcal{M}})$.

###### Proof.

We first show the case that ${\mathcal{M}}$ is finitely presented. Write
${\mathcal{M}}$ as
${\mathcal{M}}=({\mathbb{F}}\Gamma)^{n}/({\mathbb{F}}\Gamma)^{m}f$ for some
$m,n\in{\mathbb{N}}$ and $f\in M_{m,n}({\mathbb{F}}\Gamma)$. Given a map
$\sigma:\Gamma\to{\rm Sym}(d)$ for some $d\in{\mathbb{N}}$, $f$ induces linear
maps $\sigma(f):({\mathbb{F}}^{d})^{n\times 1}\to({\mathbb{F}}^{d})^{m\times
1}$ and $\bar{\sigma}(f):({\mathbb{F}}^{d})^{m\times
1}\to({\mathbb{F}}^{d})^{n\times 1}$ which are uniquely determined by the
linear maps $\sigma_{s}:{\mathbb{F}}^{d}\to{\mathbb{F}}^{d}$ and
$\bar{\sigma}_{s}:{\mathbb{F}}^{d}\to{\mathbb{F}}^{d}$ respectively for every
$s\in\Gamma$. Here $(\sigma_{s}(x))(v)=x((\sigma_{s})^{-1}v)$ and
$(\bar{\sigma}_{s}(x))(v)=x(\sigma_{s}(v))$ (see [12, Section 6] for more
details). Apply [12, Theorem 1.1] and [12, Proposition 6.1] to the length
function $\log|\cdot|$, we have

$\displaystyle h_{\Sigma,\omega}({\mathcal{M}})$
$\displaystyle=h_{\Sigma,\omega}(({\mathbb{F}}\Gamma)^{n})-h_{\Sigma,\omega}(({\mathbb{F}}\Gamma)^{m}f|({\mathbb{F}}\Gamma)^{n})$
$\displaystyle=h_{\Sigma,\omega}(({\mathbb{F}}\Gamma)^{n})-\varlimsup_{i\to\infty}\frac{\log|\operatorname{im\
}\bar{\sigma}_{i}(f)|}{d_{i}}.$

Following the similar argument as in the proof of [12, Lemma 7.4], we know
${\rm ker\ }\sigma(f)$ is isomorphic to the dual vector space of
$({\mathbb{F}}^{d})^{n\times 1}/\bar{\sigma}(f)(({\mathbb{F}}^{d})^{m\times
1})$. Hence,

$h_{\Sigma,\omega}({\mathcal{M}})=\varlimsup_{i\to\infty}\frac{\log|{\rm ker\
}\sigma_{i}(f)|}{d_{i}}.$

From [7, Lemma 4.8], we have

$h_{\Sigma}(\widehat{\mathcal{M}})=h_{\Sigma}({\rm ker\
}R(f^{\ast}))=\varlimsup_{i\to\infty}\frac{\log|{\rm ker\
}\bar{\sigma}_{i}(f^{\ast})|}{d_{i}}.$

For a given map $\sigma:\Gamma\to{\rm Sym}(d)$ for some $d\in{\mathbb{N}}$,
put ${\mathcal{W}}=\\{v\in[d]:\sigma(s^{-1})(v)=\sigma(s)^{-1}(v),{\rm for\
all\ }s\in{\rm supp}(f)\\}$. Then it is easy to verify that

${\rm ker\ }\sigma(f)\subseteq{\rm ker\
}(P_{{\mathcal{W}}}\circ\sigma(f))={\rm ker\
}(P_{{\mathcal{W}}}\circ\bar{\sigma}(f^{\ast}))\supseteq{\rm ker\
}\bar{\sigma}(f^{\ast}),$

where $P_{{\mathcal{W}}}$ is the map projecting onto the ${\mathcal{W}}$
coordinates. It follows that

$\log|{\rm ker\ }\sigma(f)|\leq\log|{\rm ker\
}(P_{{\mathcal{W}}}\circ\bar{\sigma}(f^{\ast}))|\leq
n|{\mathcal{W}}^{c}|\log|{\mathbb{F}}|+\log|{\rm ker\
}\bar{\sigma}(f^{\ast})|$

and

$\log|{\rm ker\ }\bar{\sigma}(f^{\ast})|\leq\log|{\rm ker\
}(P_{{\mathcal{W}}}\circ\sigma(f))|\leq
n|{\mathcal{W}}^{c}|\log|{\mathbb{F}}|+\log|{\rm ker\ }\sigma(f)|.$

By soficity, $|{\mathcal{W}}^{c}|/d$ approaches to zero as $d$ gets large, it
follows that

$h_{\Sigma,\omega}(\widehat{\mathcal{M}})=h_{\Sigma,\omega}({\mathcal{M}})=\varlimsup_{i\to\infty}\frac{\log|{\rm
ker\ }\sigma_{i}(f)|}{d_{i}}.$

Now for any finitely generated ${\mathbb{F}}\Gamma$-module ${\mathcal{M}}$,
write ${\mathcal{M}}$ as
${\mathcal{M}}=({\mathbb{F}}\Gamma)^{n}/\cup_{j\in{\mathbb{N}}}{\mathcal{N}}_{j}$
for some increasing sequence of finitely generated submodules
$\\{{\mathcal{N}}_{j}\\}_{j\in{\mathbb{N}}}$ of $({\mathbb{F}}\Gamma)^{n}$. By
[7, Corollary 3.6] and [12, Proposition 3.4(1)], we have

$h_{\Sigma,\omega}(\widehat{\mathcal{M}})==\inf_{j}h_{\Sigma,\omega}(\widehat{({\mathbb{F}}\Gamma)^{n}/{\mathcal{N}}_{j}})=\inf_{j}h_{\Sigma,\omega}(({\mathbb{F}}\Gamma)^{n}/{\mathcal{N}}_{j})=h_{\Sigma,\omega}({\mathcal{M}}).$

∎

Following the similar argument as in the proof of [12, Lemma 10.4], we also
have an inequality for topological entropy.

######  Lemma 4.2.

Let $X$ be a compact metrizable abelian group carrying an action of $\Gamma$
by continuous automorphisms. Then for any closed $\Gamma$-invariant subgroup
$Y$ of $X$, the induced quotient map $\pi:X\to X/Y$ satisfies

$h_{\Sigma,\omega}(X)\geq h_{\Sigma,\omega}(X/Y|X)+h_{\Sigma,\omega}(Y).$

######  Proposition 4.3.

Let ${\mathbb{F}}$ be a finite field. Then for any countable
${\mathbb{F}}\Gamma$-modules ${\mathcal{M}}_{1}\subseteq{\mathcal{M}}_{2}$, we
have
$h_{\Sigma,\omega}(\widehat{{\mathcal{M}}_{1}}|\widehat{{\mathcal{M}}_{2}})\leq
h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2})$.

###### Proof.

Case 1. ${\mathcal{M}}_{1}\subseteq{\mathcal{M}}_{2}$ are finitely generated.

Combining Lemma 4.2, Proposition 4.1, and Theorem 3.2, we have

$h_{\Sigma}(\widehat{{\mathcal{M}}_{1}}|\widehat{{\mathcal{M}}_{2}})\leq
h_{\Sigma,\omega}(\widehat{{\mathcal{M}}_{2}})-h_{\Sigma,\omega}(\widehat{{\mathcal{M}}_{2}/{\mathcal{M}}_{1}})=h_{\Sigma,\omega}({\mathcal{M}}_{2})-h_{\Sigma,\omega}({\mathcal{M}}_{2}/{\mathcal{M}}_{1})=h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2}).$

Case 2. ${\mathcal{M}}_{1}$ is finitely generated.

Write ${\mathcal{M}}_{2}$ as the union of an increasing sequence of finitely
generated ${\mathbb{F}}\Gamma$-modules
$\\{{\mathcal{N}}_{j}\\}_{j\in{\mathbb{N}}}$ of ${\mathcal{M}}_{2}$ containing
${\mathcal{M}}_{1}$. It is clear that by definition, for all
$j\in{\mathbb{N}}$, one has

$h_{\Sigma,\omega}(\widehat{{\mathcal{M}}_{1}}|\widehat{{\mathcal{M}}_{2}})\leq
h_{\Sigma,\omega}(\widehat{{\mathcal{M}}_{1}}|\widehat{{\mathcal{N}}_{j}}).$

Thus by Proposition 3.3 and Case 1, we have

$h_{\Sigma,\omega}(\widehat{{\mathcal{M}}_{1}}|\widehat{{\mathcal{M}}_{2}})\leq\lim_{j\to\infty}h_{\Sigma,\omega}(\widehat{{\mathcal{M}}_{1}}|\widehat{{\mathcal{N}}_{j}})\leq\lim_{j\to\infty}h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{N}}_{j})=h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2}).$

For general case, write ${\mathcal{M}}_{1}$ as the union of an increasing
sequence of finitely generated submodules
$\\{{\mathcal{N}}_{j}\\}_{j\in{\mathbb{N}}}$ of ${\mathcal{M}}_{1}$. Following
the similar argument as in the proof of [12, Lemma 9.10], we have

$h_{\Sigma,\omega}(\widehat{{\mathcal{M}}_{1}}|\widehat{{\mathcal{M}}_{2}})=\lim_{j\to\infty}h_{\Sigma,\omega}(\widehat{{\mathcal{N}}_{j}}|\widehat{{\mathcal{M}}_{2}}).$

Thus by Proposition 3.3 and Case 2, we have

$h_{\Sigma,\omega}(\widehat{{\mathcal{M}}_{1}}|\widehat{{\mathcal{M}}_{2}})\leq\lim_{j\to\infty}h_{\Sigma,\omega}({\mathcal{N}}_{j}|{\mathcal{M}}_{2})=h_{\Sigma,\omega}({\mathcal{M}}_{1}|{\mathcal{M}}_{2}).$

∎

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