# An Experimentally Validated Feasible Quantum Protocol for Identity-Based
Signature with Application to Secure Email Communication

Tapaswini Mohanty  Department of Mathematics, National Institute of Technology
Jamshedpur, Jamshedpur-831014, India<EMAIL_ADDRESS>Vikas Srivastava
Department of Mathematics, National Institute of Technology Jamshedpur,
Jamshedpur-831014, India<EMAIL_ADDRESS>Sumit Kumar Debnath
Department of Mathematics, National Institute of Technology Jamshedpur,
Jamshedpur-831014, India<EMAIL_ADDRESS>Debasish Roy  Department of
Mathematics, IIT Kharagpur, India<EMAIL_ADDRESS>Kouichi
Sakurai  Faculty of Information Science and Electrical Engineering, Kyushu
University, Fukuoka- 8190395, Japan<EMAIL_ADDRESS>Sourav
Mukhopadhyay  Department of Mathematics, IIT Kharagpur, India.
<EMAIL_ADDRESS>
###### Abstract

Digital signatures are one of the simplest cryptographic building blocks that
provide appealing security characteristics such as authenticity,
unforgeability, and undeniability. In 1984, Shamir developed the first
Identity-based signature (IBS) to simplify public key infrastructure and
circumvent the need for certificates. It makes the process uncomplicated by
enabling users to verify digital signatures using only the identifiers of
signers, such as email, phone number, etc. Nearly all existing IBS protocols
rely on several theoretical assumption-based hard problems. Unfortunately,
these hard problems are unsafe and pose a hazard in the quantum realm. Thus,
designing IBS algorithms that can withstand quantum attacks and ensure long-
term security is an important direction for future research. Quantum
cryptography (QC) is one such approach. In this paper, we propose an IBS based
on QC. Our scheme’s security is based on the laws of quantum mechanics. It
thereby achieves long-term security and provides resistance against quantum
attacks. We verify the proposed design’s correctness and feasibility by
simulating it in a prototype quantum device and the IBM Qiskit quantum
simulator. The implementation code in qiskit with Jupyternotebook is provided
in the Annexure. Moreover, we discuss the application of our design in secure
email communication.

quantum communication; quantum computation; identity based signature; long-
term security

##  1 Introduction

The digital signature is the most commonly utilized cryptographic building
block in modern communication networks. The signer generates their public and
secret keys in a digital signature protocol. They are distinguished from each
other by their public key. In a practical scenario, the signer can be
distinguished by their names or email addresses instead of randomly generated
keys. Public critical infrastructure (PKI) was intended to create a one-to-one
correspondence between public keys and a signer’s identity. A certificate
authority may be necessitated to connect public keys with identities with the
help of a digital certificate. However, this method of assigning public keys
with the user’s identifier has many loopholes. It lacks effectiveness, and it
is not resilient. A more efficient substitute framework is needed to make the
PKI simpler. Shamir created the identity-based signature technique (IBS), in
which a verifier uses public knowledge about the signer’s identity to confirm
the signature’s authenticity [12]. This approach overcomes the requirement of
digital certificates. In an IBS, a trustworthy negotiant or a secret key
establisher (SKG) uses the identification to generate the signer’s secret key
from a master secret key exclusively known to SKG. Many IBS schemes have been
proposed [16, 18, 9, 27, 5, 15, 19, 17, 8, 11, 4] in the area of classical
cryptography whose security relies on the number theoretic problems [10, 7, 6]
like factorization problem and discrete logarithm problem. However, these
schemes won’t be safe in the future as Shor’s algorithm [14] can solve some of
these hard problems in polynomial time with the help of an efficient quantum
computer.

There is a significant problem in preserving vulnerable information or
documents like health records and government documents for a long time. These
classically encrypted sensitive documents can be lost during transmission in a
public channel. A corrupted party may intercept this sensitive data from the
communication channel. This theft of classically encrypted information poses a
lot of risk. In the future, when large-scale quantum computers are available,
he may try to extract information from the encrypted data. Thus, we need an
appropriate solution to secure the confidentiality and integrity of user-
sensitive data. While post-quantum cryptography can withstand existing
classical and quantum algorithms, it is not a long-term security solution.
Introducing new effective or advanced classical or quantum algorithms may
break the post-quantum algorithms in the future, making them obsolete. Quantum
cryptography (QC), based on the Heisenberg uncertainty principle and the no-
cloning theorem, can solve the aforementioned security risks compared to
conventional and post-quantum cryptography. Additionally, because quantum
protocols are information-theoretically secure and computationally
unattainable, they offer security against quantum computers. Therefore, it is
necessary to use QC to construct a digital signature protocol, specifically
IBS protocols.
This paper is sketched as follows: the following section elaborates on our
contribution. Section 3 provides a preliminary background, followed by our
proposed design of QIBS. Section 5 provides the correctness of our scheme,
with Section 6 providing the security analysis and Sections 7 and 8 providing
efficiency and performance analysis. In section 9, we provided a toy example
implemented in real hardware, the code of which is given in Annexure.

##  2 Our Contribution

The conventionally used signatures employed a public key infrastructure model
that involved the management of digital certificates. Certificate management
is a computationally intensive task that makes PKI-based signatures unsuitable
for practical purposes. To tackle this problem, an identity-based signature
scheme (IBS) is a workable option. However, nearly all existing IBS protocols
rely on several theoretical assumption-based hard problems. Unfortunately,
these difficult challenges are insecure and pose a risk in the quantum realm.
Given the current circumstances, creating a quantum-secure IBS is a good idea.
Numerous quantum signature scheme designs are state-of-the-art at the moment
[26, 1, 25, 24]. However, in the context of QC, there are only a few
constructions of IBS [3, 21, 23, 22]. The security of [3] is also dependent on
choosing a one-way function as a random one-way permutation oracle. Herein, we
present the design of a quantum IBS. We utilize the quantum analogue of an OTP
(one-time pad) to design an information-theoretically secure quantum IBS. Our
proposal consists of three phases: (i) Initializing, (ii) Signing, and (iii)
Verification. Using a quantum key distribution protocol, the secret key
generator (SKG) provides the signer with a secret key corresponding to their
identity during the initializing phase. In addition, the verifier and SKG
share a secret key. In the Signing phase, the signer signs with his secret
key. Afterwards, during the verification stage, the verifier can confirm the
signature with SKG’s help. If the implicit encryption is theoretically secure,
the designed signature scheme quantum IBS will satisfy unforgeability and
undeniability. The unforgeability property ensures that no other party can do
the signature on behalf of a signer. In contrast, undeniability ensures that a
signer can not deny the signature generation if she did the signature. We
assert the suggested strategy obtains long-lived security and stays safe from
quantum attacks. The communication and computation cost of our intended design
are only $10m+2n$ qubits and $(23m+3n)\delta+(3m+n)\beta$ respectively for the
message of size $m$ qubits, where $\delta$, $\beta$ and $n$ represent the
costs of one straight forward measurement, cost of conversion of a classical
bit to a qubit, and bit length of $\phi$ respectively.

Chen et al. [3] suggested the first quantum IBS using the quantum one-way
function, encryption based on identity, and classical security token. However,
there is a major drawback in the scheme of Chen et al. [3]. The design in [3]
involves long-term quantum memory and multiple rounds of quantum swap tests,
making it inefficient. Furthermore, unlike our approach, the scheme of [3] is
not impervious to SKG’s forgery attack. Our proposed design allows anyone to
verify the signature, unlike [3]. In contrast to ideas [3, 21, 23, 22], the
suggested approach doesn’t use intricate oracle operators. Our scheme achieves
less computation cost than the existing quantum IBS schemes [3, 21]. The
proposed design performs better over [21, 22] from the communication point of
view. Since our approach relies only on single photon quantum resources and
uses straightforward measurement operators, it is more practical and viable
than the schemes of [23, 22], which uses Bell states and entangled quantum
resources. Our design does not use any Oracle one-way function compared to the
schemes of [3, 21, 23, 22]. Moreover, the suggested scheme supports signature
over classical and quantum messages. While the works of [3, 21, 23, 22]
support only signature over classical message. A summary of comparable schemes
to our suggested design with the existing quantum IBS [3, 21, 23, 22] is
tabulated in Table 1. We simulated and implemented our scheme in the IBM
Qiskit quantum simulator to test the correctness and feasibility of the
designed quantum IBS. We implemented our design on a real quantum machine to
further validate it. The detailed performance analysis can be found in section
8. We also investigated that how our proposed scheme can be employed as
building block in secure email communication.

##  3 Preliminaries

###  3.1 Quantum One-Time Pad [2]

This part explains the quantum one-time pad (OTP) [2] procedure between John,
the encryptor, and Bob, the decryptor. There are three algorithms in it: ${\sf
KeyGen}$, ${\sf Enc}$ and ${\sf Dec}$ which are described below:
In the first step, John and Bob share two randomly chosen secret bits for
every qubit. We presume that the sharing of these bits has been done in
advance. If the first bit is $0$, then John does nothing. Otherwise, he
applies $\sigma_{z}$ to the qubit. Bob doesn’t do anything if the second bit
turns out to be $0$, but if the second bit is $1$, then John operates
$\sigma_{x}$. Now John sends the resulting qubit to Bob. The protocol is
continued for the rest of the bits. Note that the quantum OTP [2] is
information-theoretically secure.

$K\leftarrow$ ${\sf KeyGen}$($n$).

On input a positive integer $n$ (length of message), ${\sf KeyGen}$ outputs a
$\kappa(\geq 2n)$ bit key $K$ shared between John and Bob using some quantum
key distribution protocol.

$E_{K}(|P\rangle)\leftarrow$ ${\sf Enc}$($|P\rangle,K$).

On input the quantum message $|P\rangle=\otimes_{i=1}^{n}|p_{i}\rangle$ with
$|p_{i}\rangle=a_{i}|0\rangle+b_{i}|1\rangle$ and $|a_{i}|^{2}+|b_{i}|^{2}=1$,
and the key $K$, John encrypts the message $|P\rangle$ in the following way:

  1. 1.

Computes $\otimes_{i=1}^{n}Z^{K_{2i-1}}|p_{i}\rangle$ i.e., if the $(2i-1)$-th
bit of $K$ is $1$ then operates the unitary operator $Z$ on the $i$-th qubit
of the message $|P\rangle$; otherwise does nothing, for $i=1,\ldots,n$.

  2. 2.

Executes
$E_{K}(|P\rangle)=\otimes_{i=1}^{n}X^{K_{2i}}Z^{K_{2i-1}}|p_{i}\rangle$ by
operating $X$ on the $i$-th qubit of
$\otimes_{i=1}^{n}Z^{K_{2i-1}}|p_{i}\rangle$ if the $(2i)$-th bit of $K$ is
$1$ for $i=1,\ldots,n$.

$|P\rangle\leftarrow$ ${\sf Dec}$($E_{K}(|P\rangle)$).

Bob decrypts $E_{K}(|P\rangle)$ by performing the following steps:

  1. 1.

To get $\otimes_{i=1}^{n}Z^{K_{2i-1}}|p_{i}\rangle$, operates the unitary
operator $X$ on the $i$-th qubit of
$E_{K}(|P\rangle)=\otimes_{i=1}^{n}X^{K_{2i}}Z^{K_{2i-1}}|p_{i}\rangle,$ if
the $(2i)$-th bit of $K$ is $1$, else does nothing, for $i=1,\ldots,n$.

  2. 2.

Computes $|P\rangle$ by operating the unitary operator $Z$ on the $i$-th qubit
of the $\otimes_{i=1}^{n}Z^{K_{2i-1}}|p_{i}\rangle$ if the $(2i-1)$-th bit of
$K$ is $1$, else does nothing.

Here $X$ and $Z$ are the unitary operators with matrix representation
$\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ and $\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$ respectively. Note that the quantum OTP is information-
theoretically secure [2].

###  3.2 U-Gate

The most versatile single-qubit quantum gate is the $U$ gate. It is a
parameterized gate with matrix representation given by:

$U(\theta,\phi,\lambda)=\begin{pmatrix}\cos(\frac{\theta}{2})&-e^{i\lambda}\sin(\frac{\theta}{2})\\\
e^{i\phi}\sin(\frac{\theta}{2})&e^{i(\phi+\lambda)}\cos(\frac{\theta}{2})\end{pmatrix}$

We take the special case where $\theta=\frac{\pi}{2}$ and $\lambda=0$. Thus,
we have $U(\frac{\pi}{2},\phi,0)=\frac{1}{\sqrt{2}}\begin{pmatrix}1&-1\\\
e^{i\phi}&e^{i\phi}\end{pmatrix}$.

##  4 Proposed Quantum Identity-Based Signature

This section outlines our suggested quantum IBS scheme.

A high-level overview: Our scheme consists of three steps: (i) Initializing
phase, (ii) Signing phase, and (iii) Verification phase. Let $Alice_{i}$ be a
signer with identity $ID_{i}$, and Bob be a verifier. In the Initializing
phase, each signer and a verifier share respective secret keys with the SKG
with the help of some quantum key distribution (QKD). During the Signing
phase, to sign a quantum message, a signer employs the quantum OTP (mentioned
in Section 3.1) as a building block. On receiving a pair of message signatures
from the signer, the verifier, Bob, communicates with SKG to verify the
cogency of the message signature couplet during the verification phase.

Initializing Phase: Let $ID_{i}=(id_{i}^{1},\dots,id_{i}^{m})\in\\{0,1\\}^{m}$
be the identity of an user $Alice_{i}$.

  1. 1.

Firstly, making use of a quantum key distribution (QKD) protocol SKG shares a
secret key $T_{i}$ (of size $\geq 2m$) with $Alice_{i}$ having identity
$ID_{i}$ for the quantum OTP mentioned in Section 3.1.

  2. 2.

In a similar manner, Bob shares a secret key $T_{u}$ (of size $\geq 2m$) with
SKG for the quantum OTP mentioned in Section 3.1.

  3. 3.

Using $T_{i}$, Alice shares a secret value $\phi\in(0,2\pi)$ with SKG
utilizing quantum authentication process [20].

See Figure 1 for a communication flow in the Initializing phase.

SKG($\displaystyle Alice_{i})$SignerVerifier (Bob)Shares a secret key
$\displaystyle T_{i}$ and $\displaystyle\phi$Shares a secret key
$\displaystyle T_{u}$

Figure 1: Communication flow in Initializing phase

Signing Phase: Let $|P\rangle=\otimes_{j=1}^{m}|p_{j}\rangle$ be the quantum
message of $m$-qubits that needs to be signed, where
$|p_{j}\rangle=a_{j}|0\rangle+b_{j}|1\rangle$ and $|a_{j}|^{2}+|b_{j}|^{2}=1$.
$Alice_{i}$ prepares two copies of $|P\rangle$. Given the quantum message
$|P\rangle$, the signer $Alice_{i}$ generates the signature by performing the
following steps:

  1. 1.

Operates $U^{\otimes m}$ (where $U=U(\pi/2,\phi,0)$) on the quantum message
state $|P\rangle$ to get $U^{\otimes m}(|P\rangle)$.

  2. 2.

Applies the quantum OTP (as mentioned in Section 3.1) on the $U^{\otimes
m}(|P\rangle)$ to get $|S\rangle=E_{T_{i}}(U^{\otimes m}(|P\rangle))$.

  3. 3.

Encodes $ID_{i}=(id_{i}^{1},\dots,id_{i}^{m})$ into
$|ID_{i}\rangle=\otimes_{j=1}^{m}|id_{i}^{j}\rangle$.

Finally, $Alice_{i}$ outputs the tuple $(|P\rangle,|S\rangle,|ID_{i}\rangle)$
as message-signature pair. The stages of the Signing phase are depicted in
Figure 2.

quantum OTPQuantum Message State $|\displaystyle
P\rangle$$\displaystyle|S\rangle=E_{T_{i}}(U^{\otimes m}(|P\rangle))$Step
IIdentity $\displaystyle ID_{i}=(id_{i}^{1},\dotsc,\ id_{i}^{m}$)Step
II$|\displaystyle
ID_{i}$$\displaystyle\rangle=\otimes_{j=1}^{m}$$\displaystyle|id_{i}^{j}$$\displaystyle\rangle$

Figure 2: Stages of Signing phase

Verification Phase: On receiving a message-signature pair
$(|P\rangle,|S\rangle,|ID_{i}\rangle)$ from $Alice_{i}$, Bob communicates with
SKG to examine the credibility of the message-signature couplet as in the next
paragraph:

  1. 1.

Bob keeps $|P\rangle$ and encrypts $(|S\rangle,|ID_{i}\rangle)$ by using the
quantum OTP (as mentioned in Section 3.1) with the help of the secret key
$T_{u}$ to get $E_{T_{u}}(|S\rangle,|ID_{i}\rangle)$. Finally, he sends
$E_{T_{u}}(|S\rangle,|ID_{i}\rangle)$ to SKG.

  2. 2.

SKG first decrypts $E_{T_{u}}(|S\rangle,|ID_{i}\rangle)$ using the secret key
$T_{u}$ to get back $(|S\rangle,|ID_{i}\rangle)$. He then retrieves
$|P\rangle$ in two steps. In the first step, SKG performs the decryption
operation on $|S\rangle$ with the help of the secret key $T_{i}$, which is
associated with the user $Alice_{i}$ with identity $|ID_{i}\rangle$.
Consequently, SKG gets $U^{\otimes m}(|P\rangle)$. Note that SKG knows secret
value $\phi$. In the second step, SKG operates $(U^{\dagger})^{\otimes m}$ on
$U^{\otimes m}(|P\rangle)$ to retrieve $|P\rangle$. In the following, SKG
encrypts $|P\rangle$ using $T_{u}$ and sends $|R\rangle=E_{T_{u}}(|P\rangle)$
to Bob.

  3. 3.

On receiving $|R\rangle$ from SKG, Bob decrypts it using the secret key
$T_{u}$. Then he verifies whether $D_{T_{u}}(|R\rangle)$ is same as the
$|P\rangle$ of the signature tuple $(|P\rangle,|S\rangle,|ID_{i}\rangle)$ . If
so, then Bob accepts the signature as valid; otherwise, he rejects the
signature.

We refer to Figure 3 for the communication flow in the Verification phase.

Verifier (Bob)SKGKeeps $\displaystyle|P\rangle$ and encrypts
$\displaystyle(|S\rangle,\ |ID_{i}\rangle)$using quantum OTP with the help of
thesecret key $\displaystyle T_{u}$ to get $\displaystyle
E_{T_{u}}$($\displaystyle|S\rangle$,
$\displaystyle|ID_{i}$$\displaystyle\rangle$$\displaystyle)$$\displaystyle
E_{T_{u}}$($\displaystyle|S\rangle$,
$\displaystyle|ID_{i}$$\displaystyle\rangle$$\displaystyle)$1\. Decrypts
$\displaystyle E_{T_{u}}$($\displaystyle|S\rangle$,
$\displaystyle|ID_{i}$$\displaystyle\rangle$$\displaystyle)$ using
$\displaystyle T_{u}$2\. Retrives $\displaystyle|P\rangle\ $by performing
decryption operation on $\displaystyle|S\rangle$ using $\displaystyle T_{i}$
and $\phi$3\. Encrypts $\displaystyle|P\rangle$ using $\displaystyle T_{u}$ to
get $\displaystyle|R\rangle=\ E_{T_{u}}(|P\rangle)$1\. Decrypts
$\displaystyle|R\rangle$ using $\displaystyle T_{u}$2\. Verifies whether
$\displaystyle D_{T_{u}}(|R\rangle)$ is same as
$\displaystyle|P\rangle$$\displaystyle|R\rangle$

Figure 3: Communication flow in Verification phase

##  5 Correctness

We can easily check the correctness of our scheme in the following way:
firstly, SKG can obtain the identity of the signer on receiving
$E_{T_{u}}(|S\rangle,|ID_{i}\rangle)$ from Bob and using $T_{u}$ he can
decrypt $E_{T_{u}}(|S\rangle,|ID_{i}\rangle)$ to get
$(|S\rangle,|ID_{i}\rangle)$. Now corresponding to the identity
$|ID_{i}\rangle$, SKG will consider $T_{i}$ and $\phi$ for decrypting
$|S\rangle$ to get $|P\rangle$. In the following, SKG encrypts $|P\rangle$
using $T_{u}$ and sends $|R\rangle=E_{T_{u}}(|P\rangle)$ to Bob. Then Bob
verifies whether $|P\rangle=D_{T_{u}}(|R\rangle)$ to judge validity of
signature. From the above analysis, it is easy to see that the correctness
holds in the proposed scheme.

##  6 Security Analysis

######  Theorem 1.

If the underlying quantum OTP is information-theoretically secure, the
proposed signature scheme quantum IBS satisfies unforgeability and
undeniability.

Unforgeability

This property ensures that no other party can do the signature on behalf of a
user, say $Alice_{i}$. Suppose one user $Alice_{j}$ is different from
$Alice_{i}$ and wants to forge a signature on behalf of the signer
$Alice_{i}$. $Alice_{i}$’s identity is $|ID_{i}\rangle$ and $Alice_{j}$’s
identity is $|ID_{j}\rangle$. $Alice_{j}$ is having $T_{j}$ as the shared key
and $\phi^{{}^{\prime}}$ as the shared secret value between him and SKG. He
has to use $T_{j}$ and $\phi^{{}^{\prime}}$ to encrypt $|P\rangle$ which
yields $|S^{\prime}\rangle=E_{T_{j}}((U(\pi/2,\phi^{{}^{\prime}},0))^{\otimes
m}|P\rangle)$. In the following, $Alice_{j}$ may send
$(|P\rangle,|S^{\prime}\rangle,|ID_{i}\rangle)$ to Bob who in turn sends
$E_{T_{u}}(|S^{\prime}\rangle,|ID_{i}\rangle)$ to SKG. As $T_{u}$ is known to
SKG, he will decrypt $E_{T_{u}}(|S^{\prime}\rangle,|ID_{i}\rangle)$. Since the
decryption yields $(|S^{\prime}\rangle,|ID_{i}\rangle)$, the SKG uses $T_{i}$
associated to $|ID_{i}\rangle$ and $\phi^{{}^{\prime}}$ for decrypting
$|S^{\prime}\rangle$. Now the decryption of $|S^{\prime}\rangle$ using $T_{i}$
and $\phi$ will produce a value $|P^{\prime}\rangle$ which is not equal to
$|P\rangle$ since $|S^{\prime}\rangle$ was encrypted using $T_{j}$ and
$\phi^{{}^{\prime}}$. Thus, no one can sign on behalf of $Alice_{i}$, i.e.,
the proposed scheme achieves unforgeability property.

Undeniability

This property ensures that a user, say $Alice_{i}$, can not deny the signature
generation if she did the signature. After the signing phase, Bob receives
$(|P\rangle,|S\rangle,|ID_{i}\rangle)$ as message-signature pair from
$Alice_{i}$. In the following, Bob shares
$E_{T_{u}}(|S\rangle,|ID_{i}\rangle)$ with SKG who in turn decrypts it and
retrieves $(|S\rangle,|ID_{i}\rangle)$. The SKG then uses $T_{i}$
corresponding to $|ID_{i}\rangle$ of $Alice_{i}$ to decrypt $|S\rangle$. This
yields $|P\rangle$, which is sent to Bob for verification. Thus, $Alice_{i}$
can not deny the generation of the signature if she does this.

######  Theorem 2.

The proposed quantum IBS is secure against communicative Pauli operator attack
[28].

###### Proof.

Suppose an attacker wants to forge the signature pair
$(|P\rangle,|S\rangle,|ID_{i}\rangle)$ using the Pauli operator, then he
applies a non-trivial Pauli operator $V$ (says), then the signature changes to
$(V|P\rangle,V|S\rangle,|ID_{i}\rangle)$. But

$V|S\rangle\equiv E_{T_{u}}VU(|P\rangle)\\\ \neq E_{T_{u}}U(V(|P\rangle))$

. Thus, this signature pair is invalid. Therefore, the attacker fails to forge
$(|P\rangle,|S\rangle,|ID_{i}\rangle)$ using the Pauli operator. ∎

##  7 Efficiency Analysis

We now present the efficaciousness of our proposed blueprint. We first discuss
the cost of communication and calculation.

Communication cost: $4m$ qubits must be transmitted among the signer, verifier
and secret key generator during the Initializing phase. During the Signing and
Verification phases, $6m$ more qubits are needed to communicate. If $\phi$ is
substituted by a bit string of length $n$, then $2n$ qubits are communicated
in the quantum authentication process. So, our proposed design’s total quantum
communication cost is $10m+2n$ qubits.

Computation cost: The entire computational expense of our suggested layout is
$(23m+3n)\delta+(3m+n)\beta$ for the message of size $m$ qubits, where
$\delta$ and $\beta$ represent the expenses of converting a conventional bit
to a qubit and of making a single, basic measurement, respectively. In
particular, $4m+3n$ simple measurements must be performed during the
initialising phase. Hence, the cost accrued during the Initializing phase is
$(4m+3n)\delta$. In addition, the total cost incurred during encryption and
decryption is $19m\delta$. Furthermore, the cost to encode classical bits into
quantum bits is $(3m+n)\beta$.

We now present the comparative analysis of our proposed design with existing
quantum IBS [3, 21, 23, 22] in Table 1.

Table 1:  Comparison with existing quantum IBS

|  Signature Space |  | Message
---
space
Key Space |  Dimension of Hilbert Space |  Oracle Used |  Quantum Communication Cost |  Quantum Computation Cost |  Classical Computation Cost |  Using Bell States
Chen et al. [3] |  Q | C |  C |  2 |  Yes |  $5m+L_{2}uw$ qubits |  $14m\delta+uw\gamma_{q}+w\gamma_{q}+uw\delta+w\delta$ |  $m\gamma+3mx_{2}$ |  No
Xin et al. [21] |  Q | C |  C |  2 |  Yes |  $11m+8l$ qubits |  $21m\delta+3m\beta+8l(\beta+\delta)$ |  $11mx_{3}+2m\gamma+27mx_{2}$ |  No
Xin et al. [23] |  Q | C |  C |  2 |  Yes |  $8m$ qubits |  $10m\delta+5m\beta$ |  $4m\gamma+4mx_{1}+18mx_{2}$ |  Yes
Xin et al. [22] |  Q | C |  C |  2 |  Yes |  $8m+l$ qubits |  $15m\delta+(2m+l)\beta+l\delta$ |  $12m\gamma+30mx_{2}$ |  Yes
Ours |  Q | Q, C |  C |  2 |  No |  $10m+2n$ qubits |  $(23m+3n)\delta+(3m+n)\beta$ |  0 |  No

Q=quantum, C = classical, $l>{}>m$ where $l$ is the total number of decoy
particles, $u$=atmost number of signature recipient, $w$=A few safety metric
acceptance thresholds utilized in the plan of [3], $L_{2}=$ size of the
quantum digest’s total qubits, $\delta$ is the computation cost for one basic
measurement, $\gamma_{q}$ is the price of preparing one quantum digital
digest, $\beta$ is the price of conversion from a single classical bit into
qubit, $x_{1}$ is computation cost for one way hash function computation,
$\gamma$ is the cost of computing one-way function, $x_{2}$ is computation
cost required for one XOR operation, $x_{3}$ cost incurred during one
evaluation of permutation function

From the Table 1, we can see that unlike [3, 21, 23, 22], our proposed design
does not use any oracle one-way functions. The scheme of [3] is inefficient as
it employs long-term quantum memory and multiple rounds of quantum swap tests.
We note that our scheme’s computational expense is lower than that of [3, 21].
Moreover, our proposed scheme is more communicationally efficient than [21,
22]. The works of [23, 22] require "entangled states" as quantum facilities
and use Bell states. Nevertheless, our plan solely uses single-photon quantum
resources, making it more feasible and practical than [23, 22]. Since the
existing technology is not feasible for preparing these resources. Moreover,
[23, 22] is less efficient as it uses Bell states and other complicated oracle
operators. Our scheme attains a lower quantum communication cost than the
works of [21, 22, 3]. The messages to be signed in the design of [23, 3, 21,
22] must be a classical message. The scheme of [23, 3, 21, 22] can’t be used
to sign a quantum message because the bits of classical message are used in
the signature generation phase and at some places in the verification phase as
well. The fact that [23, 3, 21, 22] techniques are limited to signing
classical communications is a drawback. Since our proposed design has no such
shortcoming, conventional and quantum messages can be signed. The key space of
[23, 3, 21, 22] is classical as ours. During our scheme’s Signing step, the
signer uses the conventional secret key $T_{i}$ to sign the message. In the
Verification phase, the verifier Bob uses the classical secret key $T_{u}$,
and SKG uses both $T_{u}$ and $T_{i}$ to verify the message. In [23, 3, 21,
22], the signer’s secret and public keys are classical bits. Therefore, our
proposed design has the upper hand over all other quantum IBS schemes.

##  8 Performance Analysis

We simulated and implemented the proposed quantum IBS protocol on a quantum
simulator that runs locally on a classical computer. In the next step, we also
tested our design by implementing and running it on an actual back end.

###  8.1 Implementation on a Quantum Machine

We used the quantum simulator provided by the IBM Qiskit, an open-source
software development kit, to test the correctness and feasibility of the
quantum IBS. The hardware and software specifications are listed in Table 2.

Table 2: Software and hardware specification for IBM Qiskit quantum simulation Qiskit | v0.20.2
---|---
Qiskit Element | Qiskit Aer
Simulator | QASM
Python | v3.8.5
Local OS | Linux Lite v5.2
Local Hardware | Intel Core i5, RAM 8GB

We verified the correctness of our scheme by taking a particular instance. We
take $ID_{A}={011}$ as the signer’s identity. Let $\ket{P}=\ket{010}$
represent the quantum message that requires a signature, $T_{i}={010110}$
represent the common secret code that links the SKG and the signer, and let
$T_{U}={100101}$ represents the common hidden key between the verifier and
SKG. Also, we take $U=U(\pi/2,\pi,0)$. Figure 4 displays this instance’s
elaborative circuitry. 4.

Figure 4: Quantum circuits of an instance of the proposed quantum IBS

We also tested the success probability by running an instance of quantum IBS
for $10000$ times. The results of this simulation experiment are depicted in
Figure 5. To conclude, our simulation experiments on the quantum simulator
verify the correctness and feasibility of the designed quantum IBS scheme.

Figure 5: Success probability of an instance of the proposed quantum IBS

###  8.2 Implementation on a Real Quantum Machine

We also implemented the design of quantum IBS put forward in this paper on a
real quantum machine provided by IBM Quantum Experience. The hardware and
software specifications used for performing simulation experiments are given
in Table 3. We verified the correctness of our proposed protocol by running
the same instance of quantum IBS as mentioned in section 8.1.

Table 3: Hardware specification of a real quantum machine used for simulation experiments Quantum Computer | IBM Q Lima v1.0.36
---|---
Qubits | 5
Processor type | Falcon r4T
CLOPS | 2.7K
Quantum Volume | 8
Software | Qiskit v0.20.2
Figure 6: Success probability of an instance of the proposed quantum IBS when
simulated on a real quantum machine

To find the success probability of the quantum IBS, we ran the protocol for
$1024$ times on a real quantum computer with hardware specifications mentioned
in Table 3. Figure 6 summarizes the results of this simulation experiment. We
now analyze the findings of the experiment. The success probability turns out
to be 89.2%. The error rate of 10.8 % may be attributed to facts like quantum
noise and loss of photons during transmission. For real-life applications, we
may increase the number of photons that are being transmitted. In addition, we
may use classical error-correction techniques and quantum repeater to avoid
the aforementioned problems [13]. In essence, our proposed design only uses
single photons quantum resources. As experiments suggest, it is possible and
feasible to implement it using the existing quantum back-end technologies.

##  9 Toy Example

$X\ket{0}=\ket{1},X\ket{1}=\ket{0},Z\ket{0}=\ket{0},Z\ket{1}=-\ket{1}$
Encryption

$\displaystyle E_{k}(\ket{P})$
$\displaystyle=\bigotimes_{i=1}^{3}{X^{k_{2i}}Z^{k_{2i-1}}}\ket{p_{i}}$ (1)
$\displaystyle=X^{k_{2}}Z^{k_{1}}\ket{p_{1}}\otimes
X^{k_{4}}Z^{k_{3}}\ket{p_{2}}\otimes X^{k_{6}}Z^{k_{5}}\ket{p_{3}}$
$\displaystyle=X\ket{p_{1}}\otimes X\ket{p_{2}}\otimes Z\ket{p_{3}}$
$\displaystyle=X\ket{0}\otimes X\ket{1}\otimes Z\ket{0}$
$\displaystyle=\ket{100}$

Decryption:
$\ket{e}=\ket{100}$

$\displaystyle Dec(E_{k}\ket{p})$
$\displaystyle=\bigotimes_{i=1}^{3}{Z^{k_{2i-1}}X^{k_{2i}}}\ket{e_{i}}$ (2)
$\displaystyle=Z^{k_{1}}X^{k_{2}}\ket{e_{1}}\otimes
Z^{k_{3}}X^{k_{4}}\ket{e_{2}}\otimes Z^{k_{5}}X^{k_{6}}\ket{e_{3}}$
$\displaystyle=X\ket{1}\otimes X\ket{0}\otimes Z\ket{0}$
$\displaystyle=\ket{010}$ $\displaystyle=\ket{p}$

###  9.1 QIBS

####  9.1.1 Initialization Phase

Let $ID_{A}={011}$ and $ID_{B}={100}$ denote the signer and the verifier
identities, respectively. $\ket{P}=\ket{010}$ represent the quantum message
that requires a signature. Let $T_{i}={010110}$ denote the secret key between
the signer and the SKG, and let $T_{U}={100101}$ represent the mutual hidden
key between the SKG and the person who signed it. Also, we take
$U=U(\pi/2,\pi,0)=\begin{pmatrix}1/\sqrt{2}&-1/\sqrt{2}\\\
-1/\sqrt{2}&-1/\sqrt{2}\end{pmatrix}=1/\sqrt{2}\begin{pmatrix}1&-1\\\
-1&-1\end{pmatrix}$. Note that $U\ket{0}=1/\sqrt{2}\begin{pmatrix}1&-1\\\
-1&-1\end{pmatrix}\begin{bmatrix}1\\\
0\end{bmatrix}=1/\sqrt{2}\begin{bmatrix}1\\\ -1\end{bmatrix}=\ket{-}$ and
$U\ket{1}=1/\sqrt{2}\begin{pmatrix}1&-1\\\
-1&-1\end{pmatrix}\begin{bmatrix}0\\\
1\end{bmatrix}=1/\sqrt{2}\begin{bmatrix}-1\\\ -1\end{bmatrix}=\ket{+}$

####  9.1.2 Signing Phase

In the signing phase, the signer having identity $ID_{A}={011}$, operates
$U^{\otimes 3}$ (where $U=1/\sqrt{2}\begin{pmatrix}1&-1\\\
-1&-1\end{pmatrix}$) on the quantum message state $|P\rangle=\ket{010}$ to get
$U^{\otimes 3}(|P\rangle)$. In the next step, the signer uses the quantum OTP
encryption to encrypt $U^{\otimes 3}(|P\rangle)$ to get $\ket{S}$.

$\displaystyle\ket{S}$ $\displaystyle=E_{T_{i}}(U^{\otimes 3}(\ket{P}))$
$\displaystyle=E_{T_{i}}(U^{\otimes 3}(\ket{010}))$
$\displaystyle=E_{T_{i}}(U\ket{0}\otimes U\ket{1}\otimes U\ket{0})$
$\displaystyle=E_{T_{i}}(\ket{-}\otimes\ket{+}\otimes\ket{-})$
$\displaystyle=E_{T_{i}}(\ket{-+-})$
$\displaystyle=E_{T_{i}}[(\ket{0}-\ket{1})\otimes(\ket{0}+\ket{1})\otimes(\ket{0}-\ket{1})]$
$\displaystyle=E_{T_{i}}[\ket{000}-\ket{100}+\ket{010}-\ket{110}$ (3)
$\displaystyle-\ket{001}+\ket{101}-\ket{011}+\ket{111}]$ (4)

To compute the final value of $\ket{S}$, we compute each of
$E_{T_{i}}\ket{000}$, $E_{T_{i}}\ket{100}$, $E_{T_{i}}\ket{010}$,
$E_{T_{i}}\ket{110}$,$E_{T_{i}}\ket{001}$, $E_{T_{i}}\ket{101}$,
$E_{T_{i}}\ket{011}$, and $E_{T_{i}}\ket{111}$.

$\displaystyle E_{T_{i}}\ket{000}=X\ket{0}\otimes X\ket{0}\otimes
Z\ket{0}=\ket{110}$ $\displaystyle E_{T_{i}}\ket{100}=X\ket{1}\otimes
X\ket{0}\otimes Z\ket{0}=\ket{010}$ $\displaystyle
E_{T_{i}}\ket{010}=X\ket{0}\otimes X\ket{0}\otimes Z\ket{1}=\ket{100}$
$\displaystyle E_{T_{i}}\ket{110}=X\ket{1}\otimes X\ket{1}\otimes
Z\ket{0}=\ket{000}$ $\displaystyle E_{T_{i}}\ket{001}=X\ket{0}\otimes
X\ket{0}\otimes Z\ket{1}=-\ket{111}$ $\displaystyle
E_{T_{i}}\ket{101}=X\ket{1}\otimes X\ket{0}\otimes Z\ket{1}=-\ket{011}$
$\displaystyle E_{T_{i}}\ket{011}=X\ket{0}\otimes X\ket{1}\otimes
Z\ket{1}=-\ket{101}$ $\displaystyle E_{T_{i}}\ket{111}=X\ket{1}\otimes
X\ket{1}\otimes Z\ket{1}=-\ket{001}$

Putting the values back in the original expression ($A.1)$, we get the value
of $\ket{S}$ as
$\ket{S}=\frac{1}{2\sqrt{2}}[\ket{110}-\ket{010}+\ket{100}-\ket{000}+\ket{111}-\ket{011}+\ket{101}-\ket{001}]$

Finally, the signer outputs the tuple $(|P\rangle,|S\rangle,|ID_{A}\rangle)$
as message-signature pair, where
$\ket{P}=\ket{010},\ket{S}=\frac{1}{2\sqrt{2}}[\ket{110}-\ket{010}+\ket{100}-\ket{000}+\ket{111}-\ket{011}+\ket{101}-\ket{001}],$
and $\ket{ID_{A}}=\ket{011}$.

####  9.1.3 Verification Phase

Step 1

Bob, the verifier, keeps $\ket{P}$ and encrypts $(\ket{S},\ket{ID_{A}})$ with
$T_{u}$.

$\displaystyle E_{T_{u}}$
$\displaystyle=\bigotimes_{i=1}^{3}{X^{k_{2i}}Z^{k_{2i-1}}}\ket{y_{i}}$
$\displaystyle=X^{k_{2}}Z^{k_{1}}\ket{y_{1}}\otimes
X^{k_{4}}Z^{k_{3}}\ket{y_{2}}\otimes X^{k_{6}}Z^{k_{5}}\ket{y_{3}}$
$\displaystyle=Z\ket{y_{1}}\otimes X\ket{y_{2}}\otimes X\ket{y_{3}}$
$E_{T_{u}}\ket{110}=Z\ket{1}\otimes X\ket{1}\otimes X\ket{0}=-\ket{101}$
$E_{T_{u}}\ket{010}=Z\ket{0}\otimes X\ket{1}\otimes X\ket{0}=\ket{001}$
$E_{T_{u}}\ket{100}=Z\ket{1}\otimes X\ket{0}\otimes X\ket{0}=-\ket{111}$
$E_{T_{u}}\ket{000}=Z\ket{0}\otimes X\ket{0}\otimes X\ket{0}=\ket{011}$
$E_{T_{u}}\ket{111}=Z\ket{1}\otimes X\ket{1}\otimes X\ket{1}=-\ket{100}$
$E_{T_{u}}\ket{011}=Z\ket{0}\otimes X\ket{1}\otimes X\ket{1}=\ket{000}$
$E_{T_{u}}\ket{101}=Z\ket{1}\otimes X\ket{0}\otimes X\ket{1}=-\ket{110}$
$E_{T_{u}}\ket{001}=Z\ket{0}\otimes X\ket{0}\otimes X\ket{1}=\ket{010}$

$E_{T_{u}}\ket{S}=\frac{1}{2\sqrt{2}}[-\ket{110}-\ket{010}-\ket{100}-\ket{000}-\ket{111}-\ket{011}-\ket{101}-\ket{001}]$
$E_{T_{u}}\ket{S}=-\frac{1}{2\sqrt{2}}[\ket{110}+\ket{010}+\ket{100}+\ket{000}+\ket{111}+\ket{011}+\ket{101}+\ket{001}]=\ket{T}$

$E_{T_{u}}\ket{ID_{A}}=E_{T_{u}}\ket{011}=\ket{000}=EID_{A}$

$E_{T_{u}}(\ket{S},\ket{ID_{A}})=(-\frac{1}{2\sqrt{2}}[\ket{110}+\ket{010}+\\\
\ket{100}+\ket{000}+\ket{111}+\ket{011}+\ket{101}+\ket{001}],\ket{000})$ This
is sent to SKG. For the sake of brevity, we call
$E_{T_{u}}(\ket{S},\ket{ID_{A}})$ as $\psi$.

Step 2:

SKG first decrypts $\psi$ using the secret key $T_{u}$ to get back
$(|S\rangle,|ID_{i}\rangle)$.

$\displaystyle D_{T_{u}}(\ket{\psi})$
$\displaystyle=(D_{T_{u}}\ket{T},D_{T_{u}}\ket{EID_{A}})$ $\displaystyle
D_{T_{u}}$
$\displaystyle=\bigotimes_{i=1}^{3}{Z^{k_{2i-1}}X^{k_{2i}}}\ket{y_{i}}$
$\displaystyle=Z\ket{y_{1}}\otimes X\ket{y_{2}}\otimes X\ket{y_{3}}$
$\displaystyle D_{T_{u}}(EID_{A})$
$\displaystyle=D_{T_{u}}\ket{000}=\ket{011}=\ket{ID_{A}}$ $\displaystyle
D_{T_{u}}(\ket{T})$
$\displaystyle=D_{T_{u}}[-\frac{1}{2\sqrt{2}}[\ket{110}+\ket{010}+\ket{100}+\ket{000}+$
$\displaystyle\ket{111}+\ket{011}+\ket{101}+\ket{001}],\ket{000}]$
$\displaystyle=\frac{1}{2\sqrt{2}}[\ket{110}-\ket{010}+\ket{100}-\ket{000}+$
$\displaystyle\ket{111}-\ket{011}+\ket{101}-\ket{001}]$
$\displaystyle=\ket{S}$

Now SKG performs the following steps to retrieve $\ket{P}$

$\displaystyle D_{T_{i}}$ $\displaystyle=U^{{\dagger}^{\otimes
3}}\bigotimes_{i=1}^{3}{Z^{k_{2i-1}}X^{k_{2i}}}$
$\displaystyle\bigotimes_{i=1}^{3}{Z^{k_{2i-1}}X^{k_{2i}}}$
$\displaystyle=X\otimes Z\otimes Z$ $\displaystyle U$
$\displaystyle=1/\sqrt{2}\begin{pmatrix}1&-1\\\ -1&-1\end{pmatrix}$
$\displaystyle U^{\dagger}$ $\displaystyle=1/\sqrt{2}\begin{pmatrix}1&-1\\\
-1&-1\end{pmatrix}$ $U^{{\dagger}^{\otimes
3}}=\frac{1}{2\sqrt{2}}\begin{pmatrix}1&-1&-1&1&-1&1&1&-1\\\
-1&-1&1&1&1&1&-1&-1\\\ -1&1&-1&1&1&-1&1&-1\\\ 1&1&1&1&-1&-1&-1&-1\\\
-1&1&1&-1&-1&1&1&-1\\\ 1&1&-1&-1&1&1&-1&-1\\\ 1&-1&1&-1&1&-1&1&-1\\\
-1&-1&-1&-1&-1&-1&-1&-1\\\ \end{pmatrix}$ (6)

$\bigotimes_{i=1}^{3}{Z^{k_{2i-1}}X^{k_{2i}}}\ket{S}=\frac{1}{2\sqrt{2}}[\ket{000}-\ket{100}+\ket{010}-\ket{110}\\\
-\ket{001}+\ket{101}-\ket{011}-\ket{111}]=\ket{v}(say)$

${\small U^{{\dagger}^{\otimes 3}}\ket{v}=\\\
\frac{1}{2\sqrt{2}}X\frac{1}{2\sqrt{2}}\begin{pmatrix}1&-1&-1&1&-1&1&1&-1\\\
-1&-1&1&1&1&1&-1&-1\\\ -1&1&-1&1&1&-1&1&-1\\\ 1&1&1&1&-1&-1&-1&-1\\\
-1&1&1&-1&-1&1&1&-1\\\ 1&1&-1&-1&1&1&-1&-1\\\ 1&-1&1&-1&1&-1&1&-1\\\
-1&-1&-1&-1&-1&-1&-1&-1\\\ \end{pmatrix}\begin{bmatrix}1\\\ -1\\\ 1\\\ -1\\\
-1\\\ 1\\\ -1\\\ 1\end{bmatrix}}$ (7)
$\displaystyle=\frac{1}{8}\begin{bmatrix}0&0&8&0&0&0&0&0\end{bmatrix}^{t}$
$\displaystyle=\begin{bmatrix}0&0&1&0&0&0&0&0\end{bmatrix}^{t}=\ket{010}$

In the following, SKG encrypts $|P\rangle$ using $T_{u}$ and sends
$|R\rangle=E_{T_{u}}(|P\rangle)$ to Bob.
$E_{T_{u}}(\ket{P})=E_{T_{u}}\ket{010}=\ket{001}=\ket{R}$

Step 3

On receiving $|R\rangle$ from SKG, Bob decrypts it using the secret key
$T_{u}$. That is, $D_{T_{u}}(\ket{R})=Z\ket{r_{1}}\otimes X\ket{r_{2}}\otimes
X\ket{r_{3}}=\ket{010}$. Note that $D_{T_{u}}(|R\rangle)$ is same as the
$|P\rangle$ of the signature tuple $(|P\rangle,|S\rangle,|ID_{i}\rangle)$.
Therefore, Bob accepts the signature as a valid signature.

##  10 Application of Our Proposed Quantum IBS For Secure Email

Email has become a ubiquitous form of communication, simplifying the exchange
of information globally. However, its widespread usage exposes it to various
security threats, including email spoofing and tampering. In this section, we
explore the potential usage of IBS in mitigating email security risks,
focusing on its ability to prevent spoofing and tampering. Email spoofing is
an adversary’s common technique to impersonate legitimate senders, deceiving
recipients into believing that the email originates from a trusted source. It
can lead to various forms of cybercrime, including phishing attacks, malware
distribution, and financial fraud. Moreover, email tampering involves
unauthorized modification of email content during transit, compromising its
integrity and authenticity.

Our proposed quantum identity-based signatures offer a robust solution to
address the aforementioned security challenges associated with email
communications. Unlike traditional public key infrastructure (PKI), quantum
IBS enables users to generate digital signatures using their identities, such
as email addresses or user IDs. It eliminates the need for a separate
infrastructure to manage public keys, simplifying key management processes and
enhancing scalability. Quantum IBS can prevent email spoofing by enabling
senders to sign their messages using their identities. When a sender sends an
email, they generate a digital signature using their private key. Recipients
can verify the email’s authenticity by validating the signature using the
sender’s public key, derived from their identity. If the signature is valid,
it confirms that the email was indeed sent by the claimed sender, mitigating
the risk of impersonation and spoofing. In addition to preventing spoofing,
quantum IBS ensures the integrity of email messages by detecting any
unauthorized modifications made to the content during transit. When a sender
signs an email using their private key, the signature encapsulates the entire
message, including its content and attachments. Any alteration to the message
would invalidate the signature upon verification. Thus, recipients can detect
tampering attempts and reject emails with invalid signatures, preserving the
integrity and authenticity of the communication. While traditional
cryptographic algorithms, such as RSA and ECC, are widely used to secure email
communications, they are vulnerable to quantum attacks due to the advent of
quantum computing. On the other hand, our proposed quantum IBS is immune to
the threat of quantum computers. To summarize, quantum IBS presents itself as
a robust solution for ensuring that email messages are sent and received
securely in the quantum world. Refer to Figure 7 for a schematic diagram.

Figure 7: Application of quantum IBS in secure email

##  11 Conclusion

In this work, we used the quantum analogue of a one-time pad to design an
information-theoretically secure quantum IBS. Furthermore, because our
protocol’s security is based on a fundamental aspect of quantum mechanics, it
offers lasting safety compared to conventional and post-quantum IBS systems.
Our proposed quantum IBS will satisfy unforgeability and undeniability if the
encryption is theoretically secure. The unforgeability property ensures that
no other party can do the signature on behalf of a signer. In contrast,
undeniability ensures that a signer can not deny the signature generation if
she did it. The suggested plan achieves long-term security and resists quantum
attacks. Our scheme is more practical than the existing works since it uses
only simple measurement operators and single-photon quantum resources.
Furthermore, the suggested approach has a lower communication and computing
overhead than the current quantum IBS systems [3, 21, 22]. The Jupyter
Notebook for the execution of the toy example is given in the Appendix. In
addition, the application of our IBS in secure email communication is
provided.

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Statements and Declarations

Competing Interests: The authors declare no conflicts of interest.

Data Availability Statement: Data will be available with the corresponding
author upon reasonable request.

See pages - of QIBSAppendix.pdf


# Satin Non-Woven Fabrics for Designing of Self-Regulating Breathable Building
Skins

Saied Zarrinmehr
Computer Science and Engineering Department,
Texas A&M University, College Station, TX, 77831
<EMAIL_ADDRESS>
Ergun Akleman
Visual Computing & Computational Media,
Texas A&M University, College Station, TX, 77831
<EMAIL_ADDRESS>
Tahir Cagin
Material Science and Engineering,
Texas A&M University, College Station, TX, 77831
<EMAIL_ADDRESS>
Joint with Computer Science and Engineering Department.

###### Abstract

In this paper, we introduce the concept of 2-way 2-fold genus-1 non-woven
fabrics that can be used to design self-regulating breathable building skins.
The advantage of non-woven structures over woven structures for breathable
skin design is that they can completely be closed to stop air exchange. We
have developed a theoretical framework for such non-woven structures starting
from the mathematical theory of biaxial 2-fold Genus-1 woven fabrics. By re-
purposing a mathematical notation that is used to describe 2-fold 2-way 2-fold
genus-1 woven fabrics, we identify and classify non-woven fabrics. Within this
classification, we have identified a special subset that corresponds to satin
woven fabrics and allows for maximum air exchange. Any other subset of non-
woven structures that correspond to other classical 2-way 2-fold genus-1
fabrics, such as plain or twill, will allow for less air exchange. We also
show that there exists another subset of satin non-woven fabrics that can
provide the biggest openings.

(a) A detailed view of a closed skin constructed with a non-woven fabric
structure.

(b) A detailed view of an open skin constructed with the same non-woven fabric
structure.

(c) A general view of an open skin constructed with the same non-woven fabric
structure.

Figure 1: An example of satin non-woven fabric to obtain a high amount of air
exchange. This particular non-woven structure corresponds to a specific satin
woven fabric, called $[7,1,3]$, which is one of the best possible among satin
non-woven structures for efficiency of air exchange. The notation is explained
in Section 2

##  1 Introduction and Motivation

In the US, 50% of all energy is consumed by buildings, and 75% of all
electricity is used for the operations of buildings Pérez-Lombard et al.
(2008). Reduction of energy consumption in buildings can significantly reduce
overall energy usage in the US. Therefore, there is a need for strategies to
reduce energy consumption in buildings. Energy reduction in buildings is not
an easy task since the design, construction, and operation processes of
buildings are significantly different from the processes of industrial
products. For instance, cars and/or planes are designed to be mass-produced
and once they are constructed, they are operated similarly. Therefore, when an
energy-efficient product is developed, it is relatively simple to calculate
statistically likely energy consumption of an innovative product design versus
that of a former, benchmark version.

(a) The details of the closed skin constructed with satin non-woven structure.

(b) The details of the open skin constructed with satin non-woven structure.

(c) The details of the open skin constructed with satin non-woven structure.

Figure 2: Another example of the satin non-woven structures.

(a) The details of the closed skin constructed with satin non-woven structure.

(b) The details of the open skin constructed with satin non-woven structure.

(c) The details of the open skin constructed with satin non-woven structure.

Figure 3: One more example of the satin non-woven structures.

Buildings, however, are typically designed, constructed, and operated as
unique entities. For buildings, the best strategy is to develop modular
systems that can be used for the construction of energy-efficient systems. One
of the key issues to making buildings energy efficient is to make their outer
skins self-regulated. We imagine the development of the type of building skins
that can automatically open and close based on environmental factors similar
to tiny pores called “stomata”, which are located on the epidermis of plant
leaves. These tiny pores play a key role in transpiration by opening and
closing in response to light. There is, therefore, a need to develop modules
that can play the role of stomata for building skins.

|  |
---|---|---
Biaxial (2-way) Twill Fabric | Triaxial (3-way) Sparse Fabric | Triaxial (3-way) Dense Fabric
Figure 4: Examples of 2-way 2-fold and 3-way 2-fold fabrics. 2-way, i.e.
biaxial, fabrics are always dense. On the other hand, 3-way, i.e. triaxial,
fabrics with straight yarns are usually sparse as shown in the second image.
We, therefore, need a zigzag pattern to obtain a dense fabric as shown in the
last image. All three fabrics in this example are 2-fold since there are only
2 layers in each intersection.

To play the role of stomata in buildings any self-regulatory module should (1)
completely shut down when closed, and (2) allow maximum amount of air exchange
when opened. We have observed that if we replace yarns in woven or non-woven
fabrics with bi-material strips we can automatically open and close modules.
We have also observed that woven structures cannot completely be closed since
strips must have a non-zero thickness as shown in Figure 5.

The advantage of non-woven structures over woven structures is that they can
theoretically guarantee complete shutdown as shown in Figure 1. In this paper,
we, further, demonstrate that satin non-woven fabrics can potentially play the
role of stomata by providing maximum exchange when opened. More examples of
satin non-woven structures are shown in Figures 2, and 3.

Non-woven fabrics, although not well known in the research community, have
been around since 1930’s through a significant number of patents that focus on
the engineering construction of non-woven textile structures such as Brewster
(1934); Powel (1947); Clayton et al. (1955); Graham (1956); Dupre et al.
(1960); Allison (1967). For a comparative analysis of early works on non-woven
fabrics see Casper (1975); Purdy (1983). Non-woven fabrics have been replacing
woven materials in many industrial applications from engineering to medicine
since they are more reliable and they can provide better behavior for
industrial applications Curtis and Moore (1987); Balogh et al. (2015).

(a) Top view of a woven structure.

(b) Perspective view of the center part of the same woven structure.

(c) Perspective view of the side of the same woven structure.

Figure 5: An example that demonstrates the woven structures cannot completely
shut down to avoid self-intersection if strips have non-zero thickness. The
holes have to be larger for thicker strips as shown here.

Despite such a significant amount of engineering work, we could not find any
work for the formalization of non-woven fabrics as mathematical/combinatorial
structures. This is not surprising since the first combinatorial formalization
of woven fabrics was not introduced until the 1980s Grunbaum and Shephard
(1980) although 2-way 2-fold woven fabrics such as plain, twill, and satin
have been around at least five millennia Horrocks and Anand (2000) (See Figure
4 for examples of 2-way, 3-way and 2-fold fabrics). In this work, we offer a
similar formalization that can describe all possible 2-way 2-fold genus-1 non-
woven fabrics. Our work is, in fact, based on combinatorial formalization
provided for woven fabrics by Grunbaum and Shephard Grunbaum and Shephard
(1980) in the early 1980’s.

This combinatorial formalization for non-woven fabrics is crucial in the
identification of the special subset of 2-way 2-fold genus-1 non-woven fabrics
that can provide maximum air exchange when opened. In Section 2, we provide a
classification of woven fabrics, and In Section 3, we provide our
classification of now-woven fabrics. Based on this classification, we identify
a combinatorial structure that can define a special subset that provides
maximum exchange when opened.

###  1.1 Contributions

In this paper, we have a series of contributions:

  1. 1.

We have shown that by replacing yarns with bi-material strips we can obtain
self-regulatory stomata-like behavior with both woven and non-woven fabrics.

  2. 2.

We have shown that non-woven structures are preferable to woven structures:

     * •

Woven structures cannot completely be closed. On the other hand, non-woven
structures can completely shut down when they are closed.

     * •

Woven structures can become loose when they are opened. On the other hand,
non-woven structures can still stay intact when they are opened.

  3. 3.

We have developed a classification of non-woven structures based on the
combinatorial classification of corresponding woven structures.

  4. 4.

We have shown that the property of “hanging togetherness” in woven fabrics
also applies to non-woven structures and it guarantees that non-woven
structures are completely connected.

  5. 5.

We have identified a specific subset of satin non-woven structures that can
provide maximum air exchange. This subset is also guaranteed to be connected
by having hanging together property.

##  2 Previous Work

This paper is inspired by woven fabrics that can effectively provide a
reasonable amount of air exchange through tiny openings caused by the
thickness of yarns or strips as shown in Figure 5. Most common woven fabrics
are biaxial, or 2-way and 2-fold, which consist of two strands, i.e. 2-way
that are called warp and weft. The word 2-fold originates from the behavior of
the weft strands that pass over and under warp strands, which corresponds to
the thickness of the fabric. The mathematics behind such 2-way, 2-fold woven
fabrics such as plain, satin, and twill, are first formally investigated by
Grunbaum and Shephard Grunbaum and Shephard (1980). They also coined the word,
isonemal fabrics to describe fabrics that have a transitive symmetry group on
the strands of the fabric Grunbaum and Shephard (1988). One of the important
concepts introduced by Grunbaum and Shephard is hang-together or fall-apart.
They pointed out that weaving patterns that look perfectly reasonable may not
produce interlacements that hang-together. In such patterns, some interlaced
warp and weft threads may not be interlaced with the rest of the fabric if the
fabric is woven. Such a fabric would come apart in pieces. Determining whether
or not a pattern is hang-together or fall-apart cannot be identified by simple
visual inspection of the pattern. Several procedures have been developed to
identify the hang-together property of a weaving pattern. If a weaving pattern
is hanging together, it is then called a fabric. Grunbaum and Shephard
provided images of all non-twill isonemal fabrics (i.e. hanging-together
weaves) of periods up to 17 Grunbaum and Shephard (1985, 1986). Their papers
were followed by several others giving algorithms for determining whether or
not a fabric hangs together Griswold (2004a); Clapham (1980, 1985); Enns
(1984); Delaney (1984). Roth Roth (1993), Thomas Thomas (2010, 2009a, 2009b)
and Zelinka Zelinka (1983, 1984) and Griswold Griswold (2004b, c)
theoretically and practically investigated symmetry and other properties of
isonemal fabrics.

It was recently demonstrated that any surface can be covered with plain woven
structures Akleman et al. (2009), twill woven structures Akleman et al.
(2011), and it is also possible to cover any surface with a single cycle Xing
et al. (2010). The recent mathematical work further demonstrated that surfaces
can be covered by any type of woven structure Akleman et al. (2020, 2015).

In this paper, we are interested in the non-woven generalization of certain
types of isonemal fabrics that are called genus-1 by Grunbaum and Shephard
Grunbaum and Shephard (1986). Genus-1 means that each row with length $n$ is
obtained from the row above it by a shift of $s$ units to the right, for some
fixed value of the parameter $s$. They call these fabrics $(n,s)$-fabrics
where $n$ is the period of the fabric and $n$ and $s$ are relatively prime.
The genus-1 family of fabrics includes the most common fabrics: plain, satins,
and twills. If $s=1$ the fabric is a twill and $s^{2}\equiv 1\mbox{mod}n$
corresponds to satin for the fabrics that contain exactly one black square in
every row. In Grunbaum and Shephard’s notation plain weaving in Figure 6(a) is
called $(2,1)$-fabric, twill weaves in Figure 6(b) and (c) are called $(4,1)$
and $(5,4)$-fabrics respectively, satin weaving in Figure 6(d) is called
$(8,3)$-fabric and finally the unnamed weave in Figure 6(e) now has a name and
is called $(13,4)$-fabric. Genus-1 fabrics not only include satins and twills
but also are the only isonemal fabrics that exist for every integer $n$. The
other isonemal fabrics do not exist for odd $n$ integer values Grunbaum and
Shephard (1986).

|  |  |  |
---|---|---|---|---
Plain: $[1,1,1]$ | Twill: $[2,2,1]$ | Twill: $[3,2,4]$ | Satin: $[7,1,3]$ | Unnamed: $[6,7,4]$
Figure 6: Examples of textile weaving patterns.

One problem with Grunbaum and Shephard’s $(n,s)$ notation is that it ignores
the initial row pattern. Chen et al. showed that it is useful to express the
initial pattern by two integers $a$ and $b$, where $a$ is the number of up-
crossings, and $b$ is the number of down-crossings where $a+b=n$ and an
additional integer $s$ still denotes the shift introduced in adjacent rows
Chen et al. (2010). Any such weaving pattern can be expressed by a triple
$[a,b,s]$. The Figure 6 also shows a basic block of a biaxial weaving
structure and the role of these three integers, $a>0$, $b>0$ and $0<s<a+b$
111The value of $s$ can be any integer, however, $[a,b,s]$ and
$[a,b,s+k(a+b)]$ are equivalent since $s+k(a+b))\equiv s\mbox{mod}(a+b)$.
Therefore, we assume that the value of the $s$ is between $0$ and $a+b$..
Figure 6 shows four examples of biaxial weaving structures that can be
described by the $[a,b,s]$ notation. Using these notations it is possible to
name each weaving structure uniquely.

The notation $[a,b,s]$ can represent a fabric that can fall apart. However,
unlike a general isonemal weaving case, it is easy to avoid the fabric falling
apart. A genus-1 weaving pattern produces a fabric structure that hangs
together if and only if $a+b$ and $s$ are relatively prime (i.e.
$gcd(a+b,s)=1$). In other words, if the greatest common divider of $a+b$ and
$s$ is not $1$, the resulting structure is not a fabric since it cannot hang
together and, therefore, must be avoided. This provides sufficient background
for woven fabrics. We now are ready to discuss non-woven structures.

##  3 Non-Woven Fabrics

Non-woven fabrics are obtained by bonding fibers to each other using some sort
of adhesive. We note that a non-woven structure can also be described by just
defining where adhesives are applied. For instance, the two colored grids in
Figure 6 can be considered representations of non-woven structures, if we
assume that the adhesives are applied to the blue squares. Under this
assumption, we can represent non-woven structures also with three numbers as
$[a,b,s]$ where $a$ is the length of the free region, $b$ is the length of the
bonded region given that $a+b=n$, and the additional integer $s$ still denotes
the shift introduced in adjacent rows Chen et al. (2010).

(a) A detailed view of the closed skin constructed with plain non-woven
structure.

(b) A detailed view of the open skin constructed with plain non-woven
structure.

(c) An open skin constructed with a plain non-woven structure.

Figure 7: An example that demonstrates the plain non-woven structures cannot
provide sufficient openings for air exchange.

One of the key issues with non-woven fabrics is to reduce the bonded region as
much as possible. Since we want to keep the resulting structure connected, $b$
should be larger than $0$. Since the smallest integer larger than $0$ is $1$,
it is better to keep $b$ always equal to $1$, which makes $a=n-1$. Therefore,
we are interested in non-woven fabrics only in the form of $[n-1,1,s]$. Figure
7 shows a non-woven version of the simplest woven fabrics, $[1,1,1]$, called
plain weaving. Such plain non-woven fabrics are frequently used in the
construction of many types of materials and, therefore, this structure
probably looks familiar to many people. However, as it can easily be seen
plain is not a good structure to provide air exchange since there is not
enough room for strands to curve out.

In order to get maximum air exchange, for a given $n$, $s$ has to be chosen in
such a way that there should be a maximum shift between two consecutive warps
(or wefts). In other words, $s$ must be the closest integer to $(n-1)/2$, i.e.
$s=\lfloor(n-1)/2+0.5\rfloor$. If we choose an even number, the notation
simplifies to $[2s,1,s]$. This particular structure looks promising as a first
impression, however, the other (bottom/weft) side crates problems since the
other side (bottom/weft) is not guaranteed to be the same as the warp side
(See Figure 8 for the problems in back-side).

(a) The details of front-side view of $[14,1,7]$ non-woven structure in open
position.

(b) The details of back-side view of $[14,1,7]$ non-woven structure in open
position.

(c) The details of front-side view of $[24,1,12]$ non-woven structure in open
position.

(d) The details of back-side view of $[24,1,12]$ non-woven structure in open
position.

Figure 8: An example that demonstrates the $[2n,1,n]$ non-woven structures
cannot provide sufficient openings for air exchange in the backside.

It can be proven that when the top strands follow $[n-1,1,s_{1}]$ non-woven
pattern, the bottom strands will follow $[n-1,1,s_{2}]$ non-woven pattern. It
can also be shown that $s_{1}$ and $s_{2}$ are not independent and their
dependency can be described as $s_{1}\;s_{2}\equiv 1\mbox{mod}n$. The best
solution to maximize both $s_{1}$ and $s_{2}$ is, therefore, to choose the
same integer value for $s_{1}$ and $s_{2}$. Another acceptable solution is to
choose numbers for them that only differ by one $|s_{1}-s_{2}|=1$. Assuming
these constraints, we obtain either $s^{2}=n+1$ or $s^{2}-s=n+1$. These two
equations give us an approximation for $s$ as $s\approx\sqrt{n+1}$. It is
interesting to note that these settings correspond to satin woven fabric
structures. Therefore, it is better to call the optimized solutions satin non-
woven fabrics.

(a) The details of the closed skin constructed with satin non-woven structure.

(b) The details of the open skin constructed with satin non-woven structure.

(c) The open skin constructed with satin non-woven structure.

Figure 9: An example that demonstrates the satin non-woven structures can
provide more sufficient openings than plain structures. However, the air
exchange can never get as good as $(2n+1,1,n)$ non-woven structures.

One of the important issues is the identification of the material properties
of strips such that they can automatically open and close based on some
environmental changes such as temperature or humidity. We first considered
single-material strips and found out that they do not provide automatic
opening and closing unless we use an additional scaffold base. We, therefore,
analyzed bi-material strips based on the inspiration coming from bi-metallic
strips that have been used in many applications such as thermostatic switches
to obtain self-regulating behavior Dubilier (1942); Brennan (1951).

##  4 Bi-Material Strips

In this section, we demonstrate that bi-material strips, which are shown in
Figure 10(b) as blue and yellow rectangles that are bonded together, can be
useful to automatically bend warp and wefts based on different amounts of
expansions of the yellow and blue stripes. The main idea is that the
difference in thermal expansions of two strips causes the bonded bi-material
strip to bend as shown in Figure 10. In this paper, we ignore the cause or the
nature of the expansion. Without loss of generalization and for the simplicity
of discussion, we assume that the expansions are caused only by changes in
temperature and can be explained by multiplication with an expansion ratio.

(a) Two strips with different lengths that are not bonded together.

(b) When the two strips with different lengths are bonded together they become
curved.

Figure 10: An example that demonstrates how two strips with different lengths
become curved when they are bonded together.

Now, let both stripes have the thickness denoted by $\delta$. Let
$d=d^{\prime}_{0}=d^{\prime}_{1}$ denote the original length of blue and
yellow strips when it is not curved, and let $d_{0}$ and $d_{1}$ denote the
length of curved blue and yellow strips. Without loss of generality, we can
assume that the shape of the curved bi-material strip is a circle segment as
shown in Figure 10(b) given a radius $r$ and angle $\theta$. Based on this
assumption, we can write the following two equations:

$d_{1}=r_{1}=r\theta,\;\;\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\;\;d_{0}=r_{0}\theta=(r+\delta)\theta$

Let $D$ denote the relative expansion between the two materials, which is
given as the ratio between the two stripes as $D=d_{1}/d_{0}$. Based on this,
we can compute $D$ independent of $\theta$ as and we compute the radius
$r=r_{1}$ as follows:

$D=\frac{r\theta}{(r+\delta)\theta}=\frac{r}{r+\delta}\;\;\;\;\;\;\;\;\Longrightarrow\;\;\;\;\;\;\;\;r=\frac{\delta
D}{1-D}$

We can write this equation in terms of curvature by using the fact that the
curvature $\kappa$ is inverse proportional to radius $r$. As a result, we
obtain,

$\kappa=\frac{1-D}{\delta D};\;\;\;\;\;\;\;\;\mbox{note
that}\;\;\;\;\;\;\;\;\lim_{\delta D\rightarrow 0}\kappa\longrightarrow\infty$

Using this key equation, we can design bi-material strips that can provide the
desired curvature based on changes in the temperature.

#### Remark 1:

The denominator term $\delta D$ is useful for our applications since by making
either or both of the two parameters $\delta$ and $D$, we can precisely
control curvature and we can even obtain arbitrarily high curvature values.
Because of the physical constraints neither one of them in practice can go to
zero but we can change them simultaneously to obtain the desired curvature.

#### Remark 2:

We want to point out that $D$ is not the thermal expansion of a single
material; it is rather the relative thermal expansion, which is given as a
ratio of two thermal expansions: $d_{0}/d$ and $d_{1}/d$. Using structures
from phase-changing materials such as paraffin, we can theoretically obtain up
to $25\%$ thermal expansions, which corresponds to $d_{0}/d=1.25$. If the
other material, i.e. yellow, do not extend, we get $D$ values up to
$d_{1}/d_{0}=1/1.25=0.8$, of course, such a high relative thermal expansion is
hard to get in practice. Fortunately, this is not a problem for our
applications.

#### Remark 3:

An important implication of this result is that we do not even need high $D$
values to obtain high curvature. For example, assume that we want to use two
materials with a relative expansion ratio of $D=0.999$, and we want to obtain
a radius $r=1.00\;m$, since $\delta=(1-D)r/D=0.001/0.999m\approx
0.001\;m=1\;mm$, which is not really impossible to obtain.

For designing a large opening, curvature is not the only factor. The amount of
opening is the function of areas, not the curvature. Now, let $A_{T}$ denote
the area of the isosceles triangle formed under $c_{1}$ shown in Figure 10(b),
whose length is given as $c_{1}=2rsin\theta$, then $A_{T}=r^{2}sin\theta$ and
let $A_{C}$ denote the area of the circle bounded yellow strip and the two
lines, then $A_{C}=r^{2}\theta$. Let $\Delta_{A}$ denote the open area, which
is the difference of the areas $A_{C}$ and $A_{T}$ as

$\Delta_{A}=A_{C}-A_{T}=r^{2}\;\;(\theta-\sin\theta)$

This is another interesting formula that demonstrates we do not need high
curvature to obtain large openings since the size of the opening is directly
proportional to $r^{2}$ and $(\theta-\sin\theta)$. In other words, to obtain
larger openings, we need to increase either (1) $r^{2}$ and (2) $\theta$ or
both of them.

In the previous example, let us assume that in addition to $r=1.00\;m$ we
obtain $\theta=0.5\;\mbox{radian}\approx 30\;\mbox{degree}$, then the opening
is computed as $\Delta_{A}=0.500-0.479\approx 0.02\;m^{2}$. To visualize the
area of this opening, try to mentally construct a square with sides $15\;cm$,
which is significantly large, much more than we need. If
$\theta=0.14\;\mbox{radian}$, which is approximately $10$ degree, then the
opening is computed as $\Delta_{A}=0.100-0.099\approx 0.001\;m^{2}$ To
visualize the area of this opening, try to mentally construct a square with
sides $3\;cm$, which is still significantly large. This demonstrates that it
is possible to obtain the desired amount of openings in most cases based on
environmental requirements for self-regulation.

###  4.1 The Effect of the Back Side

We also need to consider the effect of the back side in creating curved
pieces. The back side consists of $n-1$ strips. Each strip must have a width
$d/a$ when they are not curved where $a=n-1$. When they are curved the width
of the blue strip becomes $d_{0}/a$ and the width of the yellow strip becomes
$d_{1}/a$ by assuming that we use exactly the same materials. We still assume
that the shape of the thinner side of the strip is a circle segment. Since the
equation is independent of the width of the strip, if we use the same
materials with the same thickness, we obtain the same curvature $\kappa$ and
the same radius $r$.

Now, let $\theta^{\prime}$ denote the new angle and it is given
$d_{1}/a=r\theta^{\prime}$. Since the original $\theta=d_{1}/r$, the new angle
can be computed in terms of the original angle as $\theta^{\prime}=\theta/a$.
Note that there are $a$ of them and each one of them has a width
$c_{\mbox{thinner}}=2rsin\theta/2a$. Therefore, to keep both sides the same,
we need to minimize $c_{\mbox{longer}}-ac_{\mbox{thinner}}$. As a result, the
following formula must be minimized if the result is negative. Small positive
numbers are acceptable since this means an additional force is applied to keep
the structure curved. By noticing that $r=d_{1}/theta$ we obtain:

$d_{1}\left(\frac{2}{\theta}\sin\frac{\theta}{2}-\frac{2a}{\theta}\sin\frac{\theta}{2a}\right)$

#### Remark 4:

Each term in this equation are in the form of $\sin\theta/\theta$, therefore,
they both go to $1$ when $\theta$ goes to $0$. The function minimizes, but,
this is not really good for our application, since when $\theta$ goes to $0$
$r$ goes to infinity and curvature becomes $0$, which corresponds to a
straight line, which we do not want.

#### Remark 5:

We want to point out that the formulas are in the form of $\sin
0.5k\theta/(0.5k\theta)$, where $k$ is any positive integer. In other words,
we really deal with at most half of the original $\theta$. This makes the
difference smaller. For instance, For instance, for $\theta=10\mbox{degree}$,
$0.5\theta=0.075\mbox{radian}$, and $\sin 0.5\theta=0.074929$. Let us assume
that $a=10$ and $0.5\theta/a=0.0075\mbox{radian}$, then
$\sin\theta/a=0.0074999$. The difference gives us

$d_{1}\left(\frac{0.074929}{0.075}-\frac{0.0074999}{0.0075}\right)=d_{1}(0.99905333-0.99998666)\approx-0.001d_{1}$

In other words, in this example, there will still be some force to make the
structure less curved. Fortunately, this reactive force will be smaller than
the bending force.

#### Remark 6:

Increasing $a$ does not help since the second term goes to $1$ by increasing
$a$, and, as a result we get bigger difference. Therefore, it is better to
reduce $a$. However, even reducing $a$ does not make much difference. For
instance, in the previous example, let us change $a$ to $5$, then
$0.5\theta/a=0.015\mbox{radian}$ and $\sin\theta/a=0.014999$. Now, the
difference is computed as follows

$d_{1}\left(\frac{0.074929}{0.075}-\frac{0.014999}{0.015}\right)=d_{1}(0.99905333-9999625)\approx-0.001d_{1}$

In other words, optimization will be a challenging problem since we deal with
both continuous and discrete elements.

##  5 Conclusion and Future Work

In this paper, we have introduced a theoretical framework to design self-
regulatory non-woven modules that can play the role of stomata in buildings.
There is still a need for a significant amount of both theoretical and
experimental work to turn this conceptual framework into practical self-
regulating structures. We hope this paper will provide a strong interest in
using non-woven structures in architecture and construction industries.

This work only investigates the case of planar surfaces. The recent work on
cyclic woven surfaces demonstrated that any surface can be covered with woven
structures Akleman et al. (2009, 2011); Xing et al. (2010); Akleman et al.
(2020, 2015). It is straightforward to extend that work to non-woven cases.
There already exists work to create woven tiles that can be used as stomata
structures Yildiz et al. (2023); Krishnamurthy et al. (2021).

This work was partially supported by the National Science Foundation under
Grants NSF-CMMI-EAGER Award number 154824.

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