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Blind Digital Signature Schemes with Four Particle

Entanglement States

Susrutha Babu

1

, Razan Abdulhammed

2

, Khaled Elleithy

3

, and Suparshya Babu

4

Computer Science and Engineering Department

School of Engineering

University of Bridgeport

Bridgeport, CT,06604,USA

ssukhava@my.bridgeport.edu

1

, rabdulha@my.bridgeport.edu

2

, elleithy@bridgeport.edu

3

, and susukhav@my.bridgeport.edu

4

Abstract-Quantum Digital Signature (QDS) schemes provides

authenticity of information that is guaranteed by quantum

mechanics. This paper compares the blind signature schemes

with two and three-bit quantum state bits and provides a

implementation for the four quantum states. We proposed a

technique where four partial entangled states are used to

implement the blind signature scheme. The proposed technique

improves the reliability factor of the scheme in terms of its

encryption and decryption using quantum cryptosystem for

interchanging the data in a protected mode. In the

communication system, the encryption of messages was done

with the help of the sender’s private key. The third party verifies

its authenticity along with blindness of the signature.

Furthermore, future research trends were presented with a

security analysis.

Keywords— Digital Signature; QBit; Enteglment state

I. INTRODUCTION

Digital signatures are used to protect electronic documents

against forgery, either by other parties or by one of the

participating parties [1]. The classical digital signatures are

assured through complex mathematical computation and

logarithmic laws. A digital signature is considered as one of the

cryptographic protocols. A simple digital signature protocol

involves three entities [3]: (1) signer, (2) receiver, and (3)

arbitrator. The arbitrator is responsible of managing the

authentication and validation of the signed message. There are

certain criteria that must be satisfied in any digital signature

scheme for the scheme to meet security conditions

requirements[2] , [17]:

• Each user (signer) generates his own signature on the

messages that he or she intends to send.

• Each receiver is efficiently able to verify if a given

string is a signature of a different user on certain

message with the aid of the Arbitrator.

• The signer cannot deny any responsibility or support for

the message that she or he has signed.

• It is not easy or convenient to reproduce the signatures

of another user, and fake messages that he or she has

not signed.

II. QUANTUM DIGITAL SIGNATURE

A Quantum Digital Signature (QDS) is a protocol that uses

a quantum computing theory to perform the same functions as

classical digital signatures. Hence, it is equivalent to the digital

signature of the classical system. QDS can assure the security

requirements for message integrity, message originate, and

signature disavowal by utilizing quantum mechanics and

quantum laws principles. From a general perspective, quantum

digital signature scheme has two phases; the messaging phase

and the distribution phase. In the messaging phase, the message

is actually signed and then sent. In the distribution phase, the

sender sends a quantum signature with a sequence of quantum

states to either one or multiple receivers based on the QDS

schemes. DS can be grouped into two categories. The first

creates a public Qubit keys out of a private classical bit string,

while the second creates a public Qubit keys out of a

private quantum bit string. Several digital signature approaches

have been presented in the literature. Throughout the scope of

this study, we notice that the literature lacks a complete

detailed analysis study that compares and contrast among QDS

schemes. The motivation behind this study is to investigate

current quantum digital signature schemes presented in the

literature; it also will provide a complete detailed comparative

analysis among the most commonly presented schemes, with

special emphasis on its limitations, requirements, and its

applicability to classical message system.

III. RELATED WORK

Quantum signatures are the contrivances for authorizing the

authenticity of a digital message which are analogous to

original analog one. The previous foremost classical digital

signature methods rely on public key cryptographic techniques.

After some observations, digital had more legitimate power

than regular signatures. There are various categories of digital

signatures including ring, un-deniable and blind signatures). In

the ring signature method, the authenticity of the original

signer will not be publicized in the crowd of all individual

signers. The un-deniable type makes some parts of the

certification process, which needs support of the signer in order

not to deny the validity of the signature.

The most prominent digital signature is the blind signature

in which protection will be present for the signature of the

individual, although not relating the signature to its original

signer. The discovery of quantum in the field of data

computation made all the previous digital signatures

susceptible to attacks and lack the author’s uniqueness. The

algorithm proposed utilizes the techniques of the quantum

systems to generate protocols in a secured manner. The safety

of the protocol mainly depends upon the presence of the

quantum one-way functions through the quantum principles.

This study takes a glance at the previous work related to

quantum digital signature.

Zeng et al. introduced one of the first QDS that can be

applied to classical messages. His protocol used quantum one-

way functions as the basic idea behind the proposed QDS.

Later, many researchers added their contributions to further

enhance the quantum digital signature and several approaches

have been presented [3-14]. We summarized the findings in

table 1.

T

ABLE

1:

COMMON

Q

UANTUM

D

IGITAL

S

IGNATURE

A

PPROACHES

Year Authors Technique

2009 Gottesman & Chuang Blind Signature based on One-way function

2001 Zeng & Christoph Blind Signature based on Greenberger-Horne

states

2010 Q. Li Blind Signature based on Bell states

2010 Su Qi Blind Signature based on Two-State Vector

Formalism

2012 Xun-Ru Yin Blind Signature based on Four-qubit χ-type

entangled state

The schemes presented in the cited work [16, 23] require a

quantum memory in order to implement the QDS protocol.

This makes the implementation process hard to achieve due to

the fact that a quantum computing memory cannot fulfill

lasting coherence times or can take a great amount of time

[24]; the time is usually in minutes. Nonetheless, the author of

the cited work [24]; presented a QDS that can be implemented

without a quantum memory.

Another quantum digital signature scheme is the quantum

one-way function. It is simple to calculate and hard to invert.

This employs a cryptographic protocol that includes three

entities: signatory, receiver, and the Trent. The arbitrator

authorizes and verifies the signature. The safety of the

signature method relies on the arbitrator’s trueness because

only this individual will have the right of entry to original

message content.

There are certain factors that should be taken in consideration

when examining and comparing the practicality of a digital

signature scheme [24].These include: the lengths of public

keys, the key exchange messages, the signatures, the private

key lifetime, the memory, the efficiency, and the computational

cost

.

IV. BLIND DIGITAL SIGNATURE SCHEMES OF FOUR PARTICLE

ENTANGLEMENT STATE

In this paper, we propose a quantum digital signature of

four entanglement states based on blind digital signature

scheme. Fig. 1 presents the process flow of the proposed

protocol.

The protocol starts with a key generation process, which in

turn includes the distribution of the keys (within the help of

quantum key distribution protocols such as BB84 or EPR

protocols) and also produces the signature key. It later ends

with verifications. Designing this type of protocol enables the

non-reproduction of quantum bits and non-invertible nature in

measuring the quantum data. The validity of the data is done

through error correction codes in quantum.

Apart from the blind signature schemes. which were

developed in classical computing, the quantum based signature

schemes give assurance with natural intrinsic security and the

reliability to the sender. Initially, quantum digital signature

scheme were introduced with quantum entanglement states;

later schemes were with the one way functions. These quantum

blind signature schemes were implemented with single qubits,

later improved to three qubits, and have been implemented

with x-type states.

In the implementation of blind signature (especially in

entangled states), Alice initially makes a string of qubits for

the sending message. Alice converts the message to a qubit,

which is represented in superposition of two states. The states

of quantum messages can be represented with the tensor

product between the qubits in the particular string of the

message. Second, Alice will convert the private key to a

sequence of measurement operators to measure the string of

qubits to form a secret string for encryption. The encrypted

message will then be sent to Bob; now Bob is expected to

make sign or validate the message. Bob will append his

information, although Bob does not know about the message

sent by Aice. In this process, Bob will make a new string of

qubits and encrypt it similarly to the process previously made

by Alice.

Combining the secret keys of both Alice, Bob with

measurement operators, the blind signature will be created by

Bob, then the encrypted message will be sent back to Alice.

For decryption, Alice will use the private key to get back the

quantum string that will be later checked with the original

message.

The third party will now be involved by means of receiving

the encrypted message from Alice and BOb for authentication.

In the process of authentication, the third party verifies by

comparing the original message and decrypted message. The

verification can be done by both senders after receiving from

third party.

Fig. 1 Process Flow

A. Communication Startup:

Let us start the distribution of secret keys Kab, Kbc, Kac to

Alice and Bob: kab will be the secret key that is utilized two

times between Alice and Bob particularly during encryption of

the Bob and decryption of Alice in the initialization of the

communication. In order to have the communication between

the third person Trend and Alice (and also between Bob and

Trend), the keys kac, and Kbc will be used. The key Ka from

Alice will be used for encryption of the received message that

was previously signed by Bob.

The representation of the communication among Alice,

Bob and Trend is shown in the Fig. 2 The message from Alice

(which should be signed by Bob) is interpreted as {X1, X2,…

Xm }, expressing that each message had n trits and X1 will be

selected as the initial try for the blind signature start up.

Fig. 2 Quantum Key Distribution

B. Mathematical model of writing the Blind Signature:

Four particle entangled qubit array is generated by Alice, in

the form of |ΨX1> for the demanding message. The demanding

message X1 will be converted by Alice into a four particle

entangled qubits array |ΨX1>, then

|ΨX1> = {|Ψ11>, |Ψ12>,…… |Ψ1j>,……..., |Ψ1n>}

Where |Ψ1j>= α0 |0> + α1|1> + α2|2>

α0, α1, α2 are complex numbers but α0 |0> + α1 |1> + α2

|2>=1

Alice, later produces a secret array |S> and the private key

is related to measurement operators where:

Ka = {|K1a>,|K2a>,…,|Kja>,…..,|Kna>}

and the measurement operators is defined by the

following:

XKa={ X1K1,X2K2,………XjKj,……..XnKn}

The key measurement operators are now used to measure

the secret array, and the obtained value is then compared with

the value of the signer to verify while signing.

|S>={XKa|ΨX1>}={ |s1>,….|s2>,…..|sj>,…….|sn>}

In order to sign the secret message, Bob incorporates the

private data in the message and produces his own individual

data |Ψi> that is represented as

|Ψi> ={ |Ψi1>, |Ψi2>,……..|Ψij>,……….|Ψin> }

Therefore, Alice does not know Bob’s individual data and

he can no longer access it. |Ψi> can be encrypted on by

utilizing kbc; this will be mixed with a series of measurement

operators XKbc, using Kbc key:

Kbc= {|k1 bc , |K2bc_ ,………..|Kjbc_ ,…… ….|Knbc_ }

Bob will verify his |Ψi> and obtains:

|I_ = XKbc{|I1_ , |I2_ ,…… |Ij_ ,……….. |In_ }

In order to have a quantum blind signature for the secret

demanding message, Bob utilizes Kab to encrypt |S> and |I_

for the blind signature :

Ub= kab (|S>,|I_ )

At last, Bob sends Ub to Alice and he will wait for the

verification of the signature.

Fig. 3 Quantum Communication Protocol for Blind signature

C. The Signature Checking Procedure:

The procedures, as illustrated in Fig 3, involve the

following steps: Initially, (1) Alice gets Ub (Bob message) and

she decrypts it utilizing the Kab. (2) Alice will now get |S’>

and |I’_, (3) she later obtains |Ψ’X1> (Message to be signed)

by (4) decrypting the |S’> utilizing the private key Ka.

Secondly (5)comparing the |Ψ’X1> to her |ΨX1>. If the

|Ψ’X1> is not equal to the |ΨX1>, this mean that the message

data is compromise with an attacker that trying to disclose part

of the secret data message. In this case, Alice will drop of the

compromised message and start over again. If both |Ψ’X1> and

|ΨX1> are equal, this mean that the message data are not

compromised. In addition, the blind signature has started.

Hence, (6)Bob sends |I> to Trent and this can be acquired by

encrypting |Ψi> utilizing XKbc, this is sent then by Bob to

Trent via a quantum channel because no one other than Bob

and Trend know |I >, |I’> is later sent by Alice to Trend. At

last, Trend has |I > and |I’> and can authenticate the signature.

He then (7)verifies that |I >= |I’> and decrypts |I > and |I’ >

using Kbc. Trend will have |Ψi> and |Ψ’i>, to verify if |I >= |I’

>, If |Ψ’X1> = |ΨX1> and |Ψ’i> = |Ψi>, then the trying blind

signature succeeded. Trend secures communication by sending

the message to Alice and Bob. Moreover, if one of the

conditions mentioned above failed, then there will be no

communication.

V. QUBIT ENCRYPTION AND DECRYPTION:

Fig. 3 shows the process of encryption and decryption, the

n qubits can be represented as

|Ψ> = { |Ψ

1

> |Ψ

2

>,……..|Ψ

m

>,………|Ψ

1n

> }Here, Eve represent an

attacker that tries to manipulate the signature. In the case of “without Eve

Enrolment”, this clarify that there is malicious action is in progress. Hence, the

values of Qbit before and after transformation of Alice data will be the same.

A. Without Eve Evolvement

TABLE I. :

B

EFORE

T

RANSMISSION OF

A

LICE DATA

Fig. 4 Qubit Encryption and Decryption

TABLE II. B

EFORE

T

RANSMISSION OF

B

OB DATA

TABLE III.

DECRYPTION

OF

ALICE

DATA

B. With Eve Evolvement.

TABLE IV. BEFORE

TRANSMISSION

OF

ALICE

DATA

T

ABLE

IIV:

B

EFORE

T

RANSMISSION OF

B

OB

D

ATA

T

ABLE

VX:

D

ECRYPTION OF

B

OB

D

ATA BY

T

RENT

VI. SECURITY ANALYSIS, DISCUSSION AND FUTURE WORK

In this section, we analyze our proposed algorithm and

provide a simple discussion regarding the security of the

algorithm. The simulated result shows that our algorithm is

able to protect against forgery, signature disavowal and denial

of signature, which mean our algorithm is secure.

0.1 0.2 0.3 0.3 0.2 0.1 0.2 0.2 0.3 0.1 0.2 0.2

0.2 0.1 0.4 0.4 0.4 0.3 0.1 0.4 0.1 0.2 0.3 0.2

0.3 0.3 0.2 0.1 0.1 0.4 0.3 0.1 0.2 0.4 0.1 0.1

0.4 0.4 0.1 0.2 0.3 0.2 0.4 0.3 0.4 0.3 0.4 0.5

0.1 0.2 0 .3 0.3 0.2 0.1 0.2 0.2 0.3 0.1 0.2 0.2

0.2 0.1 0.4 0.4 0.4 0.3 0.1 0.4 0.1 0.2 0.3 0.2

0.3 0.3 0.2 0.1 0.1 0.4 0.3 0.1 0.2 0.4 0.1 0.1

0.4 0.4 0.1 0.2 0.3 0.2 0.4 0.3 0.4 0.3 0.4 0.5

0.1 0.2 0.30 0.3 0.2 0.1 0 .2 0.2 0.3 0 .1 0.2 0.2

0.2 0.1 0.4 0.4 0.4 0.3 0.1 0.4 0.1 0.2 0.3 0.2

0.3 0.3 0.2 0.1 0.1 0.4 0.3 0.1 0.2 0.4 0.1 0.1

0.4 0.4 0.1 0.2 0.3 0.2 0.4 0.3 0.4 0.3 0.4

0.5

0.4 0.3 0 .3 0.4 0.3 0. 2 0.2 0.4 0 .2 0.1 0.2 0. 1

0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.1 0.3 0.3 0.2 0.2

0.2 0.3 0.2 0.2 0.3 0.3 0.2 0.4 0.2 0.3 0.2 0.5

0.2 0.2 0.3 0.2 0.2 0.3 0.3 0.1 0.3 0.3 0.4 0.2

0.4 0.3 0 .3 0.4 0.3 0 .2 0.2 0.4 0.2 0.1 0.2 0.1

0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.1 0.3 0.3 0.2 0.2

0.2 0.3 0.2 0.2 0.3 0.3 0.2 0.4 0.2 0.3 0.2 0.5

0.2 0.2 0.3 0.2 0.2 0.3 0.3 0.1 0.3 0.3 0.4 0.2

0.1 0.2 0.3 0.3 0.2 0.1 0.2 0.2 0.3 0.1 0.2 0.2

0.2 0.1 0.4 0.4 0.4 0.3 0.1 0.4 0.1 0.2 0.3 0.2

0.3 0.3 0.2 0.1 0.1 0.4 0.3 0.1 0.2 0.4 0.1 0.1

0.4 0.4 0.1 0.2 0.3 0.2 0.4 0.3 0.4 0.3 0.4 0.5

A. Protecting Against Forgery

In our algorithm, all the strings of four state Quantum bit

that construct the messages mentioned in the previous section

have four entanglement states. Therefore, it is more difficult

for the attacker to succeed. To illustrate, equation 1 is present

in the form of one qubit of four state. The Shannon binary

entropy can be calculated using the following equation:

E(p1j)=

…… 1

Since

Therefore, the maximum entropy can be achieved

when

Thus

E (p

1j

) =

.

This value represent the degree of uncertainty

.

As a result, if the attacker could capture one or more of

quantum bits parts, he or she will not be able to completely

determine the original state of quantum bits

.

B. Protecting Against Signature Disavowal

In this algorithm, the protection against signature disavowal

can be accomplished due to a third party that can check to see

if the added personal information of Bob represents his

signature or not; the third party thus can determine the

authenticity of the signature. Furthermore, the mechanism of

proxy signature makes Alice prevents Bob from disavowing

the last m − 1 signature. Thus, the signatory Bob cannot

disavow the signature

.

C. Protection Against Denial of Signature by the Receiver

In this algorithm, the protection against the denial of

signature by the receiver is fulfilled in the verification phase (in

which Alice get Bob’s signature and verifies it using the shared

key between her and Bob to obtain |T and |P). In our algorithm,

the third partner can determine if |P is real or fake. If it is real,

then the third partner informs Alice and Bob that the blind

signature is authentic. The process of signature will be

discontinued immediately if the trying signature is fake; this

means that neither Alice nor Bob can deny the signature in any

way.

VII. CONCLUSION

With the concept of one way function, we have

implemented the quantum digital signature scheme with two

entangled states and four entangled states in MATLAB by

utilizing the quantum library functions (such as knor, Cnot,

tensor product operations, controlled Not gate, and Hadmard

operations). The obtained results showed similarity between

the sent messages and received messages. Furthermore, this

study suggests to use Qubit of four X-Type states to implement

the same algorithm and compare and contrast the two

implementations. In the algorithm presented in this paper, we

used CNOT gate to add certain information to the original

message. For future work, we will investigate using different

circuits such as (Fredkin gates or Toffoli gate) and comparing

the two schemes.

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