Quantum Implementation of RSA Crypto-algorithm using IBM-QISKIT

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 10 Issue: 04 | Apr 2023 www.irjet.net p-ISSN: 2395-0072
Quantum Implementation of RSA Crypto-algorithm using IBM-QISKIT
Pallavi Verma1, Dr. Palla Penchalaiah 2
1Student, VIII Semester, B.Tech(ECE), VIT University, Vellore, Tamil Nadu, India
(E-mail: pallavi.verma2019@vitstudent.ac.in)
2Associate Professor, Dept. of Micro & Nano Electronics, VIT University, Vellore, Tamil Nadu, India
---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - Quantum computers have the potential to break
classical RSA encryption, which could lead to the loss of
sensitive information, financial data, and other confidential
information. To address this issue, a new generation of
cryptography called quantum cryptography has emerged.
Quantum cryptography exploits the principles of quantum
mechanics for encryption & decryption, making it impossible
for hackers to break. However, the implementation of
quantum cryptography is still in its early stages. The
motivation behind this project is to explore the feasibility of
using quantum computing to enhance the security of
traditional cryptographic techniques. The project aims to
implement the RSA encryption algorithm on a quantum
computer such as IBM-QISKIT platform. This will evince the
potential of quantum computing in the field of cryptography.
Specifically, the aim is to investigate and compare the
efficiency, accuracy, and security of potential different
methods forquantumimplementationofRSA:(1)Montgomery
multiplication, (2) Chinese remainder theorem, (3) Shor’s
Algorithm.
Key Words: Quantum Computing, IBM-QISKIT, RSA,
Chinese Remainder Theorem, Montgomery
Multiplication, Shor’s Algorithm
1. INTRODUCTION
Quantum computing is a game-changing technology that
promises to revolutionize the world of computing as we
know it. Traditional computers work with binary digits,
known as bits, which can be either 0 or1.However, quantum
computers use quantum bits, or qubits, which can exist in
multiple states simultaneously. This property of qubits
allows quantum computers to perform certain calculations
exponentially faster than classical computers, making them
ideal for solving complex problems in fields such as
cryptography, drug discovery, and artificial intelligence. The
RSA algorithm is a widely used and trusted encryption
method that relies on the difficulty of factoring large
composite numbers. However, the security of RSA can be
compromised by quantum computers, which can efficiently
factor such numbers using Shor's algorithm. Tocounterthis,
there has been growing interest in implementing RSA using
quantum computing techniques, which can provide an
additional layer of security against quantum attacks.
In this context, the quantum implementation of RSA using
Qiskit(IBM) has emerged as a promising approach. By
leveraging the power of quantum mechanics, Qiskit can
provide efficient solutions for the complex mathematical
operations required by RSA. This quantum implementation
of RSA using Qiskit has the potential to enhance the security
of data transmission and storage, and pave the way for the
development of next-generation cryptography.[1]
1.1 What is a QUBIT?
In quantum computing, a qubit or quantum bit is a basic unit
of quantum information—the quantum version of theclassic
binary bit physically realizedwith atwo-statedevice. Aqubit
isatwo-state(ortwo-level)quantum-mechanicalsystem,one
of the simplest quantumsystemsdisplayingthepeculiarityof
quantum mechanics. Examples include the spin of the
electron in which the two levels can be taken as spin up and
spin down; or the polarization of a single photoninwhichthe
two states can be taken to be the vertical polarizationandthe
horizontal polarization.[2]
In a classical system, a bit would have to be in onestateorthe
other. However, quantum mechanicsallowsthequbittobein
a coherent superposition of both states simultaneously, a
property that is fundamental to quantum mechanics and
quantum computing. Inaddition to superposition,qubitscan
also exhibit a phenomenon called entanglement. This means
that the state of one qubit is directly related to the state of
another qubit, even if they are separated by large distances.
1.2 The BLOCH Sphere
The Bloch sphere is like a map of all the possible statesthata
single qubit can be in. Imagine a sphere, like a beach ball,
where each point on the surface represents a different state
of the qubit. The north pole of the sphere represents a qubit
that is definitely in the state "0", and the south pole
represents a qubit that is definitely in the state "1". All other
points on the sphere representa superpositionofthe"0"and
"1" states. The Bloch sphere is important because it helps us
to visualize and understand how qubits work. By looking at
the Bloch sphere, we can see how different quantum gates
(like the X, Y, and Z gates) affect the state of a qubit. We can
also see how measurements collapse the state of a qubit to
either "0" or "1".[3]
© 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 1282

2. The Quantum Phenomena
One of the most well-known quantum phenomena is
superposition, which means thata quantumobject,suchasan
atom or photon, can exist in multiplestatesorlocationsatthe
same time. This is different from classicalobjects,whichexist
in a single state or location at any given time. (This means
that an atom can be in two different states at the same time.)
Another quantum phenomenon is entanglement, which
occurs when two quantum objects become connected in a
way that their properties are correlated, even when they are
far apart from each other. This means that measuring the
property of one object will instantaneously affect the
property of the other object,even if they arelightyearsapart.
(Atoms can also become "entangled" with each other. This
means that what happens to one atom can affect what
happens to another atom, no matter how far apart they are.)
Another phenomenon is quantum tunneling, which is the
ability of a quantum object to pass through a potential
barrier, even if it does not have enough energy to overcome
the barrier according to classical physics. (When atoms are
in superposition, they can sometimes do things that seem
impossible. For example, theycanpassthroughsolidobjects,
like walls, without breakingthem.Thisiscalled"tunnelling")
Quantum mechanics also involves, which is describedbythe
Heisenberg uncertainty principle. This principle states that
the more precisely the position of a particle is known, the
less precisely its momentum can be known, and vice versa.
(It is impossible to simultaneously know the exact position
and the momentum of the particle.)[4]
3. The RSA Algorithm
RSA (Rivest-Shamir-Adleman) is a public-key cryptosystem
used for secure data transmission over the internet. It is one
of the most widely used encryption algorithms,andisusedto
encrypt sensitive information such as credit card numbers
and passwords. The RSA algorithm uses two prime numbers
to generate a public and private key pair. The security of the
algorithm relies on the difficulty of factoring large numbers
into their prime factors. Public Key encryption algorithm is
alsocalledtheAsymmetricalgorithm.Asymmetricalgorithms
are those algorithms in which sender and receiver use
different keys for encryption and decryption. Each sender is
assigned a pair of keys: Public Key & Private Key. The Public
key is used for encryption, and the Private Key is used for
decryption. Decryption cannot be done using a public key.
The two keys are linked, but the private key cannot be
derived from the public key. The public key is well known,
but the private key is secret and it is known only to the user
who owns the key. It means that everybody can send a
message to the user using user's public key. But only theuser
can decrypt the message using his private key
Choose two prime numbers p and q. These are kept secret
and are used to generate the public and private key pairs.
Compute n = p * q. This is the modulus and is part of the
public key. Euler's totient function phi(n) = (p-1)*(q-1).
Choose an integer e such that 1 < e <phi(n)andgcd(e,phi(n))
= 1. This is the public key. Compute d such that d*e = 1 mod
phi(n). This is the private key. To encrypt a message, the
sender converts the message into a number m less than n,
and raises it to the power of e mod n. To decryptthemessage,
the receiver raises the encrypted message to the power of d
mod n.[5]
4. The IBM-QISKIT
IBM Qiskit is a powerful open-source software development
kit for building quantum computing applications. It is one of
the most popular and widely used platforms for creating,
simulating, and executing quantum programs. With Qiskit,
users can write quantum algorithms and execute them on
real quantum devices provided by IBM or on simulators that
emulate the behavior of quantum systems. Qiskit also
includes a variety of powerful tools for visualizing quantum
circuits and analyzing the results of quantum computations.
Whether you are a seasoned quantum computing expert or
just starting out, Qiskit provides a robust set of tools and
resources to help you explore the exciting world of quantum
computing. Qiskit consists of four main components: Terra,
Aer, Ignis, and Aqua.[6]
6. Implementation of Quantum RSA methods
Modular exponentiation: The most straightforward way to
implement RSA is by using modular exponentiation. The key
generation algorithm generates two large prime numbers, p
and q, and calculates their product, N = p * q.Itthenchoosesa
number e such that 1 < e < (p-1)(q-1)andgcd(e,(p-1)(q-1))=
1. The public key is (N, e), and the private key is d, where d is
the modular inverse of e modulo (p-1)*(q-1). To encrypt a
message, m, the sender calculates c = mê mod N, and to
decrypt it, the receiver calculates m = c^d mod N.
Chinese remaindertheorem:AnotherwaytoimplementRSAis
by using the Chinese remainder theorem. This method is
faster than modular exponentiation, especially when the
numbers involved are very large. The key generation
algorithm is the sameas in the previousmethod.Toencrypta
message, m, the sender first calculates m1 = m mod p and m2
= m mod q. The sender then calculates c1 = m1ê mod p and
c2 = m2ê mod q. The sender then uses the Chinese
remainder theorem to calculate c such that c ≡ c1 (mod p)
and c ≡ c2 (mod q). To decrypt the message, the receiver
calculates d1 = d mod (p-1) and d2 = d mod (q-1). The
receiver then calculates m1 = c^d1 mod p and m2 = c^d2
mod q. The receiver then uses the Chinese remainder
theorem to calculate m such that m ≡ m1 (mod p) and m ≡
m2 (mod q).

Montgomery multiplication: Montgomery multiplication is a
technique used to speed up modular multiplication. It is
based on the observation that if we use the Montgomery
reduction algorithm to reduce the result of a multiplication,
then the result is already in Montgomery form, and we can
skip the final step of the algorithm. This saves time and
reduces the number of modular reductions that need to be
performed. This method is used in hardware
implementations of RSA, as it is faster than other methods.
5. The IBM-QISKIT Code
5.1 Chinese Remainder Theorem
Let's consider a system of two linear congruences:
x ≡ a mod p x ≡ b mod q, where p and q are distinct prime
numbers. The CRT tells us that there is a uniquesolutionforx
modulo pq, which can be obtained using the following steps:
1. Compute the product N = pq.
2. Find the modular inverses y and z of p and q, respectively,
such that y ≡ 1 mod p and z ≡ 1 mod q. This can be done
using the extended Euclidean algorithm.
3. Compute the values u and v, where u ≡ ay mod p and v ≡
bx mod q.
4. The solution for x is given by x ≡ (uqz + vp y) mod N.
In general, the CRT can be extended to solvesystemsoflinear
congruences withanynumberofequations,providedthatthe
moduli are pairwise coprime (i.e., they have no common
factors other than 1).[7]
Fig -1: CRT Code Snippet_1
Fig -2: CRT Code Snippet_2
Result:
The input integers are [2, 3, 4, 5]
The corresponding constants are [3, 5, 7, 11]
The moduli are [5, 7, 9, 11]
The solutions are:
x = 12640 mod 3465
5.2 Montgomery Multiplication
Generate two large prime numbers, p and q, and calculate
their product N = pq. This is the modulus for RSA encryption.
Choose an integer e such that 1 < e < (p-1)(q-1) and e is
coprime with (p-1)(q-1). This is the public key. Calculate the
modular inverse d of e modulo (p-1)(q-1). This is the private
key. Choose a random message m that is less than N. Convert
the message m to its Montgomery representation, which
involves multiplying it by a certainfactorandthenreducingit
modulo N. Encrypt the Montgomery representation of the
message m using the public key e, which involves raisingitto
the power e and reducing the result modulo N. Decrypt the
encrypted Montgomery representation of the message using
the private key d, which involves raising it tothepowerdand
reducing the result modulo N. Convert the decrypted
Montgomery representation of the message back to its
original representation.[8]

Fig -3: Montgomery Multiplication Code Snippet_1
Fig -4: Montgomery Multiplication Code Snippet_2
Result:
The first line of the output shows the result of Montgomery
multiplication of 123456 and 654321 modulo 1000000007.
The result is 128054480.
The second line of the output shows the result of
Montgomery exponentiation of 123456 to the power of
123456789 modulo 1000000007. The result is 406267049.
5.3 Shor’s Algorithm
In quantum computing, a qubit or quantum bit is a basic unit
of quantum information—the quantum version of theclassic
binary bit physically realizedwith a two-statedevice. Aqubit
isatwo-state(ortwo-level)quantum-mechanicalsystem,one
of the simplest quantumsystemsdisplayingthepeculiarityof
quantum mechanics. Examples include the spin of the
electron in which the two levels can be taken as spin up and
spin down; or the polarization of a single photoninwhichthe
two states can be taken to be the vertical polarizationandthe
horizontal polarization.[9][10][11]
Fig -5: Shor’s Algorithm Code Snippet_1

Result:
ATTEMPT 1:
Register Reading: 11000000
Corresponding Phase: 0.75
Result: r = 4
Guessed Factors: 3 and 5
*** Non-trivial factor found: {guess} ***
*** Non-trivial factor found: {guess} ***
6. CONCLUSIONS
The cryptography fieldisextensivelyresearcheduponin this
project. The potential of classical and quantum computer
and their phenomena was compared and observed. The
observation leads to the belief that quantumcryptographyis
much powerful than the traditional cryptography. This is
possible due to the fact of various potential quantum
phenomenon such superposition, entanglement, tunneling
etc.
The IBM Quantum Dashboard was accessed in order to
achieve the desired results in the project. The project was
initialized with the study of basic quantum gates and then
the same was coded in the IBM-QISKIT shell (refer to the
attached images). The histogram plots clearly explain the
presence of superposition in some case and entanglement
such as the Hadamard gate implementation & also the other
phenomena. The result might have some noise due the
quantum servers but effects are to the minimum.
The work was successful in demystifying and introducing
readers to classical and quantumcryptographyaswell asthe
fundamental concepts behind encryption. The significant
pertinent studies in the various branches of classical and
quantum cryptography were examined.
The goal of the project is to inspire to learn more about
quantum research and to serve as a solid starting point for
those just entering the subject of quantum cryptography.
Moreover, while completing theproject,itwasobservedthat
quantum cryptography for sure is a level betterthatclassical
cryptography; but the amount of development needed in
order to realize it in the practical world is still under-
developed. Be it the software or the hardware perspective
both needs their own share of development and progress.
REFERENCES
[1] Aumasson, Jean-Philippe(2017).Theimpactofquantum
computing on cryptography. Computer Fraud &
Security, 2017(6), 8–11. doi:10.1016/S1361-
3723(17)30051-9.
[2] Mavroeidis, V., Vishi, K., Zych, M. D., & Jøsang, A. (2018).
The impact of quantum computing on present
cryptography. arXiv preprint arXiv:1804.00200.
[3] Everitt, H. O. (Ed.). (2005). Experimental aspects of
quantum computing. Springer Science+Business Media.
[4] V. Padamvathi, B. V. Vardhan and A. V. N. Krishna,
"Quantum Cryptography and QuantumKeyDistribution
Protocols: A Survey," 2016 IEEE 6th International
Conference on Advanced Computing (IACC),
Bhimavaram, India, 2016, pp. 556-562, doi:
10.1109/IACC.2016.109.
[5] I.B. Djordjevic. “Conventional Cryptography
Fundamentals.” In: Physical-Layer Security and
Quantum Key Distribution.Springer,Cham.2019,pp65-
91.
[6] IBM Quantum, https://learn.qiskit.org/course.
[7] Salifu, Abdul-Mumin. (2018). Rivest Shamir Adleman
Encryption Scheme Based on the Chinese Remainder
Theorem. Advances in Networks. 6. 40.
10.11648/j.net.20180601.14.
[8] Jang, K., Song, G. J., Kim, H., Kwon, H., Lee, W. K., Hu, Z., &
Seo, H. (2021). Binary field montgomery multiplication
on quantum computers. Cryptology ePrint Archive.
[9] Chouhan, N., Saini, H. K., & Jain, S. C. (2017, February). A
novel technique to modify the SHOR'S algorithm—
Scaling the encryption scheme. In 2017 Second
International Conference on Electrical, Computer and
Communication Technologies (ICECCT) (pp. 1-4)..
[10] Experimental demonstration of Shor’s algorithm with
quantum entanglement B. P. Lanyon,T.J. Weinhold,N.K.
Langford, M. Barbieri, D. F. V. James∗ , A. Gilchrist, andA.
G. White Centre for Quantum Computer Technology
Department of Physics University of Queensland,
Brisbane QLD 4072, Australia ∗Department of Physics
Center for Quantum Information and Control University
of Toronto, Toronto ON M5S1A7, Canada.
[11] Gerjuoy, E. (2005). Shor’s factoring algorithm and
modern cryptography. An illustration of the capabilities
inherent in quantum computers. American journal of
physics, 73(6), 521-540.

Quantum Implementation of RSA Crypto-algorithm using IBM-QISKIT

More Related Content

Similar to Quantum Implementation of RSA Crypto-algorithm using IBM-QISKIT (20)

More from IRJET Journal (20)

Recently uploaded (20)

Quantum Implementation of RSA Crypto-algorithm using IBM-QISKIT