QUANTUM COMPUTER IN CRYPTOGRAPHY

1. ACP
RESEARCH PAPER
V.E.S. COLLEGE OF ARTS, SCIENCE AND
COMMERCE
QUANTUM COMPUTER IN CRYPTOGRAPHY
YEAR 2014-15
ACP CO-ORDINATOR MENTOR NAME
Prof. Aarohi Khar Prof. Shrikant Ghodke
SUBMITTED BY
Akshay M Shelake
FROM
T.Y.B.Sc (Computer Science)

2. Abstract
With the introduction of quantum computing on the horizon computer security
organizations are stepping up research and development to defend against a new kind of
computer power. Quantum computers pose a very real threat to the global information
technology infrastructure of today. Many security implementations in use are based on
the difficulty for modern-day computers to perform large integer factorization. Utilizing
a specialized algorithm such as mathematician Peter Shor’s, a quantum computer can
compute large integer factoring in polynomial time versus classical computing’s sub-
exponential time. This theoretical exponential increase in computing speed has prompted
computer security experts around the world to begin preparing by devising new and
improved cryptography methods. If the proper measures are not in place by the time full-
scale quantum computers are being produced, the world’s governments and major
enterprises could suffer from security breaches and the loss of massive amounts of
encrypted data. Cryptographers are discussing alternatives to today’s methods and have
agreed that there are four major candidates that would provide immunity from a quantum
computer attack. The four possible replacement methods include: error-correcting codes,
hash-functions, lattice cryptography systems, and multivariate public-key cryptography
system
Introduction
Computer security and protecting valuable information has long been a delicate
subject in the world of information technology. With the development of the Internet,
companies had to ensure that customer data as well as their internal private data was
protected from outside intrusions; the secure socket layer (SSL) protocol was the first
step toward allowing for the secure transmission of information from client to server
and vice versa. Data encryption became a requirement for day-to-day operations of any
organization connected to the Internet and thus the world of big-business cryptography
exploded
Cryptography, while not a new practice, grew exponentially in popularity as
more computers came “online” and companies began to realize competing in the global
market of the Internet was becoming a necessity. Encryption of data for many IT
systems today relies on public-key cryptography. The concept of public-key
cryptography was introduced by Whitfield Diffie and Martin Hellman in 1976 (Diffie
& Hellman, 1976). This new method of encryption had two main purposes, encryption
and digital signatures. It entails that each person (or communicating system) gets a pair
of keys, one was dubbed the public key and the other was named the private key. The
public key is shared between the two parties and is used for identifying the end-user
while the private key remains a secret and is never transmitted. Encrypted information
is sent using the public key to identify the source but only a receiver that possesses the
private key is able to decode the message. Unfortunately, the private key, while kept a
secret from prying eyes, is linked to the public key through a mathematical algorithm

3. therefore presenting a weakness to this system. This “weakness” in the system became
less and less of an issue due to the complexity of being able to solve the algorithm for
common computer systems. Even when utilizing a “brute force” technique (a.k.a.
systematically trying every combination of letters, numbers, and symbols), a strongly
ciphered public-key encryption system remains untouchable by even today’s most
powerful computer systems. The complexity of arriving at a solution arises due to the
inability for today’s processors to factor increasingly larger and larger integers, the basis
for the strongest breeds of encryption to date. To put this into a clearer context,
researchers were able to successfully factor a 768-bit RSA modulus cipher using the
number field sieve factoring method which would have taken a single-core 2.2 GHz
AMD Opteron processor with 2 GB of RAM over 1500 years to process (Klein Jung, et
al., 2010). It is this length of time that makes breaking today’s most heavily encrypted
data such a daunting and near impossible endeavour.
Quantum computing presents the first serious risk associated with actually
providing a means to break the most sophisticated of encryption systems in use today.
Utilizing atoms as pseudo bits of digital information, binary processing can be achieved
in a much more sophisticated fashion than is currently possible. Quantum bits, or
“qubits”, can take on the value of 1 or 0 as the traditional digital bit does; the complexity
arises as these qubits can also take on the value for everything between 0 and 1 at the
same time. This ability allows for a single qubit to represent every possible value
between 0 and 1 simultaneously, thus permitting computations to be calculated in
parallel on every one of these values as well.
In this paper, we will explore the history of quantum computing theory and the
development of varying techniques researchers are using to accomplish building a fully
functional system today. We’ll also explore the practical applications for quantum
computers and how they could affect current cryptographic systems as well as how they
are shaping the development of new systems. Cryptography will remain a central focal
point throughout delving into varying theories and methods cryptography organizations
are testing and implementing to defend against massive computing power at an atomic
scale. Current methods of cryptography are in danger due to this unique type of threat
and security organizations are realizing that they can no longer sit idle waiting for the
first quantum computer to be produced before acting. Preparation must begin now if the
world is to have any hope of converting over from the encryption techniques that we
rely on every day to new and improved quantum computer proof methods of tomorrow.

4. The History of Quantum Computing
The idea of a quantum computer began in the early 1980s and was conceived by
Paul Benioff, Charles Bennett, David Deutsch, Richard Feynman, and Yuri Manin
(Bacon & Leung, 2007). The original idea of such a system was purely theoretical and
scientists were basing these ideas on years of research into quantum theory and
information science. Scientists speculated that if technology was to continue following
Moore’s Law (the observation that steady technological improvements in
miniaturization leads to a doubling of the density of transistors on new integrated
circuits every 18 months (Moore's law)) that eventually the size of circuitry on chips
would be reduced to the size of no more than a few atoms. At this size, the workings of
an integrated circuit would be governed by the laws of quantum mechanics and thus the
researchers began to question if a new kind of computer could be created based around
the study of quantum physics.
In 1982, Richard Feynman constructed an abstract model showing how a
quantum system would function and be used for doing computational work. Basically,
a classical bit in computing is used to represent two different states of an information
processing machine. One could refer to a bit as a type of light switch; rather, a bit can
be either “on” or “off”. When a bit is “off” it is said to have the value of 0 whereas when
it is “on” it is said to have the value of 1. Quantum computer bits, or “qubits”, can be a
0 or a 1 at the same time. A particle found to be in this quantum state is said to be in
“superposition”; however, the particle will take on a single location once someone or
something observes the particle. The speed advantages of quantum computers over
classical computers were not realized until the early 1990s when David Deutsch and
Richard Jozsa demonstrated that given a quantum computer utilizing the function.
f : {0,1}
n
→{0,1}, we can be assured that the function is either constant (0 or 1), or
balanced (returns 1 for half of the results and 0 for the other half) (Deutsch & Jozsa,
1992). Deutsch and Jozsa’s algorithm provided a man by the name of Peter Shor the
basis for constructing one of the most well-known and important quantum algorithms
the computing world had ever seen.
In 1994, mathematician Peter Shor formulated a quantum algorithm designed to
be used on a quantum computer to process integer factorization computations. Integer
factorization at its basis can be taken as “Given an integer N find its prime factors”.
Shor realized that utilizing his theory a quantum computer could “efficiently factor and
compute” the solution to large integer factoring problems (Bacon & Leung, 2007).
Shor’s quantum theory works in polynomial-time unlike the classical factoring
algorithm, the general number field sieve, which factors an L-bit number N in time
O(exp(cL1/3
log2/3
L)). The algorithm works by determining the period of the function
f(x) = ax
mod N where a is a random chosen number by the quantum computer with no
factors in common with N. After obtaining this period, using number-theoretic
techniques we can now factor N with a high probability of success (Shor, 1997). Using
this method for factoring numbers results in a significant exponential difference in time

5. versus the general number field sieve method with time being O((log N)3
). These
extreme differences in time between classical computers and Shor’s algorithm on a
quantum computer can be seen in Figure 1. The NFS curve on the left is data gathered
from a previous world record, factoring a 530-bit number in one month on 104 PCs and
workstations in 2003 (Van Meter, Itoh, & Ladd, 2005). The right curve is speculative
based on 1,000 times as much computing power of these classical computers which
works out to be around 100 PCs in 2018 based on Moore’s law. We can see how Shor’s
algorithm is much more efficient in factoring than anything that’s currently possible or
will be possible in the next decade.
Figure 1. Scaling of number field sieve (NFS) on classical computers and Shor’s algorithm for
factoring on a quantum computer, using Beckman-Chari-Devabhaktuni-Preskill modular
exponentiation with various clock rates. Both horizontal and vertical axes are log scale. The
horizontal axis is the size of the number being factored (Van Meter, Itoh, & Ladd, 2005).
In 2001, Shor’s algorithm was put to the test by IBM researchers using room
temperature liquid-state nuclear magnetic resonance techniques to manipulate nuclei in
a molecule as quantum bits (Vandersypen, Steffen, Breyta, Yannoni, Sherwood, &
Chuang, 2001). As insignificant as it sounds, the researchers, utilizing a very primitive
quantum computer, were able to apply Shor’s algorithm to successfully factor 15 giving
the results of 3 and 5. They noted that their experiment could be scaled to a much larger
system with more than the 7 qubits they utilized but that it was intended to simply
demonstrate the techniques for the control and modeling of quantum computers for the
future

6. Quantum cryptography has been at the forefront of purposes for developing
quantum computers since the early 1980s. Due to the way the qubits behave when
observed, it opened up the possibility of creating a new form of quantum
communication between two parties. Where before, transmission of messages relied on
the receiver having an encryption key to decode an encoded message, researchers were
able to utilize photons to send a message and detect whether the message had been
viewed along the way. While this method does not prevent an eavesdropper from
reading the message, it created a way for both the sender and receiver to know if the
message had been intercepted.
A cryptographic application of a quantum system was one of the earliest ideas
involved with quantum computation and can be accredited to Stephen Wiesner in the
1960s. Wiesner developed a theory that was meant to prevent counterfeiting of money
using the laws of physics as a basis for protection (Bacon & Leung, 2007). His method
relied on information that is encoded in quantum states thereby being able to prevent
any outside party from accessing said information without disturbing the state. This
property of quantum information has given birth to a new method of information
exchange and other companies are investing in it to develop new products giving users
the utmost security of knowing if their critical data has been intercepted or viewed by
an unintended audience outside the exchange.
In 2002 and 2003, a Swiss company called id Quantique and an American
company called MagiQ Technologies, both developed commercial communication
products leveraging this technology for message transmission and receipt (Bacon &
Leung, 2007). These two companies are noted as marketing the very first quantum key
distribution systems. This could be the preferred method for secure communication of
the future instead of relying on a receiver held private key utilized in systems based on
the famous RSA crypto architecture for example. Larger organizations are also starting
to invest in quantum technologies, such as Hewlett-Packard, Microsoft, IBM, Lucent,
Toshiba and NEC; each have active research programs exploring how quantum
cryptography can be leveraged into their future business models (Bacon & Leung,
2007).
Regarding the aforementioned past research on quantum systems, “the short-
range business concerns of these developments remains unclear at the moment, but
experience has shown that the industry needs many years to replace legacy systems –
you cannot easily change ATMs and mainframe applications,” says Andrea Simmons
of Computer Weekly (Simmons, 2009). At this point, it’s not really an “if quantum
computers come to be” it’s “when they come to be”, will we be prepared? Research is
happening right now and scientists are getting closer to understanding the hardest
questions plaguing quantum computers; in the next section, we’ll take a look at this
research and some of the concerns and issues scientists are dealing with today.

7. Quantum Computing Today
Many advances in research being conducted on the creation of quantum
computers have lead industry and educational sector experts to believe that the
technology could be just around the corner. “Quantum computational devices with
calculating power greater than any of today’s conventional computers could be just a
decade away, says Bristol University physicist and electrical engineer Mark
Thompson,” (Docksai, 2011). Thompson aided in the development of two quantum
photonic computer chips which process photons to provide what IBM accomplished in
2001 with factoring the integer 15. This shows how technology advancements over the
past ten years has allowed past research findings to be minimized into a form factor that
can be utilized inside a much more acceptable size for consumers.
Organizations delving into quantum computer development are faced with a
number of different methods for developing quantum systems. These varying methods
each have their advantages and disadvantages regarding factors such as scalability,
longevity, and accuracy. Current forerunners in the area include ion-trap quantum
computers and NMR (nuclear magnetic resonance) quantum computers. First, we’ll
look at the ion-trap method and the current research being done to overcome the
complexities of scaling this method.
Ion-trap quantum computation is currently being researched at the National
Institute of Standards and Technology. An ion-trap can be described as using a line of
N trapped ions where each ion has two stable or metastable states; there are also N laser
beam pairs each interacting with one of the ions (a qubit) (Steane, 1996). It is these laser
beams that essentially program the qubits by providing a pulsing form of a quantum
logic gate. This is very similar to how a transistor works by switching a classical bit
from 1 to 0 and vice versa. Scientists are looking into a “quantum charge-coupled
device” (QCCD) architecture which is essentially a large collection of these ion-traps.
The QCCD method was proposed as a possible solution to the limitations researchers
faced on scaling a single ion-trap to be able to confine a large number of ions. Incredible
complexities arise technologically and “scaling arguments suggest that this scheme is
limited to computations on tens of ions (Kielpinski, Monroe, & Wineland, 2002). The
advantage of the QCCD method is that by altering voltages of the ion-traps researchers
can confine a set number of ions in each trap or even transport ions to another trap
(Kielpinski, Monroe, & Wineland, 2002). This allows for the scaling of a multiple ion-
trap quantum computer to be achieved much easier. QCCD ion-trap quantum computers
still have a long way to go to be a feasible and scalable method for developing these
machines. If we look at the size of the very first computers taking up multiple rooms
and compare this with where these quantum computers are, we can see that in a few
decades these machines will follow Moore’s law and decrease substantially in size
while increasing in power. “Build the first one and in 25 years, they will be 25% of the
size. I bet that, after the first quantum computer, the cost of one 10 years later will be
significantly reduced,” (Docksai, 2011).

8. The United States Government is funding the research at NIST on QCCD ion-
trap quantum computers because they realize the implications that could arise should a
country gain quantum computational power before them. The country that first develops
a full-scale quantum computer will have the power to crumble current encryption
methods and expose an unbelievable amount of data in a very short amount of time.
Achieving the technology to build a fully functional quantum computer “is among the
great technological races of the 21st
century, a race whose results may profoundly alter
the manner in which we compute (Bacon & Leung, 2007). In the book, The Quest for
the Quantum Computer by Julian Brown and David Deutsch, the authors note that “if
anyone could build a full-scale quantum computer, it’s possible that he or she would be
able to access everything from your bank account to the Pentagon’s most secret files.
It’s no surprise, then, that significant funds backing this line of research have come from
such organizations as the U.S. Department of Defense, the National Security Agency,
NATO, and the European Union,” (Brown & Deutsch, 2000).
Another method being explored is based on nuclear magnetic resonance (NMR)
spectroscopy. The aforementioned research done at IBM factoring 15 was done using a
NMR based quantum computer. NMR computers rely on the two spin states of a spin-
1/2 atomic nucleus in a magnetic field; different atoms in a molecule can be singled out
and thus a molecule can actually be used as a quantum computer (Jones, 1998). Each
spin-1/2 nucleus provides a single qubit that is manipulated by multiple logic gates
provided by radio frequency fields. To increase the capabilities of this quantum
computer the scientists need only make these logic gates more complex. As their
complexity increases, the qubits can affect other qubits in the computer and thus allows
them to work together as a single computational system. The advantages of this method
arise when looking at the coherence of the qubits in the system. Coherence describes
the ability for the qubits to retain their states despite “noise” from the external
environment. “Slow decoherence is one of the primary attractions of an NMR quantum
computer,” (Gershenfeld & Chuang, 1997).
There are a plethora of different methods, other than the ones mentioned here,
that scientists are working on to bring quantum computing to the next level. As research
continues in these areas, we inch closer to the day that a fully functional quantum
processing machine is in our grasp. The latest research has yielded better error
correction with the qubit states, faster quantum algorithms, and led to the evolution of
decoherence and entanglement in quantum computing

9. Computer Security Organizations and Quantum Computing
Cryptography specialists are beginning to take notice that quantum computing
may not be too far away. Many of the IT industry’s top cryptography experts have
predicted that a full-scale quantum computer could manifest in as little as 10 years
(Heger, 2009). The PQCrypto (Post-Quantum Cryptography) conference was created in
2006 to address and discuss the dangers the world of computer security may face with
the successful creation of quantum computers. Researchers from all over the world
flocked to meet in Leuven, Belgium to begin discussions on possible alternatives to the
most widely used encryption systems. The two most popular encryption methods in use
today are RSA and elliptic-curve cryptography (ECC) (Heger, 2009). These two
methods both rely on public-key architecture and digital signatures to provide a means
to communicate securely between two parties in digital communication. As we’ve
covered earlier in this paper, the entire public-key architecture is at risk of being
obliterated by the computing power of a single, albeit low-powered, quantum computer.
To gain a better understanding of where experts see cryptography technology heading,
let’s take a look at a few of the quantum-proof methods that have been discussed at past
PQCrypto conferences.
The idea of Lamport signatures was first mentioned in a paper by Leslie Lamport
from SRI International on October 18, 1979 (Lamport, 1979) where he discusses the
use of one-way hash functions to generate signatures. The idea behind this type of
encryption is that a function can be created to be irreversible in nature. The Lamport
crypto-scheme is a one-time signature scheme, therefore each time a signed message is
generated a new signature must also be created. Utilizing a hash function to implement
this type of crypto system, the text of a message is efficiently reduced to a much shorter
string of bits which is the message signature (Heger, 2009). Researchers Ray Perlner
and David Cooper (2009) from the National Institute of Standards and Technology
explain the basics to the Lamport signature in their paper on quantum resistant public-
key cryptography:
In the simplest variant of Lamport signatures, the signer generates two high-
entropy secrets, S0,k and S1,k, for each bit location, k, in the message digest that will be
used for signatures. These secrets (2n secrets are required if the digest is n bits long)
comprise the private key. The public key consists of the images of the secrets under f,
i.e., f(S0,k) and f(S1,k), concatenated together in a prescribed order (lexicographically by
subscript for example). In order to sign a message, the signer reveals half of the secrets,
chosen as follows: if bit k is a zero, the secret S0,k is revealed, and if it is one, S1,k is
revealed. The revealed secrets, concatenated together, comprise the signature. (Perlner
& Cooper, 2009)
The reason the Lamport signature is a one-time used architecture is due to the
signing method actually revealing a small amount of information about the private key.
This leak of information is not, however, enough for an attacker to build and sign a
forged message but subsequent messages must be accompanied by a newly generated

10. key to remain secure. The performance of a system like the Lamport signature is purely
dependent on the one-way function the signer chooses to implement. The original
implementation was greatly improved by others as the original requirement of running
numerous hash functions to generate signatures grew exponentially (Merkle, 1988).
Another possible candidate to replace public-key systems comes in the form of
multivariate public-key cryptosystems (MPKCs) and bases its strength on multivariable
nonlinear equations (Heger, 2009). Quantum computers share a weakness with their
classical brethren when trying to solve problems said to be “NP-complete”. Jintai Ding
(2008), a top researcher in MPKCs, notes that “It’s difficult to explain what NP-
complete means, but it just means very, very difficult. It is exponential, meaning that as
the size of a problem increases, the time to solve it increases exponentially. And
quantum computers have not yet been able to defeat NP-complete types of problems,”
(IEEE Spectrum, 2008). A computing machine is only as powerful as the mathematics
behind said machine and when the mathematics has not yet been developed to solve a
problem with efficiency, the problem is said to be NP-hard. Whereas traditional RSA
type cryptosystems rely on mathematics developed in the 17
th and 18th centuries (number theory), MPKCs use 20th century algebraic
geometry for their basis (Ding & Schmidt, 2006). Ding and Schmidt (2006) state on
MKPCs that, “the method relies on the proven theorem that solving a set of
multivariable polynomial equations over a finite field is in general an NP-hard
problem,” (Ding & Schmidt, 2006).
PQCrypto attendees also discussed lattice based cryptography systems which
researchers believe can be implemented in a way that makes for solving the algorithm
an NP-complete problem. “An n-dimensional lattice is the set of vectors that can be
expressed as the sum of integer multiples of a specific set of n vectors, collectively
called the basis of the lattice,” (Perlner & Cooper, 2009). The NP-complete issue for
cracking this type of cryptography arises when increasing the dimensions of the lattices
and trying to solve the shortest vector problem (Ajtai, 1998) as well as the closest vector
problem (van Emde Boas, 1981). Both problems revolve around the difficulty of solving
for the shortest vector to a random non-lattice vector.
The fourth and final candidate being researched is encryption schemes based on
the use of error-correcting codes. Basically, the idea behind this type of encryption is
that the sender of the message encrypts the message with noise, or random additional
information, therefore obfuscating the original message. Only the receiver has the
ability to “sift” through the information to deduce the true content of the message. The
first error-correcting encryption scheme was devised by Robert J. McEliece about one
year after the RSA encryption technique was proposed (Joye, 2009). Many refer to
error-correcting encryption schemes as “code-based cryptography” and researchers
Bernstein, Lange, and Peters (2011) state that “code-based cryptography has lately
received a lot of attention because it is a good candidate for public-key cryptography
that remains secure against attacks by a quantum computer,” (Bernstein, Lange, &

11. Peters, 2011). A major drawback in utilizing this scheme is that the public key is too
large due to the excess data for efficient communication; however, research continues
in an attempt to minimize the required information for public key creation to make this
technique a viable solution.
Of these four completely different techniques, is there a single encryption
scheme that will provide security where the aging encryption schemes fail? Jintai Ding
(2008) states on the matter, “no, I cannot really specify one area. These four systems
are all very different and each has its own advantages and disadvantages,” (IEEE
Spectrum, 2008). Researchers from all over the world continue to meet at the PQCrypto
conference, held every few years, to discuss the latest findings on each of the
aforementioned methods as well as new potential candidates. While they may disagree
on when quantum computing is coming, they all seem to agree that there is a sense of
urgency to develop encryption schemes that will be ready for when it does:
Quantum cryptography may be five years ahead; quantum computing may be 15
years away. Progress in building quantum computers tends to develop in steps as
researchers find new methods, so it is slow, but the theory is solid. We should be starting
now to evaluate the impact. (Simmons, 2009)
CONCLUSION
Quantum cryptography ensure secure communication by providing security
based on the fundamental law of physics, intead of the current state of mathematical
algorithms or computing technology unlike classical encryption algorithm quantum
cryptography does not depend factoring large integers into primes but on the
fundamental principles of quantum physics. Quantum cryptography is more secure,
because an intruder is not able to replicate the photon to recreate the key.
Integrating QKD in TLS protocol will ensure financial transaction. Instead of
using RSA, in TLS protocol .We can use Quantum Cryptography securely exchange the
secret data and avoid an attack of intruder
ACKNOWLEDGMENT
I am using this opportunity to express my gratitude to everyone who supported
me throughout the course of this seminar report. I am thankful for their aspiring
guidance, invaluably constructive criticism and friendy advice during the seminar work.
I deeply express my sincere thanks to my guide Mr. Shrikant Ghodke for his support
and guidance.

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