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)

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

IntroductionComputer 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

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.

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

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

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.

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).

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

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

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, &

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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