Quantum Security using Property of a Quantum Wave Function Wafa Elmannai, Khaled Elleithy, Varun Pande, Elham Geddeda Department of Computer Science and Engineering University of Bridgeport, USA
[email protected],
[email protected],
[email protected],
[email protected], Abstract- security over communication is totally essential especially for critical applications such as Military, Education and Financial applications. Unfortunately, many security mechanisms can be broken with new developments in classical computers. Quantum computers are capable of performing high speed simultaneous computations rather than sequentially as the case in classical computers do. In order to break a symmetric encryption algorithm, the attacker needs to find all the potential key combinations to get the right one. In this paper, we are introducing a secure protocol that considers the Qubit as a wave function. The beauty of the work is that the proposed protocol is based on time stamp that can only be broken by the correct set of time value, wave function, Qubit position, and other attributes such as: velocity, and phase of the Qubit. Moreover, using quantum tunneling makes the proposed protocol stronger by providing a very strong secure password protection mechanism using just one Qubit. Keyword: Quantum Security, Quantum Wave Function, Qubit position, quantum tunneling.
I.
INTRODUCTION
The implication of quantum mechanics has a new aspect in the physical world. In these new computing paradigms machines are composed of collection of particles whose behavior, in fact, are not deterministic but is probabilistic. Moreover, the true state of a particle cannot be known because the measurement of one aspect of the state may cause interference in the value of other aspects which is known as Heisenberg principal of uncertainty. This is considered as the basis of some security issues which are relevant to quantum communication system. The behavior of a state, although it is probabilistic, is also demonstrated as super position of values [1, 2]. In addition, the achievement of quantum computation enables us to perform calculations on all states at the same time. However, with algorithms which give a unique answer, we can obtain answers on quantum computer which is much faster than using classical computer [3]. The probabilities of the amplitude and quantum algorithms are considered as a result of obtaining the correct answer. Furthermore, the construction of quantum computer is considered as the source of quantum computing; for example, the central computation engine of a device is based on the concept of encoding information called qubits which its values
are not fixed. That can encode a superposition of states. Also, operations on quantum computer can be implemented on all states at the same time. However, quantum computer is a debatable topic due to its actual hardware implementation. The hardware implementation is based on simple quantum gates that make developing and constructing a large scale quantum computer practically overwhelming [4]. In this paper, we are presenting a new algorithm that is capable of securing data such as password or personal information by using the quantum wave function. Based on current literature, there are many security quantum mechanisms that are unsafe due to their dependency on the assumption that classical computers are not fast enough. However, the quantum security mechanisms are based on both the quantum approaches and encryption techniques which make it hard to break. “Quantum unclone theorem” and “Heisenberg inaccurate principle” could guarantee the quantum security [5]. In addition, Quantum computing is security verifiable and behaviorally predictable by several global encryption and physics groups due to its performance in detecting eavesdropper’s behavior. It is proposed in this paper is to introduce a new quantum algorithm that considers the Qubit as a wave particle in order to perform a stronger security over specific data. The algorithm is based on a stored stack of different attributes which makes it more secure than others. That is because in order to expose and match the data, all attributes in the stack need to be known. We are going to demonstrate in the following section the quantum wave function and how it is presented in such a case. A. Quantum Wave Function Overview: The function of quantum waves of a particle and in which way they act is discussed in this section. The quantum wave function is also known as quantum state, whereas its value is presented as a complex number [6]. Actually, every particle is a function of time and space and here exactly is where our protocol relies on. According to Schrodinger’s equation which discusses how a wave function transforms through time [7], the wave function is considered as water or string waves.
Therefore, this type of equation is a mathematical wave equation which points to the wave particle duality that locates a particle in a specified place and time. However, in Max Borns’ proposal, the wave function can be found by the probability density function. Furthermore, the wave function has other attributes like velocity field which depends on the actual position of N particle in the universe. The state of a particle at a given time is managed by two vectors; the first one is representing the position, and the other one is representing the momentum. Both determine a point of 6 – dimensional phase space; in other words, a particle that moves in only one diminution with position x and momentum p. Thus, the wave function depends on the position x. In addition, the physical significance of quantum wave function can be understood only if we know that they belong to a linear vector space. The linear vector space , which is called the Hilbert space of square- integral function, is discussed in functional analysis and mathematical physical. So, the study of physical interpretation of the wave function in the space represents a possible state of a quantum particle. II.
RELATED WORK
There are several attempts and experiments that have been carried out to secure the distribution of the secret key and the security in quantum computing. In this section, we summarize some of the successful attempts with our evaluation wherever possible. One of these attempts done by [8] was submitted in his thesis; its title, “Quantum Key Exchange”, is based on quantum mechanics. As the author claimed, security relies on fundamental laws of quantum mechanics. The first step in this work was the study of the “Squeezed States” which the author thinks is necessary to enable him to secure the quantum key exchange protocol. According to [10], the true transition that people need is achieved either with biometry or with digital and physical keys. Also, they found out that approaching physical keys has no pessimistic results but they are liable to hacking. The alterative solution according to [9] is application of Quantum Secure Authentication (Q S A) of none - reproducing conventional physical keys. They are applying “Quantum Secure Readout” that is limited in operation. So far an exploration has been done by experimental quantum that works with a large number of channels “represented by more than 1000 degrees with optical wave front and light modulation 13”. Such detailed process could prevent hacking which is continuously developing new techniques. However, the protocol of quantum readout of (PUF) technique challenges hackings by using non cloning property of quantum states to hide Physical Unclonable Function (PUF). Mosk et al. [11] used several steps such as highly robust cohered states of light with photon, a spatial light modulator (SLM1) and (SLM2). Also, they used a control peak in the analyzer plane for photon detection repeated the measurements 2000 times to improve the statistics. The researchers also used a histogram method to discriminate between authentic and immoral. They introduced the quantum
security parameter and photon; moreover, they used micro mirror based spatial high modulations. The results of this method shows that the readout system is rigorously protected. The paper provides security against hikers. The implementation to secure this protocol can be carried out under certain conditions and problems that occur can be solved by Quantum Key Exchange (QKE). Other implementations about securing communications by using quantum cryptography include new techniques for secure key distributions with signal photon transmissions which is called Quantum Key Distribution (QKD). The advantage of quantum cryptography offers “ease use” [11] and avoids the insider thread. Also, it avoids the harried and clumsy physical security parts of classical key distribution method. Another advantage of this method is that it provides a secure alternative to key distribution projects which are based on public key cryptography. III. PROPOSED ALGORITHM A. The Proposed Algorithm and Set Up: In this section, we are introducing a secure quantum algorithm to provide a high level of security over critical data. Our algorithm is based on Max Born’s and Louis de Broglie’s theorems. To demonstrate the security of this algorithm, we have implemented it using C sharp programming language. The idea of this algorithm is to take the wave function as the main input, the Qubit is considered as a wave particle, all over a time stamp that can only be broken by the correct set of time value, wave function, and Qubit position. Furthermore, the algorithm is based on quantum Tunneling [12]. The property of quantum tunneling is changed in our algorithmic approach using the phase velocity of one qubit particle. It has been discussed in literature that a state of a function in quantum of a particle acts quantitatively as wave function that transforms throughout the time. The equation of quality is a mathematical wave equation while density can be used to locate a particle in a given place and time, according to Max Born’s theory. Louis de Broglie invented a real value of wave function which is presented with complex wave function by a continuum of proportionality and created the De Broglie– Bohm theory. The atom velocity can be shown using the following equation: ( )
(
)(
)
In order for us to present many atoms, we consider Qk as kth particle and the following equation presents the speeds: ( )
(
)(
)
The actual position of those particles can affect their velocity fields; whereas, it can be encapsulated into an active
wave function. Thus, it provides us the ability to provide more security using one Qubit or a combination of more than one Qubit which makes the proposed algorithm stronger. Table 1 shows the initial values that we used in the implemented application. Actually, the attributes are the main input for one Qubit to present the quantum wave function. The values presented are only for one of our experimented examples.
match, the process is repeated until the desired result is achieved. Therefore, the desired results can only be achieved with the correct input data; otherwise, an infinite iterative loop occurs. This can demonstrate the strength of our algorithm for preventing compromising the password.
Table 1: Attributes of Quantum Wave Function Computation for One Qubit
Attribute
Value
Time
12:47:52:08
Wave Length ƛ
3
Frequency
3
Phase Speed
3X108
Energy
0.22
Ψ(x)
3.2745689
Φ(p)
0.9999
Spin
+1/2
B. The Experiment and the System Architecture: We have discussed in the previous sections that the proposed algorithm is designed to provide a high level of security over critical data. Therefore, our concern was to prevent attacking a user’s password over time using quantum computing efficiency. Passwords can be a combination of letters, numbers and symbols. Based on this data, the algorithm starts computing the quantum wave function as shown in Figure 1. As shown in Figure 1, we choose our data from an implemented database. In addition, we use the wave function and the given data in order to define the wave length. This information produces the frequency value and the phase. Then, we apply Schrodinger’s’ wave equation [13] to compute the particle state over the time: ( ) (
)
(
)
Where, is the imaginary part, is the h bar, is timedependent wave function, ( ) is the impending value and is the Hamiltonian operator. Based on our computation the wave function will be generated. More details are explained in the following section. In the next step of the algorithm we calculate the window size and prediction size. The algorithm starts to compare the output of the wave function that is displayed from a single Qubit to figure out whether or not they match. If they do not
Figure1: Flow Chart of the Proposed Secure Quantum Algorithm
C. The Results of the Experiment: In this section we present results of our implementation. We have used different phases and types of wave functions over the time versus the phases velocities as shown in Figure 2. Figure 2 shows the various wave functions that we use as input to determine the Qubit locations based on the time value. The used wave functions are sinusoid, sigmoid, parabola, growing sinusoid, and exponent. Based on the data that is displayed on the graph, we start to examine our application over all types of the wave function to show the efficiency of the application. Since our application stores the stack of input data, we can determine the specific probable location of a
Qubit using Schrodinger wave functions. The given values in the graph are real time positions and the probable positions are up to the computational algorithm. Figure 3 presents the application’s result for sine wave function over time where we now know the probable locations of the qubit in which the wave function has created. Also, the probable density and the velocity of each Qubit are known. Now we also know that in a certain period of space and time, the velocity, the phase’s value of the Qubit and its wavelength will be unique to that moment of space and time. Since time is a constant which is leading to infinity, the only input values of the data set leading to compute the location of a real time Qubit is done based on our novel approach. This algorithm allows us to look into the field of quantum particle which enabled the security. Therefore, as we can observe in Figure 3 we are going over infinite loop once we run the application unless the right set of data is matched. On the other hand, that makes it hard for an attacker to compromise the data. Figure 4 shows the matching of the real part and imaginary parts of the quantum wave function; where the right set of data was entered. IV. CONCLUSIONS The main objective of this paper is to a high level of security over sensitive data using a quantum wave function. To demonstrate the feasibility of this approach, we have used C sharp programming language to implement a secure algorithm. The proposed algorithm is a novel approach in looking at the way a Qubit behaves as a wave function. Every Qubit has a place in space and time in terms of wave particle duality. Schrodinger’s wave equation is used to define such a property of the wave function. Furthermore, Max Born theory is used in our algorithmic approach to define the probable location of Qubit in a wave function. Also, using De- Broglie’s theory, we can define the velocity of a Qubit.
Figure 2: The Wave Function Presentation to Determine the Location of Qubit
Figure 3: the Generation of Quantum Wave Function over the time
Moreover, the Qubit is measured as a wave particle over a time stamp which cannot be broken without the correct set of time value, wave function, Qubit position. Furthermore, our proposed algorithm is using quantum Tunneling. In summary, we presented a new approach for a security system using quantum based qubits. The advantages of such an approach can revolutionize the way we look at the system’s security replacing bounded key length Classical encryption algorithms. In addition, the proposed algorithm could prevent compromising passwords. Furthermore, the algorithm stores appropriate data of a Qubit to provide a secure feature in terms of its wave properties.
Figure 4: Matching of Real Part and Imaginary Part of Quantum Wave Function
REFERENCES [1] Bainbridge, William S., ed. Societal “Implications of nanoscience and nanotechnology”. Springer, 2001. [2] Niemiec, Marcin, and Andrzej R. Pach. "Management of security in quantum cryptography." Communications Magazine, IEEE 51, no. 8 (2013). [3] Scarani, Valerio, Helle Bechmann-Pasquinucci, Nicolas J. Cerf, Miloslav Dušek, Norbert Lütkenhaus, and Momtchil Peev. "The security of
practical quantum key distribution." Reviews of Modern Physics 81, no. 3 (2009): 1301. [4] Ursin, Rupert, and Richard Hughes. "Quantum information: Sharing quantum secrets." Nature 501, no. 7465 (2013): 37-38. [5] Baili, Sun, et al. "An Improved Method of Quantum Key Distribution Protocol."Computer Science-Technology and Applications, 2009. IFCSTA'09. International Forum on. Vol. 1. IEEE, 2009. [6] Campbell, Earl T., and Joseph Fitzsimons. "An introduction to one-way quantum computing in distributed architectures." International Journal of Quantum Information 8.01n02 (2010): 219-258. [7] Shepherd, James J., et al. "Convergence of many-body wave-function expansions using a plane-wave basis: From homogeneous electron gas to solid state systems." Physical Review B 86.3 (2012): 035111. [8] Poels, Karin, Pim Tuyls, and Berry Schoenmakers. "Generic security proof of quantum key exchange using squeezed states." Information Theory, 2005. ISIT 2005. Proceedings. International Symposium on. IEEE, 2005. [9] Vellekoop, I. M., and A. P. Mosk. "Focusing coherent light through opaque strongly scattering media." Optics letters 32.16 (2007): 23092311. [10] Goorden, Sebastianus A., Marcel Horstmann, Allard P. Mosk, Boris Škorić, and Pepijn WH Pinkse. "Quantum-Secure Authentication with a Classical Key."arXiv preprint arXiv:1303.0142 (2013). [11] Mosk, Allard P., Ad Lagendijk, Geoffroy Lerosey, and Mathias Fink. "Controlling waves in space and time for imaging and focusing in complex media." Nature photonics 6, no. 5 (2012): 283-292. [12] Datta, Ayan, and Sharmishta Karmakar. "Role of Quantum Mechanical Tunneling on the γ–effect of Silicon on Carbenes in 3Trimethylsilylcyclobutylidene." The Journal of Physical Chemistry B (2014). [13] Bortot, C. A., M. M. Cavalcanti, W. J. Corrêa, and V. N. Domingos Cavalcanti. "Uniform decay rate estimates for Schrödinger and plate equations with nonlinear locally distributed damping." Journal of Differential Equations 254, no. 9 (2013): 3729-3764.
AUTHORS Mrs. Wafa Elmannai is a PhD Candidate in the School of Engineering and Computer Science at the University of Bridgeport. She received her master degree from University Of Bridgeport, USA. She has received her Bachelor’s degree in Fall2005 at Ben Ashore College for Computer Science, Libya. She graduated with highest honors, ranking 4th in her Bachelor degree and awarded merit certificate. She also was recognized by both Phi Kappa Phi and Upsilon Pi Epsilon. She received a scholarship in 2012 based on her academic achievement from Upsilon Pi Epsilon. Mrs. Elmannai worked as a Research Assistant at University of Bridgeport from 2010-2011 and Graduate Assistance from 2012- 2014 at University of Bridgeport. She worked in the management department as assistant administrator in education department from 2005-2009 in Libya. She also worked as a teacher in Algamaheria School from 2004-2005.
Mrs. Elmannai is interested in network areas, mobile communications, and some software applications. She has published several papers in international journals and conferences. She has made presentation of her researches in different conferences.
Dr. Khaled Elleithy is the Associate Dean for Graduate Studies in the School of Engineering at the University of Bridgeport. He has research interests in the areas of network security, mobile communications, and formal approaches for design and verification. He has published more than two hundred and fifty research papers in international journals and conferences in his areas of expertise. Dr. Elleithy is the co-chair of the International Joint Conferences on Computer, Information, and Systems Sciences, and Engineering (CISSE). CISSE is the first Engineering/Computing and Systems Research E-Conference in the world to be completely conducted online in real-time via the internet and was successfully running for four years. Dr. Elleithy is the editor and co-editor of 12 books published by Springer for advances on Innovations and Advanced Techniques in Systems, Computing Sciences and Software. Mr. Varun Pande is a Graduate Research Assistant currently attending the University of Bridgeport as a PhD candidate in Computer Science and Engineering. He graduated from the University of Bridgeport with a Masters in Computer Science in May of 2012. He had worked as a CSR representative at TATA Power during his Bachelor in Computer Science and Information Technology. Currently and for the past two years, he has been a Graduate Assistant and taught Labs on Wireless Sensor Communication using MICA z Motes. His research interests are Computer Vision, Image Processing, Parallel processing and Wireless Sensor Networks. He hopes to share my experiences, research and knowledge with other graduates and professionals in order to work in a collaborative research for a Better tomorrow! Miss. Elham Geddeda was graduated from Faculty of Information Technology - Computer science - , and she worked as a teacher in her specialty for five years. Also, she is presently undergoing her Masters in the field of Computer Science at University of Bridgeport. She is interested in network area, mobile communication.
