BA, Oberlin College, 2018, Mathematics

At its core, cryptography relies on problems that are simple to construct but difficult to solve unless certain information (the “key”) is known. Many of these problems come from number theory and group theory. One method of obtaining groups from which to build cryptosystems is to define algebraic curves over finite fields and then derive a group structure from the set of points on those curves. This thesis serves as an exposition of Elliptic Curve Cryptography (ECC), preceded by a discussion of some basic cryptographic concepts and followed by a glance into one generalization of ECC: cryptosystems based on hyperelliptic curves.

Committee: Benjamin Linowitz (Advisor) Subjects: Computer Science; Mathematics

MS, Kent State University, 2024, College of Arts and Sciences / Department of Computer Science

Cryptography, derived from the Greek word meaning "to hide information," involves techniques for converting readable plaintext into unreadable ciphertext through a process called encryption. Cryptography algorithms are broadly categorized into two types: symmetric key cryptography and asymmetric key cryptography. Symmetric key cryptography is further divided into block ciphers and stream ciphers. Block ciphers, based on their structure, can be classified into two main categories: Substitution-Permutation Networks (SPN) and Feistel Networks (FN). This research focuses on SPN-based block ciphers. In 1949[1], Claude Shannon introduced two fundamental operations required for a robust cryptosystem: substitution and permutation. Substitution, the core component of SPN-based cryptography, is implemented through substitution boxes (S-Boxes), where each element in the plaintext is mapped to another element to achieve nonlinearity and provide the confusion property crucial for security. With the rise of constrained devices, such as the Internet of Things (IoT), there is an increasing demand for lightweight symmetric-key algorithms. However, in many cases, the S-Box contributes the most to the hardware complexity and computational load compared to other linear components. This research addresses this challenge by designing and optimizing a lightweight cryptosystem suitable for resource-limited environments. The thesis makes two key contributions to the field of lightweight cryptography. The first contribution is the development of chaos-based S-Boxes tailored for devices with restricted computational capabilities. By leveraging chaotic maps, the proposed S-Boxes achieve a high degree of nonlinearity and security while maintaining a minimal computational and hardware footprint, making them ideal for IoT and other constrained devices. These chaos-based S-Boxes introduce dynamic, unpredictable substitution patterns that enhance resistance to cryptanalysis techniques such as l (open full item for complete abstract)

Committee: Maha Allouzi Dr (Advisor); Younghun Chae Dr (Committee Member); Lei Xu Dr (Committee Member) Subjects: Computer Engineering; Computer Science

PhD, University of Cincinnati, 2021, Arts and Sciences: Mathematical Sciences

It has been known that the rapid development of large-scale quantum computers gives rise to threats to widely-deployed number theory based cryptography such as RSA, DSA, ECDH, etc. The goal of post-quantum cryptography is to develop cryptosystems that can resist quantum computer attacks. Multivariate public key cryptography is believed to be one of the choices for quantum-safe cryptography. At the end of 2017, 10 multivariate public key cryptosystems participated the round one of the National Institute of Standards and Technology (NIST) post-quantum standardization. The Himq-3 signature scheme proposed by Kyung-Ah Shim et al. is one of those NIST post-quantum standardization candidates. The Himq-3 signature scheme can be classified into the oil vinegar signature scheme family. Similar to the rainbow signature scheme, the Himq-3 signature scheme uses a multilayer structure to shorten the key size and the signature size. Moreover, the signing process is very fast due to a special system called L-invertible cycle system in its central map. The authors of the Himq-3 signature scheme claim that the scheme can resist all known attacks. The main result of this dissertation is a new attack method on the Himq-3 signature scheme. We will first discuss the urgency of post-quantum cryptography. Next multivariate public key cryptography will be introduced. We will also present some useful attacks on multivariate public key cryptography. Then the Himq-3 scheme will be described and the security against all known attacks will be analyzed. Afterward, we will show our new attack method called the singularity attack on the Himq-3 scheme and its variant Himq-3F. This new attack is based on the fact that some variables in the central map cannot be equal to zero in any valid signature. Based on this observation, we are able to filter out those linear combinations of variables that would be equal to zero for some signature, leaving the true ones we want provided that a large number of si (open full item for complete abstract)

Committee: Jintai Ding Ph.D. (Committee Chair); Seungki Kim (Committee Member); Benjamin Vaughan Ph.D. (Committee Member) Subjects: Mathematics

MS, University of Cincinnati, 2017, Engineering and Applied Science: Computer Engineering

This thesis focuses on elliptic curve arithmetic over the prime field GF (p) and elliptic curve cryptography (ECC). ECC over GF(p) has its own arithmetic which is done over elliptic curves of the form y2; ≡ x3;+ax+b (mod p), where p is prime. ECC is gaining importance in security because it uses smaller keys to provide the same security level as the popular RSA. It is the superior cryptographic scheme based on time efficiency and resource utilization. It is more suitable than RSA for DNSSEC and IoT systems and devices. Unlike RSA, which is easily understood, ECC is complicated because of the arithmetic involved. It is not widely understood. We provide a tutorial on elliptic curve arithmetic and also explain the working of the ElGamal cryptosystem. We also describe general hardware-efficient methods to implement ECC such as Montgomery multiplication and projective coordinates. These methods are challenging to understand. Essentially, projective coordinates help reduce the number of inversions required in doing scalar multiplication. If Montgomery multiplication is used, a time-consuming operation like reduction modulo a prime p can be simplified. In this work, we also present a user-friendly Java GUI application to provide education in elliptic curve arithmetic and its applications in cryptosystems. Lastly, we provide a module of questions and solutions to do the same and also enable senior students and graduate students to use ECC in their project work.

Committee: Carla Purdy Ph.D. (Committee Chair); Wen-Ben Jone Ph.D. (Committee Member); George Purdy Ph.D. (Committee Member) Subjects: Computer Engineering

Doctor of Philosophy, The Ohio State University, 2017, Electrical and Computer Engineering

Vehicular Adhoc Networks (VANETs) promises to empower the future autonomous vehicles with a cooperative awareness facility that will help in avoiding accidents and alleviating traffic congestion. The foreseen collective awareness requires the vehicles to communicate with their neighbors and with the infrastructure; such communication will need the fulfillment of many requirements such as security, privacy, and efficiency. The Dedicated Short-Range Communication (DSRC) standard has been formulated to afford these requisites. On one hand, when focusing on the application layer, DSRC adopts the successful Internet-based Public Key Infrastructure (PKI) framework to safeguard the vehicles. However, PKI alone cannot comprehensively meet all of the security and privacy requirements. On the other hand, the DSRC 's Medium Access Control (MAC) layer adopts the IEEE 802.11p access mode, which also needs augmentation to fulfill the efficiency of communication when collisions arise for safety beacons. Since many issues have not been well addressed in DSRC, academic, industrial, and governmental research has flourished over the last two decades to complement the standard. As being part of such large research community, we also have been incentivized to contribute with our own solutions. Our contributions have been ranging between two limits: either finding solutions to acclimate with the available DSRC shortcomings or disregarding the bias that DSRC has towards using only specific standards by bringing other alternative frameworks into scene. With the first direction in mind, our efforts are a mixture of high-level re-arrangement protocols such as grouping and overhead omissions to minimize the PKI and Carrier Sense Multiple Access - Collision Avoidance (CSMA/CA) privacy and efficiency shortcomings. For the other direction, we especially address the application layer level. Since some frameworks have small communication overhead while others have high anonymous traits, we have at (open full item for complete abstract)

Committee: Fusun Ozguner Professor (Advisor); Can Emre Koksal Professor (Committee Member); Xiaorui Wang Professor (Committee Member) Subjects: Computer Engineering; Computer Science; Electrical Engineering; Transportation

Master of Mathematical Sciences, The Ohio State University, 2015, Mathematics

Modern cryptography relies heavily on concepts from mathematics. In this thesis we will be discussing several cryptographic ciphers and discovering the mathematical applications which can be found by exploring them. This paper is intended to be accessible to undergraduate or graduate students as a supplement to a course in number theory or modern algebra. The structure of the paper also lends itself to be accessible to a person interested in learning about mathematics in cryptography on their own, since we will always give a review of the background material which will be needed before delving into the cryptographic ciphers.

Committee: James Cogdell (Advisor); Rodica Costin (Committee Member) Subjects: Mathematics; Mathematics Education

Doctor of Philosophy, Case Western Reserve University, 2009, EECS - Computer Engineering

In order to provide security in ubiquitous, passively powered systems, especially RFID tags in the supply chain, improved asymmetric key cryptographic processors are presented, tested and compared with others from the literature. The proposed processors show a 12%-20% area and a 31%-45% time improvement. A secure protocol is also presented to minimize cryptographic effort and communication between tag and reader. A set of power management techniques is also presented to match processor performance to available power, resulting in greater range and responsiveness of RFID tags.

Committee: Christos Papachristou PhD (Committee Chair); Francis L. Merat PhD (Committee Member); Swarup Bhunia PhD (Committee Member); Xinmiao Zhang PhD (Committee Member); Francis G. Wolff PhD (Committee Member) Subjects: Computer Science; Electrical Engineering

Master of Science, The Ohio State University, 2005, Graduate School

Committee: Not Provided (Other) Subjects:

Bachelor of Science (BS), Ohio University, 2024, Mathematics

The existence of post-quantum cryptography alludes to the existence of its predecessor: pre-quantum cryptography. In this thesis, we walk the reader through this progression from classical to quantum cryptography. We cover the two most promising forms of PQC and make it a point to emphasize the strength of lattices. Above all, we stress the gravity of encouraging the general public to be aware of this emerging field. In pursuing this goal, we are joining the consensus in academic circles about the importance of bringing this information to contemporary students. It is worth noting the efforts being made at Ohio University to introduce these topics into the curriculum.

Committee: Sergio Lopez-Permouth (Advisor) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2024, Physics

Quantum information science holds the promise of revolutionizing information processing and communication through advancements in quantum computing and quantum communications. Quantum key distribution (QKD) stands out as a method offering unconditional security in communications, particularly with the looming threat that quantum computers pose to traditional cryptographic systems. While existing QKD systems predominantly focus on long-distance communication, future quantum networks are likely to involve the integration of fixed nodes with highly mobile platforms to facilitate the ``last mile'' communication between users. In this dissertation, I present a full system design of a QKD system specifically developed for implementation on highly mobile platforms such as drones and cars. Designing a QKD system for highly mobile platforms poses a formidable challenge, demanding careful consideration of the trade-space of system performance and size, weight, and power (SWaP). In the first part of this thesis, I discuss the design considerations for developing such a platform, culminating in a system using a polarization-based prepare-and-measure BB84 QKD protocol with decoy states. This system is designed with two field-programmable gate arrays, three resonant-cavity LEDs, and off-the-shelf passive optics and detectors. The transmitter platform has a SWaP of 1803 cm^3, 2017 g, and 2453 mW. The receiver has a SWaP of 2728 cm^3, 2678 g, and 5886 mW. The usage of multiple sources can increase the risk of side channel attacks in a QKD system due to the indistinguishability of the quantum states' spectral, temporal and spatial degrees-of-freedom. Other similar studies, have relaxed system security requirements to achieve SWaP metrics with the idea that these attack vectors will be addressed in the future. In this work, great effort is dedicated to ensuring that all sources are as indistinguishable as possible. The remaining distinguishability is quantified as the mutual (open full item for complete abstract)

Committee: Daniel Gauthier (Advisor); Louis DiMauro (Committee Member); David Bromwich (Other); Douglass Schumacher (Committee Member); Jay Gupta (Committee Member) Subjects: Experiments; Physics; Quantum Physics

PhD, University of Cincinnati, 2022, Arts and Sciences: Mathematical Sciences

In this digital age, well tested public-key cryptography is vital for the continuing function of society. An example of one of the uses of cryptography is signature schemes which allow us to digitally sign a document. However, quantum computers utilizing Shor's algorithm threaten the security of all the cryptosystem currently in use. What is needed is post-quantum cryptography: classical cryptographic algorithms able to resist quantum attacks. In 2016, NIST put out a call for proposals for post-quantum cryptosystems for standardization. We are currently in the third round of the “competition,” with many different types of schemes being proposed. In 2017, Ward Beullens et al. submitted the Lifted Unbalanced Oil and Vinegar signature scheme to the NIST competition, which is a modification to the Unbalanced Oil and Vinegar Scheme by Patarin. The main modification is called lifting, which is to take a polynomial over a small finite field and view it as a map over some extension field. LUOV made it into the second round of the competition, but two attacks by Ding et al. showed a flaw in the modifications of LUOV. The first attack was the Subfield Differential Attack (SDA) which prompted a change of parameters by the authors of LUOV. The second was the Nested Subset Differential Attack (NSDA), which broke half of the parameters put forward by the authors of LUOV again. Due to the strengths of these attacks and the possibility stronger ones of a similar nature exist, LUOV did not go into the third round. This dissertation shows that such a stronger attack, which will be called NSDA+, is possible. All three of the attacks SDA, NSDA, and NSDA+ are straightforward but powerful in application against the lifting modification. First in chapter 1, we will discuss what is a public key cryptosystem by looking at the original definition of Diffie and Hellman. Then we will talk of the NIST Post-Quantum Standardizat (open full item for complete abstract)

Committee: Jintai Ding Ph.D. (Committee Member); Seungki Kim Ph.D. (Committee Member); Robert Buckingham Ph.D. (Committee Member) Subjects: Mathematics

Master of Sciences, Case Western Reserve University, 2021, EECS - Computer and Information Sciences

We prove some new results regarding binary and n-ary totally symmetric and medial (TSM) quasigroups and explore their applications to cubic curves and cryptography. We first generalize to the n-ary case Etherington's result that a product in binary TSM quasigroup is symmetric in factors whose depths differ by a multiple of 2 in the corresponding product tree. We then demonstrate that there are an equal number of the four following labeled structures over any finite set: abelian groups, n-ary abelian groups, TSM quasigroups, and n-ary TSM quasigroups. Next we explore applications of quasigroups in cryptography and discuss how our maps between abelian groups, TSM quasigroups, and n-ary TSM quasigroups can be used to generate and easily calculate products in quasigroups for use in cryptosystems. Finally, we discuss the TSM quasigroup of points on a cubic curve and prove some properties of iterated squaring in prime-order TSM quasigroups.

Committee: Harold Connamacher (Advisor); Shuai Xu (Committee Member); David Singer (Committee Member) Subjects: Computer Science; Mathematics

Master of Science, The Ohio State University, 2020, Electrical and Computer Engineering

In this thesis, I present an integrated Quantum key distribution (QKD) system designed to be low-weight, small-size and low-power. The system is designed to be stationed on drones, hence enabling the system to be deployed to remote locations, new scenarios where information security is of highest priority. Previous implementations used bulky time-taggers and high-power laser sources which are impractical for extended periods of operation. Our system integrates the bulky time-tagger into an FPGA and replaces the high-power laser source with low-power resonant cavity LEDs. The resonant cavity LEDs are driven directly by the electrical pulses generated from the FPGA. The FPGAs also handle the transmission and reception processes. The transmission frequency for the system is capped at 25 MHz. The frequency cap is purely a limitation of the single-photon detectors used. The duration of the wave-packets produced by the LEDs are 10 ns. When mounted on an optical bench, a successful key transmission yielded a Quantum bit error rate (QBER) of 7.5%. This test was done without a spectral filter at the output of the transmitter, and the input of the receiver. The QBER increases significantly when spectral filters are introduced into the system. The spectral filter is important in the creation of indistinguishable light sources in our system. Since having indistinguishable light sources are a key part of QKD security, the system as presented is not fully secure.

Committee: Daniel Gauthier Dr (Advisor); Tawfiq Musah Dr (Committee Member) Subjects: Computer Engineering; Electrical Engineering; Physics

Master of Science in Cyber Security (M.S.C.S.), Wright State University, 2019, Computer Science

Elliptic Curve Cryptography (ECC) is a public-key cryptography system. Elliptic Curve Cryptography (ECC) can achieve the same level of security as the public-key cryptography system, RSA, with a much smaller key size. It is a promising public key cryptography system with regard to time efficiency and resource utilization. This thesis focuses on the software implementations of ECC over finite field GF(p) with two distinct implementations of the Big Integer classes using character arrays, and bit sets in C++ programming language. Our implementation works on the ECC curves of the form y^2 = x^3 + ax + b (mod p). The point addition operation and the scalar multiplication are implemented on a real SEC (Standards for Efficient Cryptography) ECC curve over a prime field with two different implementations. The Elliptic Curve Diffie-Hellman key exchange, the ElGamal encryption/decryption system, and the Elliptic Curve Digital Signature Algorithm (ECDSA) on a real SEC ECC curve with two different implementations of the big integer classes are tested, and validated. The performances of the two different implementations are compared and analyzed.

Committee: Meilin Liu Ph.D. (Advisor); Junjie Zhang Ph.D. (Committee Member); Keke Chen Ph.D. (Committee Member) Subjects: Computer Science; Information Technology

PhD, University of Cincinnati, 2019, Arts and Sciences: Mathematical Sciences

In 2011, Gao and Heindl proposed a family of Multivariate Public Key Cryptosystems by combining the triangular scheme and the oil-vinegar scheme. The new design was claimed to be secured under known attacks. Besides that, they also used the Medium-Field Multivariate Public Key Cryptosystem as an example of their general frame and explained how it works. Later, by introducing several Diophantine equations into their design, they presented the Diophantine Equations Multivariate Public Key Cryptosystem (DEMPKC) with three sets of suggested parameters and the claimed security level were high. In this paper, we present our cryptanalysis on DEMPKC. Our cryptanalysis uses embedded surfaces associated with the DEMPKC and shows the attack can break the system efficiently. Our work provides an example of more general embedded surfaces other than linearization type of equations can be very useful to attack cryptosystems.

Committee: Jintai Ding Ph.D. (Committee Chair); Seungki| Kim Ph.D. (Committee Member); Benjamin Vaughan Ph.D. (Committee Member) Subjects: Mathematics

Master of Science, The Ohio State University, 2019, Electrical and Computer Engineering

The Advanced Encryption Standard (AES) is the current standard for symmetric key ciphers and is algorithmically secure. Side channel attacks that target power consumption can reveal the secret key in AES implementations. Masking data with random variables is one of the main methods used to thwart power analysis attacks. Data can be masked with multiple random variables to prevent higher-order attacks at the cost of a large increase in area. This thesis tests the plausibility of using varied finite field construction to prevent power analysis attacks as an alternative to masking. Initially, a design using finite field architecture as the sole countermeasure was investigated. This was followed by varied field construction in conjunction with a low entropy masking scheme. Neither approach provided an acceptable trade off between security and area. Analysis then turned to a combined Boolean and multiplicative masking scheme. Varied construction provided little gain for multiplicative masking. However, varied constructions were found to greatly increase security when used in conjunction with a Boolean random mask. A novel masking scheme for AES resistant to second-order attacks is proposed. Instead of an additional mask, variation in finite field construction is exploited to increase resistance to second-order attacks in Boolean masked shares. As a result, the area requirement is substantially reduced. For an example AES encryption, the proposed design is 12% smaller compared to the previous best design, with a small drop in achievable security level.

Committee: Zhang Xinmiao (Advisor); Steven Bibyk (Committee Member) Subjects: Engineering

Master of Science, University of Toledo, 2018, Electrical Engineering

Similar to wireless software updates for the smartphones, over-the-air (OTA) methods are used to update the software and firmware for the various electronic control units (ECUs) in the car. It is an efficient and convenient approach to update the software in the car and it will help customers save their money and time by reducing time to visit the service center to repair small bugs in the software. However, OTA updates open a new attack vector for hackers, who can use the OTA mechanism to steal OEM firmware, to reprogram ECUs, or to control the vehicle remotely. In this thesis, we perform a comprehensive security analysis for the current OTA mechanism to understand its associated threats. We also propose an approach to secure the original OTA software update method by adding a biometric iris scan and cryptographic hash function before updating the software and firmware over the air. With these security enhancements, vehicle owners can ensure that the car can be secure from unwanted and potentially malicious software updates.

Committee: Weiqing Sun (Advisor); Ahmad Javaid (Committee Co-Chair); Hong Wang (Committee Member) Subjects: Automotive Engineering; Information Technology

PhD, University of Cincinnati, 2017, Engineering and Applied Science: Computer Science and Engineering

A ubiquitous pervasive network incorporates today's Internet of Things/Internet of Everything Paradigm: Everything becomes smart with at least one microprocessor and a network interface. All these are under an umbrella of IoT/IoE paradigm where everything is network capable and connected. In most of the cases, these devices have multiple microprocessors and network interfaces at their disposal. In such a scenario, bringing every application to specific network on the same platform is critical, specifically for Sensor Networks, Cloud, WPANs and VANETs. While, enforcing and satisfying the requirements of CIA triad with non-repudiation universally is critical as this can solve multiple existing problems of ISM band exhaustion, leading to excessive collisions and contentions. Cooperative Interoperability also enables universal availability of data across all platforms which can be reliable and fully synchronized. Plug and play universal usability can be delivered. Such a network necessitates robust security and privacy protocols, spanning uniformly across all platforms. Once, reliable data access is made available, it leads to an accurate situation aware decision modeling. Simultaneous multiple channel usage can be exploited to maximize bandwidth otherwise unused. Optimizing Content delivery in hybrid mode which will be the major chunk of network traffic as predicted for near future of IoE. Now, such a proposed hybrid network does sound very complicated and hard to establish and maintain. However, this is the future of networks with huge leaps of technological advancement and ever dropping prices of hardware coupled with immensely improved capabilities, such a hybrid ubiquitous network can be designed and deployed in a realistic scenario. In this work, we go through not only looking into the issues of the large scale hybrid WMN, but also minutely discovering every possible scenario of direct mesh clients or sub-nets (VANET, Cloud or BAN) associated to it. Further, we pr (open full item for complete abstract)

Committee: Dharma Agrawal D.Sc. (Committee Chair); Richard Beck Ph.D. (Committee Member); Yizong Cheng Ph.D. (Committee Member); Rashmi Jha Ph.D. (Committee Member); Wen-Ben Jone Ph.D. (Committee Member); Marepalli Rao Ph.D. (Committee Member) Subjects: Computer Science

Doctor of Philosophy, The Ohio State University, 1976, Graduate School

Committee: Not Provided (Other) Subjects: Computer Science

PhD, University of Cincinnati, 2015, Arts and Sciences: Mathematical Sciences

The concept of semi-regular sequences was introduced in order to assess the complexity of Groumlbner basis algorithms such as F4 for the solution of polynomial equations. Despite the experimental evidence that semi-regular sequences are common, it was unknown whether there existed semi-regular sequences for all n, except in extremely trivial situations. In the present work I prove some results on the existence and non-existence of semi-regular sequences. It was observed by J. Schlather and T. Hodges that if an element of degree d in Β(n)-variables is semi-regular, then we must have n≤3d. In this thesis, I establish precisely when the elementary symmetric polynomial of degree d is semi-regular. In particular, when d=2t and n=3d, the elementary symmetric polynomial of degree d is semi-regular establishing that the bound given by J. Schlather and T. Hodges is sharp for infinitely many n. For the general case of existence of semi-regular sequences Bardet, Faugère and Salvy conjecture that the proportion ϖ(n, m, d1, . . . , dm) of semi-regular sequences over F2 in the set Ε(n, m, d1, . . . , dm) of algebraic systems of m equations of degrees d1, . . . , dm in n-variables tends to 1 as n tends to infinity. In this work, I show that for a fixed choice of (m, d1, . . . , dm), we have that limn→∞ ϖ(n, m, d1, . . . , dm ) — 0 showing that the conjecture is false in this case.

Committee: Timothy Hodges Ph.D. (Committee Chair); Donald French Ph.D. (Committee Member); Tara Smith Ph.D. (Committee Member) Subjects: Mathematics