Browsing by Author "Phatak, Dhananjay"
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ItemAnalytic Proof of the recent Baseline Primality Conjecture(2021-12-12) Phatak, DhananjayThis document presents an analytic proof of the Baseline Primality Conjecture (BPC) that was recently unveiled in [1, Part I]. The BPC identifies a new small set of conditions that are sufficient to decide the primality of any input integer N under test (see Section 2 for the exact statement of the BPC in the original form using algebraic integers; and Section 3 for an equivalent polynomial domain reformulation). The practical significance of the BPC is that it directly leads to ultra low complexity primality testing algorithms, wherein the number of bit-operations required is essentially a quadratic function of the bit length of the input N . More specifically, the Baseline Primality Result (BPR) demonstrates that after an/any integer in the closed interval [2,N −2] which is a Quadratic Non Residue (QNR) modulo-N is found; exactly 2 (Two, which is a small O(1) constant, independent of the bit-length of the input N) specific modular exponentiations are sufficient to determine whether N is a composite or a prime. The BPC was (and to this day continues to be) extensively tested numerically.1 Additionally, analytic proofs of the BPC for several specific forms of the input N were also presented in , wherein the BPR was first unveiled. However, at the time of the original publication , we were not able to complete a general analytic proof of the BPC that covered all possible cases (i.e., forms) of the input N. We have now completed that vital task by developing a general analytic proof of the BPC using its polynomial domain reformulation. A concise presentation of that analytic proof is the main and narrow focus as well as the main new contribution of this article. An auxiliary contribution is a clear and precise explanation of the intuition behind our approach and the illustration of how it leads to the new theoretical results developed in . ItemThe CATS Hackathon: Creating and Refining Test Items for Cybersecurity Concept Inventories(2019-01-29) Sherman, Alan T.; Oliva, Linda; Golaszewski, Enis; Phatak, Dhananjay; Scheponik, Travis; Herman, Geoffrey L.; Choi, Dong San; Offenberger, Spencer E.; Peterson, Peter; Dykstra, Josiah; Bard, Gregory V.; Chattopadhyay, Ankur; Sharevski, Filipo; Verma, Rakesh; Vrecenar, RyanFor two days in February 2018, 17 cybersecurity educators and professionals from government and industry met in a "hackathon" to refine existing draft multiple-choice test items, and to create new ones, for a Cybersecurity Concept Inventory (CCI) and Cybersecurity Curriculum Assessment (CCA) being developed as part of the Cybersecurity Assessment Tools (CATS) Project. We report on the results of the CATS Hackathon, discussing the methods we used to develop test items, highlighting the evolution of a sample test item through this process, and offering suggestions to others who may wish to organize similar hackathons. Each test item embodies a scenario, question stem, and five answer choices. During the Hackathon, participants organized into teams to (1) Generate new scenarios and question stems, (2) Extend CCI items into CCA items, and generate new answer choices for new scenarios and stems, and (3) Review and refine draft CCA test items. The CATS Project provides rigorous evidence-based instruments for assessing and evaluating educational practices; these instruments can help identify pedagogies and content that are effective in teaching cybersecurity. The CCI measures how well students understand basic concepts in cybersecurity---especially adversarial thinking---after a first course in the field. The CCA measures how well students understand core concepts after completing a full cybersecurity curriculum ItemDesign and Evaluation of a Common Access Point for Bluetooth, 802.11 and Wired LANs(2002-06-24) Bethala, Bhagyalaxmi; Joshi, Anupam; Phatak, Dhananjay; Avancha, Sasikanth; Goff, TomDevices using wireless technologies like Bluetooth and 802.11b will be ubiquitous in the near future. The technologies supporting these devices will need to co-exist in order to support smooth exchange of information. We envision the existence of an Access Point (AP) , that can sport all these wireless technologies and bridge wireless networks with wired networks. We present the design and preliminary performance evaluation of two types of APs - a Bluetooth Access Point (BAP) and a Common Access Point (CAP) for Bluetooth, WLAN and Ethernet networks. ItemNew Proof of the recent Baseline Primality Conjecture(2023-06-08) Phatak, DhananjayThis document presents a theoretical proof of the Baseline Primality Conjecture (BPC) that was recently unveiled in [4, Part I]. The BPC identifies a new small set of conditions that are sufficient to decide the primality of any input integer N under test (see Section 2 for the exact statement of the BPC in the original form using algebraic integers; and Section 3 for an equivalent polynomial domain reformulation). The practical significance of the BPC is that it directly leads to ultra low complexity primality testing algorithms, wherein the number of bit-operations required is essentially a quadratic function of the bit-length of the input N . More specifically, the Baseline Primality Result (BPR) demonstrates that after an/any integer in the closed interval [2,N −2] which is a Quadratic Non Residue (QNR) modulo-N is found; exactly 2 (Two, which is a small O(1) constant, independent of the bit-length of the input N) specific modular exponentiations are sufficient to determine whether N is a composite or a prime. The BPC was (and to this day continues to be) extensively tested numerically. Additionally, theoretical proofs of the BPC for several specific forms of the input N were also presented in . However, at the time of the original publication , we were not able to complete a general theoretical proof of the BPC that covered all possible cases (i.e., forms) of the input N. A preliminary version of a general proof of the BPC was posted in the IEEE TechRxiv . However, after the release of that document, the author solicited feedback from the renowned number theory expert Prof. Carl Pomerance in Jan. 2022. He pointed out a mistake  in that version of the proof (which appears in ). Since then, after more than an year of effort, a completely new proof has been developed in the period approximately from the 28th of January 2023 to the end of April 2023. To the best of our knowledge, this proof  is error free (and therefore will replace  in the near future.) A concise presentation of that new theoretical proof is the main focus as well as the main new contribution of this article. An auxiliary contribution is a clear and precise explanation of the intuition behind our approach and the illustration of how it leads to the new theoretical results developed in . ItemPPT : New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues(2019-08-20) Phatak, Dhananjay; Sherman, Alan T.; Houston, Steven D.; Henry, AndrewIn this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as lemmas/corollaries/claims whenever we have complete analytic proof(s); otherwise the results are introduced as conjectures. In Part/Article 1 , we start with the Baseline Primality Conjecture (PBPC) which enables deterministic primality detection with a low complexity = O (logN) 2 (polylog(logN)) ; when an explicit value of a Quadratic NON Residue ( QNR ) modulo-N is available (which happens to be the case for an overwhelming majority = 11 12 = 91.67% of all odd integers). We then demonstrate Primality Lemma 1 , which reveals close connections between the state-of-the-art Miller-Rabin method and the renowned Euler-Criterion. This Lemma, together with the Baseline Primality Conjecture enables a synergistic fusion of Miller-Rabin iterations and our method(s), resulting in hybrid algorithms that are substantially better than their components. Next, we illustrate how the requirement of an explicit value of a QNR can be circumvented by using relations of the form: Polynomial(x) mod N ≡ 0 ; whose solutions implicitly specify NON-RESIDUEs modulo-N. We then develop a method to derive low-degree canonical polynomials that together guarantee implicit NON-RESIDUEs modulo-N ; which along with the Generalized Primality Conjectures enable algorithms that achieve a worst case deterministic polynomial complexity = O (logN) 3 (polylog(logN)) ; unconditionally, for any/all values of N. In Part/Article 2 , we present substantial experimental data that corroborate all the conjectures3 . For a striking/interesting example, wherein a combination of well-known state-of-the-art probabilistic primality tests applied together fail to detect a composite (which is easily/instantly detected correctly to be a composite number by the Baseline Primality Conjecture), see Section 23 in Part/Article 2. The data show that probabilistic primality tests in software platforms such as Maple are now extremely robust. I have not yet found a single composite number N that fools Maple’s Probabilistic primality tests; but gets correctly identified as a composite by our deterministic tests. Construction of (or a search for) such integers (assuming that those exist) is an interesting open problem which we plan to investigate in the future. Finally in Part/Article 3 , we present analytic proof(s) of the Baseline Primality Conjecture that we have been able to complete for some special cases (i.e. for particular types of inputs N). We are optimistic that full analytic proofs of all conjectures introduced herein will be generated in the near future. That will be a big step toward moving the problem of primality detection into the category of problems that have been “solved and retired ItemStudent Misconceptions about Cybersecurity Concepts: Analysis of Think-Aloud Interviews(DigitalCommons@Kennesaw State University, 2018) Thompson, Julia D.; Herman, Geoffrey L.; Scheponik, Travis; Oliva, Linda; Sherman, Alan; Golaszewski, Ennis; Phatak, DhananjayWe conducted an observational study to document student misconceptions about cybersecurity using thematic analysis of 25 think-aloud interviews. By understanding patterns in student misconceptions, we provide a basis for developing rigorous evidence-based recommendations for improving teaching and assessment methods in cybersecurity and inform future research. This study is the first to explore student cognition and reasoning about cybersecurity. We interviewed students from three diverse institutions. During these interviews, students grappled with security scenarios designed to probe their understanding of cybersecurity, especially adversarial thinking. We analyzed student statements using a structured qualitative method, novice-led paired thematic analysis, to document patterns in student misconceptions and problematic reasoning that transcend institutions, scenarios, or demographics. Themes generated from this analysis describe a taxonomy of misconceptions but not their causes or remedies. Four themes emerged: overgeneralizations, conflated concepts, biases, and incorrect assumptions. Together, these themes reveal that students generally failed to grasp the complexity and subtlety of possible vulnerabilities, threats, risks, and mitigations, suggesting a need for instructional methods that engage students in reasoning about complex scenarios with an adversarial mindset. These findings can guide teachers’ attention during instruction and inform the development of cybersecurity assessment tools that enable cross-institutional assessments that measure the effectiveness of pedagogies.