Certificates Intro

Technical Certificate Programs

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Once you enrolled in the courses, please use this link to apply for the chosen certificate.  Enrollment Request Form using your primary curriculum, and then adding curriculum number (200, 280 or 283 as identified below).

Academic Certificate in Network Science  - Curriculum 200      

See website here

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Program Manager
Prof. Ralucca Gera, Ph.D.
260 Spanagel Hall
Phone: (831) 656-2230
Fax: (831) 656-2355
rgera@nps.edu

Brief Overview
The Academic Certificate in Network Science provides education in the use of mathematical methods for the analysis, understanding, and exploitation of complex networks. Network Science has emerged as an area critical to the success of the mission of the Navy and the Department of Defense because of the central role it plays in cyber-security, network-centric warfare, and other related areas of critical interest. A thorough understanding of the underlying mathematics is essential for the correct interpretation and further development of practical methods, models, and approaches to problems involving complex networks. The certificate program is designed to provide that very background. Upon successful completion of the course-work, students will be awarded a certificate of accomplishment in keeping with standard practices of the Naval Postgraduate School.

Requirements for Entry
Prospective students must have taken a discrete mathematics class at NPS (or equivalent from other universities): MA1025 or MA2025 

Entry Date
Program entry dates are flexible, preferred entry is the Fall quarter.

Program Length
Three quarters.

MA Academic Certificate Requirements
To earn the academic certificate students must pass all four courses with a C+ (2.3 Quality Point Rating (QPR)) or better in each course and an overall QPR of 3.0 or better. Students earning grades below these standards will need to retake the courses to bring their grades within standards or they will be withdrawn from the program.

Required Courses:

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offered in the Fall quarters
Advanced topics in the theory of graphs and digraphs. Topics include graph coloring, Eulerian and Hamiltonian graphs, perfect graphs, matching and covering, tournaments, and networks. Application areas with DoD/DoN relevance range from mathematics to computer science and operations research, including applications to coding theory, searching and sorting, resource allocation, and network design. Prerequisites: MA3025/MA2025.

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(4-0) offered in the Winter quarters
The course focuses on the emerging science of complex networks and their applications, through an introduction to techniques and models for understanding and predicting their behavior.  The topics discussed will be building mainly on graph theory concepts, and they will address the mathematics of networks, their applications to the computer networks and social networks, and their use in research.  The students will learn the fundamentals of dynamically evolving complex networks, study current research in the field, and apply their knowledge in the analysis of real network systems through a final project.  DoD applications include security of critical communication infrastructure.

One of the following courses:

MA3025 provides a rigorous foundation in logic and elementary discrete mathematics to students of mathematics and computer science. Topics from logic include modeling English propositions, propositional calculus, quantification, and elementary predicate calculus. Additional mathematical topics include elements of set theory, mathematical induction, relations and functions, and elements of number theory.

Prerequisite: MA1025 or MA2025

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Offered any quarter that there are enough requests. This course focuses on the emerging field of network science. The students will learn about the ongoing research in the field by listening to conference presentations (prerecorded or in person if possible to attend conferences), and applying the learned knowledge to a research project. If the research project is used towards a funded research, then (1) students' visit to sponsors will be included during the quarter (if feasible) or virtual conversations with sponsors are facilitated, and (2) a final presentation to the sponsor is anticipated upon the completion of the project. There is no set content for students, as they will be exposed to current topics in the network science field at the time. Prerequisites: consent of instructor (Prof. Ralucca Gera).

Offered in the Summer quarters The course will develop game theoretic concepts in evaluations of the importance of players in bargaining situations and of elements in networks. Topics covered include cooperative and noncooperative games, bargaining, the Shapley Value, and coalitions. The course will study applications to military problems and applications to economics, political science, and biology. There will be extensive reading from the literature. Prerequisites: MA3042, OA3201, and an introductory course in probability.

Offered in the Fall quarters Explores fundamentals of packet-switched network traffic analysis at the network layer and above as applied to problems in traffic engineering, economics, security, etc. Explores the design and integration of analytic tools and techniques into the fabric of the network including: spatial and temporal anomaly detection, origin-destination matrix estimation, application mix determination, deep-packet inspection, fingerprinting, intrusion detection and insider threat mitigation. Finally, the course covers active defense and offensive methods reliant on traffic analysis. Prerequisites: CS3502 and CS4550 or equivalent.

Offered in the Summer quarters Introduction to formulation and solution of problems involving networks, such as maximum flow, shortest route, minimum cost flows, and PERT/CPM. Elements of graph theory, data structure, algorithms, and computational complexity. Applications to production and inventory, routing, scheduling, network interdiction, and personnel management. Prerequisite: OA3201.

Examines the impact of the radio frequency environment on the security of wireless communications networks. Specifically, examines security primitives including confidentiality assurance, integrity and authentication and encryption mechanisms and their impact on different network topologies. Various types of ciphers such as block and stream ciphers and their principles/implementation are studied. Basic encryption algorithms including the Data Encryption Standard (DES) and Advanced Encryption Standard (AES) are examined along with various key establishment and authentication protocols. Public Key Cryptography (PKC) and its variants (RSA, Diffie-Hellman) are discussed. Discusses approaches to integrity assurance in the form of cryptographic hash functions. Security protocols for the TCP/IP protocol stack are discussed including the EAP, SSL/TLS and IPSec. Examines all of the above factors in the context of a variety of topologies including personal area networks (PAN-Security in Bluetooth), wireless local area networks (WLAN), metropolitan area networks (MAN-WiMax and LTE), and wide area networks (WAN). Analyzes and compares protocol implementations such as Wired Equivalent Privacy (WEP), Wi-Fi Protected Access (WPA), and the Mobile Application Part (MAP) of Signaling System No. 7 (SS7). Also analyzes security in telephony, including 2G (GSM) and 3G (UMTS). Discuss general aspects of wireless communication vulnerabilities with example attacks. PREREQUISITES: EC2010, EC3710.

This course deals with the problems faced by a rational decision-maker, trying to maximize some payoff in a social setting. A distinction will be made between Type I behavior (optimization in a game against nature,) Type II (optimization when faced with agents who react against the decision-maker¹s perceived behavior,) Type III (optimizations against strategic agents,) and Type IV (cooperation with other agents). Applications include arms race models, treaty inspections problems, monopolistic behavior, coalition formation, and pursuit games. The computer is used as a modeling tool. PREREQUISITE: DA2410

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This course focuses at how to illuminate networked adversaries. The course develops a deep understanding on how to use social network analysis metrics (e.g., topology, centrality, cohesion, brokerage) to better understand the relational aspects of dark networks and how to design intervention strategies for disrupting, destabilizing and possibly destroying dark networks. The course's first objective is to teach students how to identify and describe these networks and their dynamics. We use various software packages (e.g. Inflow, e.g. Analyst's Notebook) to aid in the identification and description effort. The second objective is to design intervention strategies to disrupt, destabilize and possibly destroy dark networks once they have been identified and described. Students are expected to focus their interventions at the tactical and operational levels, although some attention is given to the strategic and policy levels.

Academic Certificate in the Mathematics of Secure Communication - Curriculum 280

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Program Manager
Prof. Pantelimon Stanica, Ph.D.
268 Spanagel Hall
Phone: (831) 656-2714
Fax: (831) 656-2355
pstanica@nps.edu

Brief Overview
The Mathematics of Secure Communication certificate program provides mathematics education to US and allied military officers and DoD civilians in the broad area of cryptography and secure communications. As such it satisfies a Knowledge, Skills, Abilities (KSA) requirement in the Applied Technology field of Fundamentals of Cryptography, Cryptanalysis, Coding and Information Theory for Professional Military Education. Upon successful completion of the required three courses (described below), students will be awarded a certificate of accomplishment in keeping with standard practices of the Naval Postgraduate School.



 

Requirements for Entry
Prospective students must have taken a discrete mathematics class at NPS (or equivalent from other universities): MA1025, MA2025, or MA3025 

Entry Date
Program entry dates are flexible, preferred entry is the Winter quarter.

Program Length
Three quarters.

MA Academic Certificate Requirements
To earn the academic certificate students must pass all three courses with a C+ (2.3 Quality Point Rating (QPR)) or better in each course and an overall QPR of 3.0 or better. Students earning grades below these standards will need to retake the courses to bring their grades within standards or they will be withdrawn from the program.

Required Courses:

This course is devoted to aspects of modern algebra and number theory that directly support applications, principally in communication. The algebraic emphasis is on ring and field theory, with special emphasis on the theory of finite fields, as well as those aspects of group theory that are important in the development of coding theory. Elements of number theory include congruences and factorization. Applications are drawn from topics of interest to DoN/DoD. These include error correcting codes and cryptography.

Prerequisite: MA3025

Mathematical analysis of the codes used over communication channels is made. Techniques developed for efficient, reliable and secure communication are stressed. Effects of noise on information transmission are analyzed and techniques to combat their effects are developed. Linear codes, finite fields, single and multiple error-correcting codes are discussed. Codes have numerous applications for communication in the military, and these will be addressed.

Prerequisite: MA3560

MA4570 - Cryptography (4-1)
The methods of secret communication are addressed. Some simple cryptosystems are described and classical techniques of substitution and transposition are considered. The public-key cryptosystems, RSA, Discrete Logarithm and other schemes are introduced. Applications of cryptography and cryptanalysis.
Prerequisite: MA3560

Additionally, the students in the program can register in MA4026 Combinatorial Mathematics (4-0).

Additionally, the students in the program can register in: 

Advanced techniques in enumerative combinatorics and an introduction to combinatorial structures. Topics include generating functions, recurrence relations, elements of Ramsey theory, theorems of Burnside and Polya, and balanced incomplete block designs. Application areas with DoD/DoN relevance range from mathematics to computer science and operations research, including applications in probability, game theory, network design, coding theory, and experimental design. Prerequisites: MA3025.

Advanced topics in the theory of graphs and digraphs. Topics include graph coloring, Eulerian and Hamiltonian graphs, perfect graphs, matching and covering, tournaments, and networks. Application areas with DoD/DoN relevance range from mathematics to computer science and operations research, including applications to coding theory, searching and sorting, resource allocation, and network design. Prerequisites: MA3025.

The course will discuss the Fourier analysis of Boolean functions and the relevant combinatorics with an eye toward cryptography and coding theory. Particular topics will include avalanche features of Boolean functions, correlation immunity and resiliency, bentness, trade-offs among cryptographic criteria and real-life applications in the designs of stream and block ciphers.

Prerequisite: MA3025

Academic Scientific Computation Certificate - Curriculum 283 

  

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Program Manager
Prof. Frank Giraldo, Ph.D.
240B Spanagel Hall
Phone: (831) 656-4490
Fax: (831) 656-2355
fxgirald@nps.edu

Brief Overview
The Scientific Computation Certificate provides a foundation in mathematical analysis and numerical methods for modeling scientific and engineering problems on modern computers. As a core pillar of scientific discovery alongside theory and experimentation, scientific computing is essential for applications such as weather prediction, ocean modeling, advanced weapons design, and materials development, making it critical to the missions of the Navy and the Department of Defense.  The program emphasizes the mathematical principles underlying computational algorithms, enabling students to apply, interpret, and develop advanced scientific computing methods. The certificate consists of four courses: two core courses and two electives selected from a range of specialized topics, allowing students to tailor the program to their interests. Students who successfully complete the program receive a Naval Postgraduate School Certificate in Scientific Computation

Requirements for Entry
Prospective students must take MA2043.

Entry Date
Program entry dates are flexible, preferred entry is the Winter quarter.

Program Length
Four quarters.

MA Academic Certificate Requirements
To earn the academic certificate students must pass all four courses with a C+ (2.3 Quality Point Rating (QPR)) or better in each course and an overall QPR of 3.0 or better. Students earning grades below these standards will need to retake the courses to bring their grades within standards or they will be withdrawn from the program.

Required Courses:

This course provides students in the engineering and physical sciences curricula with an applications-oriented coverage of major topics of matrix and linear algebra. Matrix factorizations (LU, QR, Cholesky), the Singular Value Decomposition, eigenvalues and eigenvectors, the Schur form, subspace computations, structured matrices. Understanding of practical computational issues such as stability, conditioning, complexity, and the development of practical algorithms. Prerequisites: MA2043 and EC1010.

  • Provides the basic numerical tools for understanding more advanced numerical methods. Topics for the course include: Sources and Analysis of Computational Error, Solution of Nonlinear Equations, Interpolation and Other Techniques for Approximating Functions, Numerical Integration and Differentiation, Numerical Solution of Initial and Boundary Value Problems in Ordinary Differential Equations, and Influences of Hardware and Software. Prerequisites: MA1115, MA2121 and ability to program in MATLAB and MAPLE.

And any two from:

The subject matter will vary according to the abilities and interest of those enrolled. Applications of the subject matter to DoD/DoN are discussed. Prerequisites: Consent of instructor.

Adams formulas, Runge-Kutta formulas, extrapolation methods, implicit formulas for stiff equations; convergence and stability, error estimation and control, order and stepsize selection, applications. Prerequisites: MA3232.

Finite difference methods for parabolic, elliptic, and hyperbolic equations, multi-grid methods; convergence and stability, error estimation and control, numerical solution of finite difference equations, applications. Prerequisites: MA3132, MA3232 suggested.

Variational formulation of boundary value problems, finite element and boundary element approximations, types of elements, stability, eigenvalue problems. Prerequisites: MA3132, MA3232 or equivalent.

Development of algorithms for matrix computations. Rounding errors and introduction to stability analysis. Stable algorithms for solving systems of linear equations, linear least squares problems and eigen problems. Iterative methods for linear systems. Structured problems from applications in various disciplines. Prerequisites: MA3046, or consent of instructor, advanced MATLAB programming.

General principles of parallel computing, parallel techniques and algorithms, solution of systems of linear equations, eigenvalues and singular value decomposition, domain decomposition and application (e.g., satellite orbit determination and shallow water fluid flow). Prerequisites: MA3042 or MA3046, MA3132, and MA3232.

Euler equation, Weierstrass condition, Legendre condition, numerical procedures for determining solutions, gradient method, Newton method, Transversability condition, Rayleigh Ritz method, conjugate points. Concepts are related to geometric principles whenever possible. Prerequisites: MA2121 (programming experience desirable).

Advanced course in the application of approximate methods to the study of integrals and differential equations arising in physical problems. Topics covered include: asymptotic sequences and expansions, integrals of a real variable, contour integrals, limit process expansions applied to ordinary differential equations, multiple variable expansion procedures and applications to partial differential equations. Prerequisites: MA3132.

This course provides an introduction to the theory of dynamical systems providing a basis for the analysis and design of systems in engineering and applied science. It includes the following topics: Second order linear systems; contraction mapping, existence and uniqueness of solutions; continuous dependence on initial conditions; comparison principle; Lyapunov stability theorems; LaSalle's theorem; linearization methods; nonautonomous systems; converse theorems; center manifold theorems; and stationary bifurcations of nonlinear systems. Prerequisites: MA2121.
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