MAT5123

MAT 5123. Introduction to Cryptography. (3-0) 3 Credit Hours.

Prerequisite: MAT 4213. Congruences and residue class rings, Fermat’s Little Theorem, the Euler phi-function, the Chinese Remainder Theorem, complexity, symmetric-key cryptosystems, cyclic groups, primitive roots, discrete logarithms, one-way functions, public-key cryptosystems, digital signatures, finite fields, and elliptic curves. Differential Tuition: $150. Course Fees: GS01 $90.

Textbook: J. Hoffstein, J. Pipher, J. H. Silverman, An Introduction to Mathematical Cryptography (2nd Ed.) Springer Undergraduate Mathematics Series, Springer-Verlag (2014). ISBN: 978-1-4939-1711-2.

Week	Sections	Topics	Student Learning Outcomes
1	1.2 & 1.3	Substitution ciphers and basic theory of divisibility.	Caesar's and more general substitution ciphers. Greatest common divisor. The extended Euclidean algorithm.
2	1.4, 1.5.	Modular arithmetic and finite fields.	Primes and integer factorizations. The Fundamental Theorem of Arithmetic. Modular arithmetic and shift ciphers. Modular rings and finite fields 𝔽ₚ. Powers and primitive roots in finite fields. Fast exponentiation.
3	1.7, 2.1–2.3.	Public and private-key cryptosystems. Cyclic groups. Discrete Logarithms. Diffie-Hellman key exchange.	Symmetric and asymmetric ciphers. Encoding schemes. Perfect secrecy. Vernon's cipher. Examples of symmetric ciphers. Cyclic groups. The Discrete Logarithm Problem. The Diffie-Hellman key exchange.
4	4.2	Examples of power series and their formal manipulation.	Review of Taylor coefficients and Taylor series. Radius of convergence. Power series of rational functions. Power series defining the complex exponential, trigonometric and hyperbolic functions.
5	4.3, 4.5 & 4.5	Complex natural logarithms. Multivalued holomorphic functions. Singularities.	Definition of the multivalued complex natural logarithm, its principal branch, and other branches. Complex powers via logarithms. Definition of branch point and branches. Functions holomorphic in punctured neighborhoods. Poles and other singularities. Examples of branching and singularities (complex logarithms, inverse trigonometric/hyperbolic functions, and complex powers).
6	None	Review. First midterm exam.
7	5.2 & 5.3	Parametric curves. Line integrals.	Parametric representation of piecewise smooth curves. Arc-length. Rectifiable curves. Line integrals: Definition, examples, and elementary properties. Line integrals of holomorphic functions. Fundamental Theorem.
8	5.4 & 5.5	Estimation and convergence of line integrals.	Majorization of path integrals by arclength and bound on magnitude of integrand. Antiderivatives of complex functions with path-independent line integrals. Uniform and non-uniform convergence of sequences and series of complex functions. Continuous uniform limits of continuous sequences and series, and their integrals.
9	6.1, 6.2, 6.3	Cauchy's Theorem and its basic consequences.	Statement of Cauchy's Theorem. Proof of Cauchy's Theorem. The Deformation Theorem.
10	7.1 & 7.2	Cauchy's Integral Formula. Taylor series.	Statement and proof of Cauchy's Integral Formula. Existence, uniqueness, and general theory of Taylor series of holomorphic functions. Rigorous definition of and proof that complex logarithms are holomorphic.
11	None	Review. Second midterm exam.
12	8.1–8.3	Isolated singularities and Laurent series. The Residue Theorem.	Definition of Laurent series about an isolated singularity. Examples. Types of isolated singularities: Removable, polar, essential. The Cassorati-Weierstrass Theorem. Statement and proof of the Residue Theorem. Elementary techniques to evaluate residues.
13	Chapter 9.	Calculus of residues.	Evaluation of integrals of real analytic functions using residues. Evaluation of series of real analytic functions using residues.
14	11.1–11.3	Conformal mappings.	Preservation of angles and conformal mappings of the plane. Conformal mappings yield pairs of conjugate harmonic functions. Dirichlet's Problem on a planar region. The Riemann Mapping Theorem. Möbius transformations and their use in solving elementary Dirichlet Problems.
15	Chapter 10. (At instructor's discretion, week 15 may be used to wrap-up and review instead.)	Complex integration and geometric properties of holomorphic functions	Rouché's Theorem. The Open Mapping Theorem. Winding numbers.