MAT5123

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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.
  • Discrete Logarithms.
  • The Diffie-Hellman key exchange.

4

2.4, 2.5. 2.6, 2.7.

Elgamal public-key cryptosystem. Cyclic groups. Collision algorithms.

  • Theory of finite cyclic groups.
  • The Discrete Logarithm Problem (DLP).
  • Shanks’ Babystep-Giantstep DLP algorithm.

5

2.8, 2.9, 2.10

Rudiments of ring theory. The Chinese Remainder Theorem. The Pohlig-Hellman Algorithm.

  • Rings. Polynomial rings. Quotient rings.
  • Systems of congruences. The Chinese Remainder Theorem.
  • The Pohlig-Hellman Algorithm.

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.