MAT5123 - Revision history

Jose.iovino at 04:06, 26 March 2023

2023-03-26T04:06:55Z

Eduardo.duenez: Typo…

2023-03-24T17:00:58Z

Typo…

Eduardo.duenez: Added FLT and Euler's Thm.

2023-03-24T16:59:30Z

Added FLT and Euler's Thm.

Eduardo.duenez: Better formatting of Topics column.

2023-03-24T16:55:50Z

Better formatting of Topics column.

Eduardo.duenez: Better formatting reference.

2023-03-24T16:54:00Z

Better formatting reference.

Eduardo.duenez: Better formatting reference.

2023-03-24T16:52:59Z

Better formatting reference.

Eduardo.duenez: Better formatting reference.

2023-03-24T16:52:34Z

Better formatting reference.

Eduardo.duenez: Added line break.

2023-03-24T16:51:20Z

Added line break.

Eduardo.duenez: Fixed typo.

2023-03-24T16:50:33Z

Fixed typo.

Eduardo.duenez: MAT 5123 Crypto: Wiki done!

2023-03-24T16:48:06Z

MAT 5123 Crypto: Wiki done!

@@ Line 1: / Line 1: @@
 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.
+== Textbook ==
 {| class="wikitable sortable"

@@ Line 10: / Line 10: @@
 |1
 ||
-.2 & 1.3
+.2, 1.3
 ||
 Substitution ciphers and basic theory of divisibility.

@@ Line 28: / Line 28: @@
 * Modular arithmetic and shift ciphers.
 * Modular rings and finite fields 𝔽ₚ.
-* Powers and primitive roots in finite fields.
+* Powers and primitive roots in finite fields. Fermat's Little Theorem.
 * Fast exponentiation.
 |-  <!-- START ROW -->
@@ Line 101: / Line 101: @@
 ||  <!-- SLOs -->
 * Modular groups 𝑈ₙ.
 * Powers and roots modulo 𝒑𝒒.
 * The Rivest-Shamir-Adleman (RSA) cryptosystem.

@@ Line 36: / Line 36: @@
 .7, 2.1–2.3.
 ||  <!-- Topics -->
-Public and private-key cryptosystems. Cyclic groups. Discrete Logarithms. Diffie-Hellman key exchange.
+Public and private-key cryptosystems.
 ||  <!-- SLOs -->
 * Symmetric and asymmetric ciphers.
@@ Line 50: / Line 56: @@
 .4, 2.5. 2.6, 2.7.
 ||  <!-- Topics -->
-Elgamal public-key cryptosystem (EGPKC). Cyclic groups. Collision algorithms.
+Elgamal public-key cryptosystem (EGPKC).
 ||  <!-- SLOs -->
 * Theory of finite cyclic groups.
@@ Line 61: / Line 71: @@
 .8, 2.9, 2.10
 ||  <!-- Topics -->
-Rudiments of ring theory. The Chinese Remainder Theorem. The Pohlig-Hellman Algorithm.
+Rudiments of ring theory.
 ||  <!-- SLOs -->
 * Rings. Polynomial rings. Quotient rings.
@@ Line 80: / Line 94: @@
 .1, 3.2, 3.3.
 ||  <!-- Topics -->
-Modular groups of units. The RSA cryptosystem. Practical considerations of security in implementation.
+Modular groups of units.
 ||  <!-- SLOs -->
 * Modular groups 𝑈ₙ.
@@ Line 148: / Line 166: @@
 .4, 6.7
 ||  <!-- Topics -->
-Elliptic-Curve Cryptography (ECC). Elliptic curves in characteristic 2.
+Elliptic-Curve Cryptography (ECC).
 ||  <!-- SLOs -->
 * EC Diffie-Hellman key exchange.

@@ Line 161: / Line 161: @@
 .6
-Atkin-Morain's “ECs and Primality Proving” (Math. Comp. 61 (1993) 29-68.
+Atkin-Morain's “ECs and Primality Proving” (Math. Comp. 61 (1993) 29–68.
 [https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/])
 ||  <!-- Topics -->

@@ Line 160: / Line 160: @@
 || <!-- Sections -->
 .6
 Atkin-Morain's “ECs and Primality Proving” (Math. Comp. 61 (203) 29-68.)
 [https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/]

@@ Line 97: / Line 97: @@
 * Fermat's Little Theorem and Carmichael numbers.
 * The Miller-Rabin probabilistic primality test.
-* Pollard's “𝒑-1” factorization algorithm.
+* Pollard's “𝒑−𝟣” factorization algorithm.
 |-  <!-- START ROW -->
 | <!-- Week# -->

@@ Line 50: / Line 50: @@
 .4, 2.5. 2.6, 2.7.
 ||  <!-- Topics -->
-Elgamal public-key cryptosystem. Cyclic groups. Collision algorithms.
+Elgamal public-key cryptosystem (EGPKC). Cyclic groups. Collision algorithms.
 ||  <!-- SLOs -->
 * Theory of finite cyclic groups.
@@ Line 113: / Line 113: @@
 || <!-- Sections -->
-.1 & 7.2
+.1, 5.3, 5.6, 5.7.
 ||  <!-- Topics -->
-Cauchy's Integral Formula. Taylor series.
+Probability, entropy, information theory and complexity.
 ||  <!-- SLOs -->
-* Statement and proof of Cauchy's Integral Formula.
+* Rudiments of combinatorics and probability.
-* Existence, uniqueness, and general theory of Taylor series of holomorphic functions.
+* Bayes's Formula.
-* Rigorous definition of and proof that complex logarithms are holomorphic.
+* Random variables and expected values.
 |-  <!-- START ROW -->
 | <!-- Week# -->
@@ Line 132: / Line 134: @@
 || <!-- Sections -->
-.1–8.3
+.1, 6.2., 6.3
 ||  <!-- Topics -->
-Isolated singularities and Laurent series. The Residue Theorem.
+Elliptic curves and discrete logarithms.
 ||  <!-- SLOs -->
-* Definition of Laurent series about an isolated singularity. Examples.
+* Introduction to elliptic curves (ECs).
-* Types of isolated singularities: Removable, polar, essential. The Cassorati-Weierstrass Theorem.
+* Elliptic curves over finite fields.
-* Statement and proof of the Residue Theorem.
+* Fast multiples (“powers”) in ECs. The Double-and-Add algorithm.
-* Elementary techniques to evaluate residues.
+* The Elliptic Curve Discrete Logarithm Problem (ECDLP).
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-Chapter 9.
+.4, 6.7
 ||  <!-- Topics -->
-Calculus of residues.
+Elliptic-Curve Cryptography (ECC). Elliptic curves in characteristic 2.
 ||  <!-- SLOs -->
-* Evaluation of integrals of real analytic functions using residues.
+* EC Diffie-Hellman key exchange.
-* Evaluation of series of real analytic functions using residues.
+* EC Elgamal PKC.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-.1–11.3
+.6
 ||  <!-- Topics -->
-Conformal mappings.
+EC-based primality testing and factorization techniques.
 ||  <!-- SLOs -->
-* Preservation of angles and conformal mappings of the plane.
+* Lenstra's EC factorization algorithm.
-* Conformal mappings yield pairs of conjugate harmonic functions.
+* EC primality certification.
 |-  <!-- START ROW -->
 | <!-- Week# -->
 || <!-- Sections -->
-Chapter 10. (At instructor's discretion, week 15 may be used to wrap-up and review instead.)
+None.
 ||  <!-- Topics -->
-Complex integration and geometric properties of holomorphic functions
+Student Presentations. Wrap-up and review.
 ||  <!-- SLOs -->
 |-
 |}