MAT 599-61:

Applications in Number Theory

Summer 2017

• Syllabus
• Presentations:
  1. Chinese Remainder Theorem – Rocca ✓
  2. Euler’s \(\phi\)-Function and Euler’s Theorem ✓
  3. Pseudoprimes, Miller’s Test, Theorem 6.8, Rabin’s Probabilistic Test (Theorem 6.11) – D. Ciskowski ✓
  4. Mersenne Primes, Lucas-Lehmer Test – K. Ciskowski ✓
  5. Lucas’s Converse to Fermat’s Little (Theorem 9.18) and Corollary 9.18.1 – D. Cook ✓
  6. Theorem 9.19 and Big O-Notation – Rocca ✓
  7. Pocklington’s and Proth’s Primality Tests – D. Ciskowski ✓
  8. Sections 11.1-11.3 – Rocca ✓
  9. Euler Pseudoprimes, Solovay-Strassen Probabilistic Test – K. Ciskowski ✓
  10. Fermat Factorization – D. Cook
  11. Pollard’s Rho Factorization Method – D. Ciskowski
  12. Continued Fractions and theorem 12.8 – Rocca ✓
  13. Theorem 12.7 – K. Ciskowski ✓
  14. \(k^{th}\)-Convergent, Theorem 12.9 – D. Cook ✓
  15. Theorem 12.15 – Rocca
  16. Theorem 12.24 (including Lemma 12.6), Continued Fraction Factorization – D. Ciskowski
  17. Rabin Cryptosystem (p. 329 and exercise 49 in section 11.1) – K. Ciskowski
  18. Knapsack Problem and Super Increasing Sequences (pp. 331-333) – Rocca
  19. Knapsack Ciphers (pp. 334-336) – D. Cook
  20. Electronic Poker – Rocca
  21. El Gamal Cryptosystem – Rocca
• Programing Assignments: These are listed on CoCalc.com
• Text Exercises: Do 8 of the following
  • p.170 15 and 19
  • p. 170 27-29 (use the result from #19, change periods to 23, 27, and 31 (why?))
  • ???