LEC # | TOPICS | REFERENCES |
---|---|---|

1 | Introduction to Elliptic Curves | No readings. |

2 | The Group Law, Weierstrass and Edwards Equations | [Washington] Sections 2.1–3, and 2.6.3. Bernstein, Daniel, and Lange, Tanja. Faster Addition and Doubling on Elliptic Curves. |

3 | Integer Arithmetic | Gathen, Joachim von zur, and Jürgen Gerhard. Chapter 8 in Modern Computer Algebra. Cambridge University Press, 2003. ISBN: 9780521826464. [Preview with Google Books] |

4 | Finite Field Arithmetic | Gathen, Joachim von zur, and Jürgen Gerhard. Chapter 3, Sec. 9.1, and Sec. 11.1 in Modern Computer Algebra. Cambridge University Press, 2003. ISBN: 9780521826464. [Preview with Google Books] Cohen, Henri, Gerhard Frey, and Roberto Avanzi. Chapter 9 in Rabin, Michael O. "Probabilistic Algorithms in Finite Fields." |

5 | Isogenies and Endomorphisms | [Washington] Section 2.9. [Silverman] Section III.4. |

6 | Division Polynomials and Torsion Subgroups | [Washington] Section 3.2. |

7 | Endomorphism Rings and Hasse's Theorem | [Washington] Section 4.2. [Silverman] Section III.6. |

8 | Point Counting | [Washington] Section 4.3. |

9 | Schoof's Algorithm | [Washington] Sections 4.2, and 4.5. Schoof, Rene. "Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p." (PDF) |

10 | Discrete Logarithms: Generic Algorithms | [Washington] Section 5.2. Pohlig, S., and M. Hellman. "An Improved Algorithm for Computing Logarithms Over GF(p) and Its Cryptographic Significance (Corresp.)." Pollard, J. M. "Monte Carlo Methods for Index Computation (mod p)." |

11 | Discrete Logarithms: Lower Bounds, Index Calculus | Shoup, V. "Lower Bounds for Discrete Logarithms and Related Problems." (PDF) Lecture Notes in Computer Science 1233 (1997): 256–66. [Washington] Section 5.1. Granville, Andrew. "Smooth Numbers: Computational Number Theory and Beyond." (PDF) In |

12 | Elliptic Curve Factorization Method (ECM) | [Washington] Section 7.1. Lenstra, H. W. "Factoring Integers with Elliptic Curves." (PDF - 1.3MB). Annals of Mathematics, Mathematical Sciences Research Institute, 1986. Montgomery, Peter L. "Speeding the Pollard and Elliptic Curve Methods of Factorization." Bernstein, Daniel J., Peter Birkner, et al. "ECM Using Edwards Curves." |

13 | Elliptic Curve Primality Proving (ECPP) | [Washington] Section 7.2. Goldwasser, S., and J. Killan. "Almost all Primes can be Quickly Certified." Pomerance, Carl. "Very Short Primality Proofs." |

14 | Endomorphism Algebras | [Silverman] Section III.9. |

15 | Ordinary and Supersingular Curves, The j-invariant | [Silverman] Sections III.1, and V.3. [Washington] Sections 2.7, and 4.6. |

16 | Elliptic Functions, Eisenstein Series, Weierstrass p-function | [Cox] Chapter 10. [Silverman] Sections VI.2–3. [Washington] Sections 9.1–2. |

17 | Complex Tori, Elliptic Curves over C, Lattice j-invariants | [Cox] Chapters 10, and 11. [Silverman] Sections VI.4–5. [Washington] Sections 9.2–3. |

18 | Uniformization Theorem, Complex Multiplication | [Cox] Chapter 11. [Silverman] Section VI.5. [Washington] Section 9.3. |

19 | Orders, Ideals, Class Groups, Isogenies over C | [Cox] Chapter 7. [Silverman (Advanced Topics)] Section II.1.1. |

20 | Riemann Surfaces and the Modular Curve X(1) | [Silverman (Advanced Topics)] Section I.2. [Milne] Section V.1. |

21 | Modular Functions and the Modular Equation | [Cox] Chapter 11. [Milne] Section V.2. |

22 | The Main Theorem of Complex Multiplication | [Cox] Chapters 8, and 11. |

23 | CM Method and Isogeny Volcanoes | Sutherland, Andrew V. Isogeny Volcanoes. (PDF) 2012. |

24 | Modular Forms and L-functions | [Milne] Sections V.3–4. |

25 | Fermat's Last Theorem | [Milne] Sections V.7–9. [Washington] Chapter 15. Cornell, Gary, Joseph H. Silverman, and Glenn Stevens. |