SCHEDULE
| # | Date | Topic (references) | Materials |
|---|---|---|---|
| 1 | 9/7 | Introduction | notes, slides |
| 2 | 9/12 | The group law, Weierstrass and Edwards equations (Washington 2.1-3, 2.6.3, Bernstein-Lange 2007, Bernstein-Lange 2010) | notes, worksheet 1, worksheet 2 |
| 3 | 9/14 | Finite field arithmetic (Modern Computer Algebra 3.2, 8.1-4, 9.1, 11.1, 14.2-6, HEHCC 9.1-2, Rabin 1980, Cantor-Zassenhaus 1981) | notes, slides, worksheet, bonus video |
| 4 | 9/21 | Isogenies (Washington 2.9, Silverman III.4) | notes, slides |
| 5 | 9/26 | Isogeny kernels and division polynomials (Washington 3.2, 12.3, Silverman III.4) | notes, slides, worksheet |
| 6 | 9/28 | Endomorphism rings (Washington 4.2, , Silverman III.6) | notes, slides |
| 7 | 10/3 | Hasse's Theorem, point counting (Washington 4.3) | notes, slides |
| 8 | 10/5 | Schoof's algorithm (Washington 4.2, 4.5, Schoof 1985) | notes, slides, worksheet |
| 9 | 10/12 | Generic algorithms for the discrete logarithm problem (Washington 5.2, Pohlig-Hellman 1978, Pollard 1978, Shor 1994, Shoup 1997) | notes, slides |
| 10 | 10/17 | Index calculus, smooth numbers, factoring integers (Washington 5.1, 7.1, Granville, Lenstra) | notes, slides, worksheet 1, worksheet 2, worksheet 3 |
| 11 | 10/19 | Elliptic curve primality proving (ECPP) (Washington 7.2, Goldwasser-Kilian, Pomerance) | notes, slides |
| 12 | 10/24 | Endomorphism algebras (Silverman III.9) | notes, slides |
| 13 | 10/26 | Ordinary and supersingular curves (Silverman III.1,V, Washington 2.7, 4.6) | notes, slides |
| 14 | 10/31 | Elliptic curves over C (part I) (Cox Sec. 10, Silverman VI.2-3, Washington 9.1-2) | notes, slides |
| 15 | 11/2 | Elliptic curves over C (part II) (Cox Sec. 10-11, Silverman VI.4-5, Washington 9.2-3) | notes, slides |
| 16 | 11/7 | Complex multiplication (CM) (Cox Sec. 11, Silverman VI.5, Washington 9.3) | notes, slides |
| 17 | 11/9 | The CM torsor (Cox Sec. 7, Silverman (advanced topics) II.1.1) | notes, slides |
| 18 | 11/14 | Riemann surfaces and modular curves (Silverman (advanced topics) I.2, Milne V.1) | notes, slides |
| 19 | 11/16 | The modular equation (Cox Sec. 11, Milne V.2, Washington pp. 273-274) | notes, slides |
| 20 | 11/21 | The Hilbert class polynomial (Cox Sec. 8, 11) | notes, slides |
| 21 | 11/28 | Ring class fields and the CM method (Cox 8, 11) | notes, slides |
| 22 | 11/30 | Isogeny volcanoes (Sutherland) | |
| 23 | 12/5 | The Weil pairing (Miller, Washington 11, Silverman III.8) | notes, slides |
| 24 | 12/7 | Modular forms and L-functions (Milne V.3-4) | notes, slides |
| 25 | 12/12 | Fermat's last theorem (Milne V.7-9, Washington 15, Cornell-Stevens-Silverman I) | notes, slides |
