18.783 - Elliptic Curves


# Date Topic (references) Materials
11/31Introductionnotes, slides
22/2The group law, Weierstrass and Edwards equations (Washington 2.1-3, 2.6.3, Bernstein-Lange 2007, Bernstein-Lange 2010)notes, worksheet 1, worksheet 2
32/7Finite 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
42/9Isogenies (Washington 2.9, Silverman III.4)notes, slides
52/14Isogeny kernels and division polynomials (Washington 3.2, 12.3, Silverman III.4)notes, slides, worksheet
62/16Endomorphism rings (Washington 4.2, , Silverman III.6)notes, slides
72/22Hasse's Theorem, point counting (Washington 4.3)notes, slides
82/23Schoof's algorithm (Washington 4.2, 4.5, Schoof 1985)notes, slides, worksheet
92/28Generic algorithms for the discrete logarithm problem (Washington 5.2, Pohlig-Hellman 1978, Pollard 1978, Shor 1994, Shoup 1997)notes, slides
103/2Index calculus, smooth numbers, factoring integers (Washington 5.1, 7.1, Granville, Lenstra)notes, slides, worksheet 1, worksheet 2, worksheet 3
113/7Elliptic curve primality proving (ECPP) (Washington 7.2, Goldwasser-Kilian, Pomerance)notes, slides
123/9Endomorphism algebras (Silverman III.9)notes, slides
133/14Ordinary and supersingular curves (Silverman III.1,V, Washington 2.7, 4.6)notes, slides
143/16Elliptic curves over C (part I) (Cox Sec. 10, Silverman VI.2-3, Washington 9.1-2)notes, slides
153/28Elliptic curves over C (part II) (Cox Sec. 10-11, Silverman VI.4-5, Washington 9.2-3)notes, slides
163/30Complex multiplication (CM) (Cox Sec. 11, Silverman VI.5, Washington 9.3)notes
174/4The CM torsor (Cox Sec. 7, Silverman (advanced topics) II.1.1)notes
184/11Riemann surfaces and modular curves (Silverman (advanced topics) I.2, Milne V.1)slides, notes
194/13The modular equation (Cox Sec. 11, Milne V.2, Washington pp. 273-274)slides, notes
204/20The Hilbert class polynomial (Cox Sec. 8, 11)slides, notes
214/25Ring class fields and the CM method (Cox 8, 11)slides, notes
224/25Isogeny volcanoes (Sutherland)slides, notes
235/2The Weil pairing (Miller, Washington 11, Silverman III.8)slides, notes
245/4Modular forms and L-functions (Milne V.3-4)slides, notes
255/9Fermat's last theorem (Milne V.7-9, Washington 15, Cornell-Stevens-Silverman I)slides, notes