18.783 - Elliptic Curves


# Date Topic (references) Materials
12/3Introduction to elliptic curvesslides
22/5The group law, Weierstrass and Edwards equations (Washington 2.1-3, 2.6.3, Bernstein-Lange)notes, worksheet 1, worksheet 2
32/10Finite fields and integer arithmetic (Modern Computer Algebra Ch. 8)notes
42/12Finite field arithmetic (Modern Computer Algebra: Sec. 3.2, 9.1, 11.1, HEHCC Ch. 9, Rabin)notes
52/19Isogenies (Washington 2.9, Silverman III.4)notes
62/24Isogeny kernels and division polynomials (Washington 3.2, 12.3, Silverman III.4))notes, worksheet
72/28Endomorphism rings (Washington 4.2, , Silverman III.6)notes
83/3Hasse's Theorem, Point counting (Washington 4.3)notes
93/5Schoof's algorithm (Washington 4.2, 4.5, Schoof)notes, worksheet
103/10Generic algorithms for discrete logarithms (Washington 5.2, Pohlig-Hellman, Pollard, Shoup)notes
113/12Index calculus, smooth numbers, factoring integers (Washington 5.1, 7.1, Granville, Lenstra)notes, worksheet
123/17Elliptic curve primality proving (ECPP) (Washington 7.2, Goldwasser-Kilian, Pomerance)notes
133/19Endomorphism algebras (Silverman III.9)notes
143/31Ordinary and supersingular curves (Silverman III.1,V, Washington 2.7, 4.6)notes
154/2Elliptic curves over C (part 1) (Cox Sec. 10, Silverman VI.2-3, Washington 9.1-2)notes
164/7Elliptic curves over C (part 2) (Cox Sec. 10-11, Silverman VI.4-5, Washington 9.2-3)notes
174/9Complex multiplication (CM) (Cox Sec. 11, Silverman VI.5, Washington 9.3)notes
184/14The CM action (Cox Sec. 7, Silverman (advanced topics) II.1.1)notes
194/16Riemann surfaces and modular curves (Silverman (advanced topics) I.2, Milne V.1)notes
204/23The modular equation (Cox Sec. 11, Milne V.2, Washington pp. 273-274)notes
214/28The Hilbert class polynomial (Cox Sec. 8, 11)notes
224/30Ring class fields and the CM method (Cox 8, 11)notes
235/5Isogeny volcanoes (Sutherland)notes
245/7The Weil pairing (Miller, Washington 11, Silverman III.8)notes
255/12Modular forms and L-series (Milne V.3-4)notes
265/14Fermat's last theorem (Milne V.7-9, Washington 15, Cornell-Stevens-Silverman I)notes