18.783 - Elliptic Curves


# Date Topic (references) Materials
19/7Introductionnotes, slides
29/12The group law, Weierstrass and Edwards equations (Washington 2.1-3, 2.6.3, Bernstein-Lange 2007, Bernstein-Lange 2010)notes, worksheet 1, worksheet 2
39/14Finite 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
49/21Isogenies (Washington 2.9, Silverman III.4)notes, slides
59/26Isogeny kernels and division polynomials (Washington 3.2, 12.3, Silverman III.4)notes, slides, worksheet
69/28Endomorphism rings (Washington 4.2, , Silverman III.6)notes, slides
710/3Hasse's Theorem, point counting (Washington 4.3)notes, slides
810/5Schoof's algorithm (Washington 4.2, 4.5, Schoof 1985)notes, slides, worksheet
910/12Generic algorithms for the discrete logarithm problem (Washington 5.2, Pohlig-Hellman 1978, Pollard 1978, Shor 1994, Shoup 1997)notes, slides
1010/17Index calculus, smooth numbers, factoring integers (Washington 5.1, 7.1, Granville, Lenstra)notes, slides, worksheet 1, worksheet 2, worksheet 3
1110/19Elliptic curve primality proving (ECPP) (Washington 7.2, Goldwasser-Kilian, Pomerance)notes, slides
1210/24Endomorphism algebras (Silverman III.9)notes, slides
1310/26Ordinary and supersingular curves (Silverman III.1,V, Washington 2.7, 4.6)notes, slides
1410/31Elliptic curves over C (part I) (Cox Sec. 10, Silverman VI.2-3, Washington 9.1-2)notes, slides
1511/2Elliptic curves over C (part II) (Cox Sec. 10-11, Silverman VI.4-5, Washington 9.2-3)notes, slides
1611/7Complex multiplication (CM) (Cox Sec. 11, Silverman VI.5, Washington 9.3)notes, slides
1711/9The CM torsor (Cox Sec. 7, Silverman (advanced topics) II.1.1)notes, slides
1811/14Riemann surfaces and modular curves (Silverman (advanced topics) I.2, Milne V.1)notes, slides
1911/16The modular equation (Cox Sec. 11, Milne V.2, Washington pp. 273-274)notes, slides
2011/21The Hilbert class polynomial (Cox Sec. 8, 11)notes, slides
2111/28Ring class fields and the CM method (Cox 8, 11)notes, slides
2211/30Isogeny volcanoes (Sutherland)notes, >slides
2312/5The Weil pairing (Miller, Washington 11, Silverman III.8)notes, slides
2412/7Modular forms and L-functions (Milne V.3-4)notes, slides
2512/12Fermat's last theorem (Milne V.7-9, Washington 15, Cornell-Stevens-Silverman I)notes, slides