Readings | Elliptic Curves | Mathematics

There is no required text, but lecture notes are provided. We make reference to material in the five books listed below. In addition, there are citations and links to other references.

[Washington] = Washington, Lawrence C. Elliptic Curves: Number Theory and Cryptography. Chapman & Hall / CRC, 2008. ISBN: 9781420071467. ( This resource may not render correctly in a screen reader. Errata (PDF)) [Preview with Google Books]. Online version.

[Milne] = Milne, James S. Elliptic Curves. BookSurge Publishing, 2006. ISBN: 9781419652578. ( This resource may not render correctly in a screen reader. Addendum / erratum (PDF)). Online version (PDF - 1.5MB).

[Silverman] = Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. ( This resource may not render correctly in a screen reader. Errata (PDF)) [Preview with Google Books]. Online version.

[Silverman (Advanced Topics)] = Silverman, Joseph H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, 1994. ISBN: 9780387943251. ( This resource may not render correctly in a screen reader. Errata (PDF)). Online version.

[Cox] = Cox, David A. Primes of the Form x² + ny²: Fermat, Class Field Theory, and Complex Multiplication. Wiley-Interscience, 1989. ISBN: 9780471506546. ( This resource may not render correctly in a screen reader. Errata (PDF)). Online version.

Course readings.
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." Lecture Notes in Computer Science 4833 (2007): 29–50.
3	Finite Fields and 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. Sections 3.2, 9.1, and 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 Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall / CRC, 2005. ISBN: 9781584885184. [Preview with Google Books] Rabin, Michael O. "Probabilistic Algorithms in Finite Fields." Society for Industrial and Applied Mathematics 9, no. 2 (1980): 273–80.
5	Isogenies	[Washington] Section 2.9 [Silverman] Section III.4
6	Isogeny Kernels and Division Polynomials	[Washington] Sections 3.2 and 12.3 [Silverman] Section III.4
7	Endomorphism Rings	[Washington] Section 4.2 [Silverman] Section III.6
8	Hasse's Theorem, 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 - 1.1MB) Mathematics of Computation 44, no. 170 (1985): 483–94.
10	Generic Algorithms for Discrete Logarithms	[Washington] Section 5.2 Pohlig, Stephen, and Martin Hellman. "An Improved Algorithm for Computing Logarithms Over GF(p) and Its Cryptographic Significance." (PDF) IEEE Transactions on Information Theory 24, no. 1 (1978): 106–10. Pollard, John M. "Monte Carlo Methods for Index Computation (mod p)." Mathematics of Computation 32, no. 143 (1978): 918–24. Shoup, Victor. "Lower Bounds for Discrete Logarithms and Related Problems." Lecture Notes in Computer Science 1233 (1997): 256–66.
11	Index Calculus, Smooth Numbers, Factoring Integers	[Washington] Sections 5.1 and 7.1 Granville, Andrew. "Smooth Numbers: Computational Number Theory and Beyond." (PDF) In Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. Cambridge University Press, 2008. ISBN: 9780521808545. Lenstra, H. W. "Factoring Integers with Elliptic Curves." (PDF - 1.3MB). Annals of Mathematics, Mathematical Sciences Research Institute 126 (1986): 649–73.
12	Elliptic Curve Primality Proving (ECPP)	[Washington] Section 7.2 Goldwasser, Shafi, and Joe Killan. "Almost all Primes can be Quickly Certified." (PDF - 1MB). STOC'86 Proceedings of the 18^th Annual ACM Symposium on Theory of Computing (1986): 316–29. Pomerance, Carl. "Very Short Primality Proofs." Mathematics of Computation 48, no. 177 (1987): 315.
13	Endomorphism Algebras	[Silverman] Section III.9
14	Ordinary and Supersingular Curves	[Silverman] Section III.1 and Chapter V [Washington] Sections 2.7 and 4.6
15	Elliptic Curves over C (Part 1)	[Cox] Chapter 10 [Silverman] Sections VI.2–3 [Washington] Sections 9.1–2
16	Elliptic Curves over C (Part 2)	[Cox] Chapters 10 and 11 [Silverman] Sections VI.4–5 [Washington] Sections 9.2–3
17	Complex Multiplication	[Cox] Chapter 11 [Silverman] Section VI.5 [Washington] Section 9.3
18	The CM Action	[Cox] Chapter 7 [Silverman (Advanced Topics)] Section II.1.1
19	Riemann Surfaces and Modular Curves	[Silverman (Advanced Topics)] Section I.2 [Milne] Section V.1
20	The Modular Equation	[Cox] Chapter 11 [Milne] Section V.2 [Washington] pp. 273–74
21	The Hilbert Class Polynomial	[Cox] Chapters 8 and 11
22	Ring Class Fields and the CM Method	[Cox] Chapters 8 and 11 (cont.)
23	Isogeny Volcanoes	Sutherland, Andrew V. "Isogeny Volcanoes." The Open Book Series. 1, no. 1 (2013): 507–530.
24	Divisors and the Weil Pairing	Miller, Victor S. "The Weil Pairing, and Its Efficient Calculation." Journal of Cryptology: The Journal of the International Association for Cryptologic Research (IACR) 17, no. 4 (2004): 235–61. [Washington] Chapter 11 [Silverman] Section III.8
25	Modular Forms and L-Functions	[Milne] Sections V.3–4
26	Fermat's Last Theorem	[Milne] Sections V.7–9 [Washington] Chapter 15 Cornell, Gary, Joseph H. Silverman, and Glenn Stevens. Chapter 1 in Modular Forms and Fermat's Last Theorem. Springer, 2000. ISBN: 9780387989983. Online version.