Readings

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] = Buy at Amazon 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]

[Milne] = Buy at Amazon Milne, J. S. Elliptic Curves. BookSurge Publishing, 2006. ISBN: 9781419652578. (addendum / erratum (PDF))

[Silverman] = Buy at Amazon 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]

[Silverman (Advanced Topics)] = Buy at Amazon 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))

[Cox] = Buy at Amazon 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))

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 Buy at Amazon 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

Buy at Amazon 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]

Buy at Amazon 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 of Indian Automobile Manufactures Journal on Computing 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. This resource may not render correctly in a screen reader."Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p." (PDF) Mathematics of Computation 44, no. 170 (1985): 483–94.
10 Generic Algorithms for Discrete Logarithms

[Washington] Section 5.2

Pohlig, S., and M. Hellman. "An Improved Algorithm for Computing Logarithms Over GF(p) and Its Cryptographic Significance (Corresp.)." IEEE Transactions on Information Theory 24, no. 1 (1978): 106–10.

Pollard, J. M. "Monte Carlo Methods for Index Computation (mod p)." Mathematics of Computation 32, no. 143 (1978): 918–24.

Shoup, V. "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

Buy at Amazon Granville, Andrew. This resource may not render correctly in a screen reader."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. This resource may not render correctly in a screen reader."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, S., and J. Killan. "Almost all Primes can be Quickly Certified." STOC'86 Proceedings of the 18th 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. This resource may not render correctly in a screen reader."Isogeny Volcanoes." (PDF) 2012.
24 The Weil Pairing

Miller, Victor S. "The Weil Pairing, and Its Efficient Calculation." Journal of Cryptology: The Journal of the International Association for Cryptologie 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

Buy at Amazon Cornell, Gary, Joseph H. Silverman, and Glenn Stevens. Chapter 1 in Modular Forms and Fermat's Last Theorem. Springer, 2000. ISBN: 9780387989983.