Elliptic Curves

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] = Washington, Lawrence C. Elliptic Curves: Number Theory and Cryptography. Chapman & Hall / CRC, 2008. ISBN: 9781420071467. (Errata (PDF)) [Preview with Google Books]. Online version.

[Milne] = Milne, James S. Elliptic Curves. BookSurge Publishing, 2006. ISBN: 9781419652578. (Addendum / erratum (PDF)). Online version (PDF - 1.5MB).

[Silverman] = Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. (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. (Errata (PDF)). Online version.

[Cox] = Cox, David A. Primes of the Form \(x^2 + ny^2\): Fermat, Class Field Theory, and Complex Multiplication. Wiley-Interscience, 1989. ISBN: 9780471506546. (Errata (PDF)). Online version.

Lecture 1: Introduction to Elliptic Curves

• No readings assigned

Lecture 2: The Group Law and Weierstrass and Edwards Equations

• [Washington] Sections 2.1–3 and 2.6.3
• Bernstein, Daniel, and Tanja Lange. “Faster Addition and Doubling on Elliptic Curves.” Lecture Notes in Computer Science 4833 (2007): 29–50.
• Bernstein, Daniel, and Tanja Lange. “A Complete Set of Addition Laws for Incomplete Edwards Curves.” (PDF)

Lecture 3: Finite Field Arithmetic

• Gathen, Joachim von zur, and Jürgen Gerhard. Sections 3.2, 8.1–4, 9.1, 11.1, and 14.2–6 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.

Lecture 4: Isogenies

• [Washington] Section 2.9
• [Silverman] Section III.4

Lecture 5: Isogeny Kernels and Division Polynomials

• [Washington] Sections 3.2 and 12.3
• [Silverman] Section III.4

Lecture 6: Endomorphism Rings

• [Washington] Section 4.2
• [Silverman] Section III.6

Lecture 7: Hasse’s Theorem and Point Counting

• [Washington] Section 4.3

Lecture 8: 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.

Lecture 9: Generic Algorithms for the Discrete Logarithm Problem

• [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).” (PDF) Mathematics of Computation 32, no. 143 (1978): 918–24.
• Shor, Peter W. “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer.” SIAM J.Sci.Statist.Comput. 26 (1997): 1484.
• Shoup, Victor. “Lower Bounds for Discrete Logarithms and Related Problems.” Lecture Notes in Computer Science 1233 (1997): 256–66.

Lecture 10: Index Calculus, Smooth Numbers, and 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.

Lecture 11: Elliptic Curve Primality Proving (ECPP)

• [Washington] Section 7.2
• Goldwasser, Shafi, and Joe Kilian. “Almost All Primes Can Be Quickly Certified.” STOC'86 Proceedings of the 18^th Annual ACM Symposium on Theory of Computing (1986): 316–29.
• Pomerance, Carl. “Very Short Primality Proofs.” (PDF) Mathematics of Computation 48, no. 177 (1987): 315–22.

Lecture 12: Endomorphism Algebras

• [Silverman] Section III.9

Lecture 13: Ordinary and Supersingular Curves

• [Silverman] Section III.1 and Chapter V
• [Washington] Sections 2.7 and 4.6

Lecture 14: Elliptic Curves over C (Part I)

• [Cox] Chapter 10
• [Silverman] Sections VI.2–3
• [Washington] Sections 9.1–2

Lecture 15: Elliptic Curves over C (Part II)

• [Cox] Chapters 10 and 11
• [Silverman] Sections VI.4–5
• [Washington] Sections 9.2–3

Lecture 16: Complex Multiplication (CM)

• [Cox] Chapter 11
• [Silverman] Section VI.5
• [Washington] Section 9.3

Lecture 17: The CM Torsor

• [Cox] Chapter 7
• [Silverman (Advanced Topics)] Section II.1.1

Lecture 18: Riemann Surfaces and Modular Curves

• [Silverman (Advanced Topics)] Section I.2
• [Milne] Section V.1

Lecture 19: The Modular Equation

• [Cox] Chapter 11
• [Milne] Section V.2
• [Washington] pp. 273–74

Lecture 20: The Hilbert Class Polynomial

• [Cox] Chapters 8 and 11

Lecture 21: Ring Class Fields and the CM Method

• [Cox] Chapters 8 and 11 (cont.)

Lecture 22: Isogeny Volcanoes

• Sutherland, Andrew V. “Isogeny Volcanoes.” The Open Book Series. 1, no. 1 (2013): 507–530.

Lecture 23: 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

Lecture 24: Modular Forms and L-Functions

• [Milne] Sections V.3–4

Lecture 25: 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.