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18.783 | Fall 2025 | Undergraduate

# Elliptic Curves

There is no required textbook; 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. Second edition. Chapman & Hall / CRC, 2008. ISBN: 9781420071467. (Errata (PDF)) Online version.

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

[Silverman] = Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. (Errata (PDF)) 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.” (PDF)
- 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. Sections 9.1–2 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.
- Cantor, David G. and Hans Zassenhaus. “A New Algorithm for Factoring Polynomials Over Finite Fields.” (PDF)

### 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) 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.” (PDF)

### 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) 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 18th 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.

## Course Info

##### Instructor

- Andrew V. Sutherland

##### Departments

- Mathematics

##### As Taught In

Fall 2025

##### Level

Undergraduate

##### Topics

- Engineering
  - Computer Science
- Mathematics
  - Algebra and Number Theory
  - Computation

##### Learning Resource Types

notes Lecture Notes
menu_book Open Textbooks
assignment Problem Sets
auto_stories Readings

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