1. Introduction & Purposes

Cryptography is an essential technology in modern society for ensuring confidentiality, authentication, and integrity of information. Cryptographic schemes are designed using mathematical structures, and their security is evaluated through mathematical and computational analysis.

Elliptic curve cryptography (ECC), which is based on the arithmetic of elliptic curves over finite fields, is widely used in practice and is trusted for its strong security and efficiency.

On the other hand, many existing public-key cryptography, including RSA and ECC, are known to be vulnerable to large-scale quantum computers. This has motivated extensive research on post-quantum cryptography (PQC). One important research direction in this area is isogeny-based cryptography, which uses isogenies between supersingular elliptic curves.

2. Outline & Descriptions

This course begins with an overview of cryptography and then introduces ECC and number-theoretic algorithms, including point-counting algorithms and algorithms for the elliptic curve discrete logarithm problem.

We will also discuss the background of post-quantum cryptography and introduce isogeny-based cryptography, together with the related arithmetic of elliptic curves and isogenies.

If time permits, we will discuss recent developments and further topics in isogeny-based cryptography.

The course will be given using slides. The slides will be distributed to the participants as needed.

Recommended Textbooks and References:
General topics in mathematical cryptography
--Neal Koblitz, A Course in Number Theory and Cryptography, Graduate Texts in Mathematics, Springer, 1994.
--Steven D. Galbraith, Mathematics of Public Key Cryptography, Cambridge University Press, 2012.
Elliptic curve cryptography
--Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography (2nd Ed.), CRC Press, 2008.
Isogeny-based cryptography
--Denis X. Charles, Eyal Z. Goren, and Kristin E. Lauter, Cryptographic hash functions from expander graphs, Journal of Cryptology, 22, 93--113, 2009.
--Luca De Feo, David Jao, and J'er^ome Pl^ut, Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies, Journal of Mathematical Cryptology, 8(3), 209--247, 2014.
--Wouter Castryck and Thomas Decru, An efficient key recovery attack on SIDH, EUROCRYPT 2023, 2023.

3. Prerequisites

The course assumes basic knowledge of algebra, in particular finite fields, and elementary number theory. Familiarity with elliptic curves is helpful but not strictly required, since the necessary background will be reviewed during the course.

Prior knowledge of public-key cryptography is not required. The basic concepts and examples will be introduced in the course.

4. Registration

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