Universitas Gadjah Mada, Yogyakarta, Indonesia
July 21, 2025
Lecturing Session:
The seventh day of CIMPA SCHOOL 2025 began with Dr. Any Muanalifah delivering her first lecture on Cryptography. She introduced the main problems in cryptography and, using fundamental properties from number theory, explained classical cryptographic methods such as the Caesar Cipher, Vigenère Cipher, and RSA. She then discussed integer factorization methods including trial division, Fermat’s factorization, and Pollard’s rho algorithm. Dr. Any Muanalifah concluded that these classical cryptographic methods are no longer secure due to the effectiveness of integer factorization algorithms in attacking these systems.
The next session was delivered by Dr. Samuele Anni, who continued his lecture series on Modular Forms. He began by defining congruence subgroups, the slash operator, and modular forms for subgroups, providing interesting examples and results. The lecture continued with discussions on the theta function and the four-square theorem, and Hecke operators on modular forms of level 1, including key properties of Hecke operators. Dr. Anni introduced Maeda’s conjecture on Hecke operators and concluded with an insightful discussion on the relationship between elliptic curves, Galois representations, and Hecke operators.
SageMath Session:
Participants continued their practical work on modular forms and elliptic curves using SageMath, reinforcing the concepts learned during the lectures through computational exploration.
Extra Presentation:
The day concluded with three participant presentations on their work in algebraic number theory, number theory, and cryptography, allowing them to share their research and receive feedback from peers and faculty, fostering a collaborative learning environment.
