Summer 2022

Extended Reunion — Lattices: Algorithms, Complexity, and Cryptography

May 23Jun. 24, 2022

This extended reunion is for long-term participants in the program "Lattices: Algorithms, Complexity, and Cryptography" held in the Spring 2020 semester. It will provide an opportunity to meet old and new friends. Moreover, we hope that it will give everyone a chance to reflect on the progress made during the semester and since, and sketch which directions the field should go in the future. In an effort to keep things informal and to encourage open discussion, none of the activities will be recorded. Participation in the reunion is by invitation only.

The study of integer lattices serves as a bridge between number theory and geometry and has for centuries received the attention of illustrious mathematicians, including Lagrange, Gauss, Dirichlet, Hermite, and Minkowski. In computer science, lattices made a grand appearance in 1982 with the celebrated work of Lenstra, Lenstra, and Lovász, who developed the celebrated LLL algorithm to find short vectors in integer lattices. The role of lattices in cryptography has been equally, if not more, revolutionary and dramatic, with lattices first playing a destructive role as a potent tool for breaking cryptosystems and later serving as a new way to realize powerful and game-changing notions such as fully homomorphic encryption. These exciting developments over the last two decades have taken us on a journey through such diverse areas as quantum computation, learning theory, Fourier analysis, and algebraic number theory. 

The promise of practical lattice-based cryptosystems together with their apparent quantum resistance is generating a tremendous amount of interest in deploying these schemes at Internet scale. However, before lattice cryptography goes live, we need major advances in understanding the hardness of lattice problems that underlie the security of these cryptosystems. Significant, groundbreaking progress on these questions requires a concerted effort by researchers from many areas: (algebraic) number theory, (quantum) algorithms, optimization, cryptography, and coding theory.
The goal of this extended reunion for the "Lattices: Algorithms, Complexity, and Cryptography" program is to bring back together experts in these areas to review progress, to continue to attack outstanding open questions, and to continue the exploration of connections among lattices, computer science, and mathematics. The need to thoroughly understand the computational landscape and cryptographic capabilities of lattice problems is greater now than ever given the possibility that secure communication on the Internet and secure collaboration on the cloud might soon be powered by lattices.

This program was supported in part by the Alfred P. Sloan Foundation.



Shafi Goldwasser (UC Berkeley; co-chair), Vinod Vaikuntanathan (Massachusetts Institute of Technology; co-chair), Daniele Micciancio (UC San Diego), Oded Regev (Courant Institute, NYU), Hoeteck Wee (École Normale Supérieure, Paris)

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