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## Section: Partnerships and Cooperations

### National Initiatives

#### ANR Alambic – AppLicAtions of MalleaBIlity in Cryptography

Participant : Guilhem Castagnos.

The Alambic project is a research project formed by members of the Inria Project-Team CASCADE of ENS Paris, members of the AriC Inria project-team of ENS Lyon, and members of the CRYPTIS of the university of Limoges. G. Castagnos is an external member of the team of Lyon for this project.

Non-malleability is a security notion for public key cryptographic encryption schemes that ensures that it is infeasible for an adversary to modify ciphertexts into other ciphertexts of messages which are related to the decryption of the first ones. On the other hand, it has been realized that, in specific settings, malleability in cryptographic protocols can actually be a very useful feature. For example, the notion of homomorphic encryption allows specific types of computations to be carried out on ciphertexts and generate an encrypted result which, when decrypted, matches the result of operations performed on the plaintexts. The homomorphic property can be used to create secure voting systems, collision-resistant hash functions, private information retrieval schemes, and for fully homomorphic encryption enables widespread use of cloud computing by ensuring the confidentiality of processed data.

The aim of the Alambic project to investigate further theoretical and practical applications of malleability in cryptography. More precisely, this project focuses on three different aspects: secure computation outsourcing and server-aided cryptography, homomorphic encryption and applications and << paradoxical >> applications of malleability.

#### ANR CLap–CLap – The $p$-adic Langlands correspondence: a constructive and algorithmical approach

Participant : Xavier Caruso.

The $p$-adic Langlands correspondence has become nowadays one of the deepest and the most stimulating research programs in number theory. It was initiated in France in the early 2000's by Breuil and aims at understanding the relationships between the $p$-adic representations of $p$-adic absolute Galois groups on the one hand and the $p$-adic representations of $p$-adic reductive groups on the other hand. Beyond the case of ${\text{GL}}_{2}\left({ℚ}_{p}\right)$ which is now well established, the $p$-adic Langlands correspondence remains quite obscure and mysterious new phenomena enter the scene; for instance, on the ${\text{GL}}_{n}\left(F\right)$-side one encounters a vast zoology of representations which seems extremely difficult to organize.

The CLap–CLap ANR project aims at accelerating the expansion of the $p$-adic Langlands program beyond the well-established case of ${\text{GL}}_{2}\left({ℚ}_{p}\right)$. Its main originality consists in its very constructive approach mostly based on algorithmics and calculations with computers at all stages of the research process. We shall pursue three different objectives closely related to our general aim:

1. draw a conjectural picture of the (still hypothetical) $p$-adic Langlands correspondence in the case of ${\text{GL}}_{n}$,

2. compute many deformation spaces of Galois representations and make the bridge with deformation spaces of representations of reductive groups,

3. design new algorithms for computations with Hilbert and Siegel modular forms and their associated Galois representations.

This project will also be the opportunity to contribute to the development of the mathematical software SageMath and to the expansion of computational methodologies.