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Functional encryption and distributed signatures based on projective hash functions, the benefit of class groups

Ida Tucker 1, 2, 3, 4 
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
4 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : One of the current challenges in cryptographic research is the development of advanced cryptographic primitives ensuring a high level of confidence. In this thesis, we focus on their design, while proving their security under well-studied algorithmic assumptions.My work grounds itself on the linearity of homomorphic encryption, which allows to perform linear operations on encrypted data. Precisely, I built upon the linearly homomorphic encryption scheme introduced by Castagnos and Laguillaumie at CT-RSA'15. Their scheme possesses the unusual property of having a prime order plaintext space, whose size can essentially be tailored to ones' needs. Aiming at a modular approach, I designed from their work technical tools (projective hash functions, zero-knowledge proofs of knowledge) which provide a rich framework lending itself to many applications.This framework first allowed me to build functional encryption schemes; this highly expressive primitive allows a fine grained access to the information contained in e.g., an encrypted database. Then, in a different vein, but from these same tools, I designed threshold digital signatures, allowing a secret key to be shared among multiple users, so that the latter must collaborate in order to produce valid signatures. Such signatures can be used, among other applications, to secure crypto-currency wallets. Significant efficiency gains, namely in terms of bandwidth, result from the instantiation of these constructions from class groups. This work is at the forefront of the revival these mathematical objects have seen in cryptography over the last few years.
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Submitted on : Tuesday, November 24, 2020 - 2:16:08 PM
Last modification on : Monday, May 16, 2022 - 4:58:02 PM
Long-term archiving on: : Thursday, February 25, 2021 - 8:06:11 PM


Version validated by the jury (STAR)


  • HAL Id : tel-03021689, version 1


Ida Tucker. Functional encryption and distributed signatures based on projective hash functions, the benefit of class groups. Cryptography and Security [cs.CR]. Université de Lyon, 2020. English. ⟨NNT : 2020LYSEN054⟩. ⟨tel-03021689⟩



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