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LattAC Report Summary

Project ID: 335086
Funded under: FP7-IDEAS-ERC
Country: France

Mid-Term Report Summary - LATTAC (Lattices: algorithms and cryptography)

Contemporary cryptography, with security relying on the factorisation and discrete logarithm problems, is ill-prepared for the future: It will collapse with the rise of quantum computers, its costly algorithms require growing resources, and it is utterly ill-fitted for the fast-developing trend of externalising computations to the cloud. The emerging field of *lattice-based cryptography* (LBC) addresses these concerns: it resists would-be quantum computers, trades memory for drastic run-time savings, and enables computations on encrypted data, leading to the prospect of a privacy-preserving cloud economy. LBC could supersede contemporary cryptography within a decade.
A major goal of this project is to enable this technology switch.

To achieve it, the project aims at studying all computational aspects of lattices, with cryptography as the driving motive, and with a strong focus on lattice algorithms. Indeed, assessing the limits of current lattice algorithms and attempting to find novel algorithmic approaches are crucial towards establishing the security of lattice-based cryptography.

The project contains three main components: the conception of lattice-based cryptographic primitives, the study of the hardness assumptions underlying LBC, and the design, analysis and implementation of lattice algorithms.

Concerning the *expressiveness of LBC*, our main result in the first half of the project was the first cryptanalysis of a multi-linear map, severely impacting the bounty of their cryptographic applications. This was followed by results from other teams breaking many other multi-linear maps. Today, the single surviving application seems to be indistinguishability obfuscation, which itself has many applications, but whose security foundations are questionable. In the second half of the project, we plan to study the security of indistinguishability obfuscation constructions and of some of its applications.

We have significantly simplified the landscape of *hardness assumptions* used in LBC. We showed that the Approximate GCD problem and the more classical LWE problem reduce to one another, we exhibited a serious weakness in the (overstretched) NTRU hardness assumptions, and we studied the hardness of variants of LWE. In the context of the LWE variants, we showed the relevance of the Renyi Divergence to quantify the closeness of two distributions, when it comes to studying lattice problems. In the second half of the project, we plan to focus on the quantum hardness of lattice problems, in particular those arising in LBC.

We made progress on various *algorithmic objectives* related to lattices. We proposed the asymptotically fastest LLL-type lattice reduction algorithm, which allows to efficiently compute a basis of decent quality. We designed a (heuristic) algorithm for computing a shortest non-zero vector in a lattice, which is exponential-time like the best known algorithms, but requires less memory. On the implementation side, we contributed to a massive overhaul of the fplll lattice reduction library, which now contains the fastest available implementations of most lattice reduction algorithms.

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