There has been a recent revolution in cryptography due to the introduction of lattice-based constructions. These are cryptographic schemes whose security relies on the presumed hardness of certain computational problems over ubiquitous (and beautiful) geometric objects called lattices. Their many applications (e.g., fully homomorphic encryption) and security against adversaries with quantum computers has created some urgency to deploy lattice-based schemes widely over the next few years. For example, the National Institute of Standards and Technology is in the process of standardizing lattice-based cryptography, and Google has already implemented such a scheme in its Canary browser.

The security of the proposed schemes relies crucially on the assumption that our current best algorithms (both classical and quantum) for the relevant computational lattice problems cannot be improved by even a relatively small amount. I will discuss the state of the art in the study of this assumption. In particular, I will describe the fastest known algorithms for these problems (and potential directions to improve them) as well as a recent series of hardness results that use the tools of fine-grained complexity to provide strong evidence for the security of lattice-based cryptography.

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