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I am a last-year master student in pure mathematics and I am working on my thesis. I am working on a connection between lattice-based encryption and Ring LWE and between Ring LWE and Homomorphic encryption. For the second part, I manage to find an appropriate paper to provide me with some information. However, with the connection of the lattice-based encryption and LWE things seems to be messier. Apart from the assumptions of LWE that we have seen in a course of mine (search-LWE and decisional-LWE) I was able to find a Regev's definition where it presents that LWE is a sequence of approximations and the problem is to find the suitable vector to solve those approximations and also that this problem can be extended in the R-LWE to polynomials. I was also able to find a matrix form of the LWE definition. So let's say that I can understand how these three parts are connected, I still don't see how to connect them with lattices, so I kept searching. In many papers, I found that the hardness of the lattice-based problems, Shortest Vector Problem (SVP) and Closest Vector Problem (CVP), is connected with the LWE but nothing more. So I was wondering if anyone is familiar with any book/paper that explains how lattice-based cryptography is connected with LWE and explains the structure of R-LWE or at least guides me to a less chaotic path?

Thank you in advance.

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I highly recommend this survey as a summary of all things lattice-based.

And to answer your question, LWE and it's variants have hardness reductions to certain lattice problems (e.g. GapSVP). That is to say that breaking an encryption scheme like LWE is at least as hard as solving the corresponding lattice problems (for certain lattices).

The security of schemes like LWE depend on the hardness of lattice problems.

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