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I've been reading How to estimate the hardness of SIS instances? and following some of its sources, and I want to confirm a few things.

LLL algorithm runs in polynomial time, but isn't capable of producing arbitrarily small basis, and is therefore not very relevant in cryptanalysis in most lattice-based schemes. Is that correct?

BKZ algorithm invokes a SVP (Shortest Vector Problem) oracle polynomial number of times, and the lower-bounded of the norm of the basis it produces is inversely proportional to the block size the SVP oracle operates on, regardless of the norm of the input basis; and the SVP oracle it uses runs in time exponential to the block size. Is that correct?

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Yes, if you instantiate the SVP oracle in BKZ using sieving then the cost is exponential in time and memory. (If you instantiate it using enumeration then the cost is $k^{1/8\,k + o(k)}$). The norm of the output vector is expected to be $\delta_k^{d-1} \cdot \operatorname{Vol}(\Lambda)^{1/d}$ with $\delta_k \approx GH(k)^{1/(k-1)}$. Here $\operatorname{Vol}(\Lambda)$ is the volume of the lattice, and $GH(k) \approx \sqrt{k/(2\pi e)}$ is the Gaussian heuristic for a lattice with volume 1. This does not depend on the norm of the input basis.

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  • $\begingroup$ $\operatorname{Vol}(\Lambda)$ - > $\operatorname{Vol}(\Lambda)^{\mathbf 1/d}$ $\endgroup$ – LeoDucas Oct 7 '20 at 7:06
  • $\begingroup$ Thanks for this Léo $\endgroup$ – Martin R. Albrecht Oct 7 '20 at 11:05
  • $\begingroup$ Nevertheless LLL is important in cryptanalysis, serving as generic function to find short vectors under certain conditions. There is plenty of work that simply requires a polynomial time lattice reduction, where the lattice can be constructed in such a way that LLL is successful. And since LLL was the first, I believe it often takes this role as "generic lattice reduction". $\endgroup$ – Fleeep Oct 8 '20 at 7:27

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