├── README.md └── note.pdf /README.md: -------------------------------------------------------------------------------- 1 | # A note on a Claim of Eldar & Hallgren: LLL already solves it 2 | 3 | following up on the presentation of Sean Hallgren at the Simons Institute 4 | https://simons.berkeley.edu/events/efficient-quantum-algorithm-lattice-problems-achieving-subexponential-approximation-factor 5 | 6 | Wessel and myself wrote a note (available on this repository) 7 | on what the state of the art has to say on those specific 8 | BDD instances studied by Lior and Sean. 9 | https://github.com/lducas/BDD-note/blob/main/note.pdf 10 | 11 | The first remark is that they are already considered easy in 12 | the average-case for standard classical lattice reduction 13 | algorithms. With due diligence, we can in fact prove this is 14 | also true in the worst-case. This requires no new idea. The 15 | algorithm is just straight up LLL+Babai. 16 | 17 | # FAQ 18 | 19 | ** Does the note contain a new result on lattice problems ? 20 | 21 | No. There is only a sudden focus on specific worst-case instances 22 | that were previously undocumented, simply because the average-case 23 | was the case of interest in crypto. 24 | 25 | ** Does the fact that it is a worst-case result invalidate the 26 | underlying worst-case to average-case reasoning ? 27 | 28 | No. Ajtai's and Regev worst-case reduction ranges over *all* 29 | lattices. This polynomial time algorithm applies to a small 30 | class of easy lattices. 31 | 32 | ** The note only deal with the case k=1, and q = c^n. What about 33 | the general case ? 34 | 35 | We will generalize the note as soon as we find the time to do so. 36 | In the mean time, the best guess is that it would behaves as in 37 | the random-case (LWE): BDD in deterministic poly-time for this 38 | class of lattices when parameters satisfies: 39 | 40 | k * log(q) / log^2(α) < cste . 41 | -------------------------------------------------------------------------------- /note.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/lducas/BDD-note/5e658c469f27e29466935adac9c46b93b6b55e94/note.pdf --------------------------------------------------------------------------------