Abstract
Let F_{q} be the finite field of size q and let l: F_{q}^{n} > F_{q} be a linear function. We introduce the Learning From Subset problem LFS(q,n,d) of learning l, given samples u in F_{q}^{n} from a special distribution depending on l: the probability of sampling u is a function of l(u) and is non zero for at most d values of l(u). We provide a randomized algorithm for LFS(q,n,d) with sample complexity (n+d)^{O(d)} and running time polynomial in log q and (n+d)^{O(d)}. Our algorithm generalizes and improves upon previous results [Friedl et al., 2014; Gábor Ivanyos, 2008] that had provided algorithms for LFS(q,n,q1) with running time (n+q)^{O(q)}. We further present applications of our result to the Hidden Multiple Shift problem HMS(q,n,r) in quantum computation where the goal is to determine the hidden shift s given oracle access to r shifted copies of an injective function f: Z_{q}^{n} > {0, 1}^{l}, that is we can make queries of the form f_{s}(x,h) = f(xhs) where h can assume r possible values. We reduce HMS(q,n,r) to LFS(q,n, qr+1) to obtain a polynomial time algorithm for HMS(q,n,r) when q=n^{O(1)} is prime and qr=O(1). The best known algorithms [Andrew M. Childs and Wim van Dam, 2007; Friedl et al., 2014] for HMS(q,n,r) with these parameters require exponential time.
BibTeX  Entry
@InProceedings{ivanyos_et_al:LIPIcs:2018:9529,
author = {G{\'a}bor Ivanyos and Anupam Prakash and Miklos Santha},
title = {{On Learning Linear Functions from Subset and Its Applications in Quantum Computing}},
booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)},
pages = {66:166:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770811},
ISSN = {18688969},
year = {2018},
volume = {112},
editor = {Yossi Azar and Hannah Bast and Grzegorz Herman},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9529},
URN = {urn:nbn:de:0030drops95299},
doi = {10.4230/LIPIcs.ESA.2018.66},
annote = {Keywords: Learning from subset, hidden shift problem, quantum algorithms, linearization}
}
Keywords: 

Learning from subset, hidden shift problem, quantum algorithms, linearization 
Collection: 

26th Annual European Symposium on Algorithms (ESA 2018) 
Issue Date: 

2018 
Date of publication: 

14.08.2018 