Abstract
The recent line of study on randomness extractors has been a great success, resulting in exciting new techniques, new connections, and breakthroughs to long standing open problems in several seemingly different topics. These include seeded nonmalleable extractors, privacy amplification protocols with an active adversary, independent source extractors (and explicit Ramsey graphs), and nonmalleable codes in the split state model. Previously, the best constructions are given in [Xin Li, 2017]: seeded nonmalleable extractors with seed length and entropy requirement O(log n+log(1/epsilon)log log (1/epsilon)) for error epsilon; tworound privacy amplification protocols with optimal entropy loss for security parameter up to Omega(k/log k), where k is the entropy of the shared weak source; twosource extractors for entropy O(log n log log n); and nonmalleable codes in the 2split state model with rate Omega(1/log n). However, in all cases there is still a gap to optimum and the motivation to close this gap remains strong.
In this paper, we introduce a set of new techniques to further push the frontier in the above questions. Our techniques lead to improvements in all of the above questions, and in several cases partially optimal constructions. This is in contrast to all previous work, which only obtain close to optimal constructions. Specifically, we obtain:
1) A seeded nonmalleable extractor with seed length O(log n)+log^{1+o(1)}(1/epsilon) and entropy requirement O(log log n+log(1/epsilon)), where the entropy requirement is asymptotically optimal by a recent result of Gur and Shinkar [Tom Gur and Igor Shinkar, 2018];
2) A tworound privacy amplification protocol with optimal entropy loss for security parameter up to Omega(k), which solves the privacy amplification problem completely;
3) A twosource extractor for entropy O((log n log log n)/(log log log n)), which also gives an explicit Ramsey graph on N vertices with no clique or independent set of size (log N)^{O((log log log N)/(log log log log N))}; and
4) The first explicit nonmalleable code in the 2split state model with constant rate, which has been a major goal in the study of nonmalleable codes for quite some time. One small caveat is that the error of this code is only (an arbitrarily small) constant, but we can also achieve negligible error with rate Omega(log log log n/log log n), which already improves the rate in [Xin Li, 2017] exponentially.
We believe our new techniques can help to eventually obtain completely optimal constructions in the above questions, and may have applications in other settings.
BibTeX  Entry
@InProceedings{li:LIPIcs:2019:10850,
author = {Xin Li},
title = {{NonMalleable Extractors and NonMalleable Codes: Partially Optimal Constructions}},
booktitle = {34th Computational Complexity Conference (CCC 2019)},
pages = {28:128:49},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771160},
ISSN = {18688969},
year = {2019},
volume = {137},
editor = {Amir Shpilka},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10850},
URN = {urn:nbn:de:0030drops108507},
doi = {10.4230/LIPIcs.CCC.2019.28},
annote = {Keywords: extractor, nonmalleable, privacy, codes}
}
Keywords: 

extractor, nonmalleable, privacy, codes 
Collection: 

34th Computational Complexity Conference (CCC 2019) 
Issue Date: 

2019 
Date of publication: 

16.07.2019 