We conjecture that the smallest possible share size for binary secrets for the t-out-of-n and (n-t+1)-out-of-n access structures is the same for all 1 ≤ t ≤ n. This is a strenghtening of a recent conjecture by Csirmaz (J. Math. Cryptol., 2020). We prove the conjecture for t = 2 and all n. Our proof gives a new (n-1)-out-of-n secret sharing scheme for binary secrets with share alphabet size n.
@InProceedings{bogdanov:LIPIcs.ITC.2023.3, author = {Bogdanov, Andrej}, title = {{Csirmaz’s Duality Conjecture and Threshold Secret Sharing}}, booktitle = {4th Conference on Information-Theoretic Cryptography (ITC 2023)}, pages = {3:1--3:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-271-6}, ISSN = {1868-8969}, year = {2023}, volume = {267}, editor = {Chung, Kai-Min}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2023.3}, URN = {urn:nbn:de:0030-drops-183317}, doi = {10.4230/LIPIcs.ITC.2023.3}, annote = {Keywords: Threshold secret sharing, Fourier analysis} }
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