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DOI: 10.4230/LIPIcs.ITC.2021.10
URN: urn:nbn:de:0030-drops-143299
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Damgård, Ivan Bjerre ; Larsen, Kasper Green ; Yakoubov, Sophia

Broadcast Secret-Sharing, Bounds and Applications

LIPIcs-ITC-2021-10.pdf (0.7 MB)


Consider a sender 𝒮 and a group of n recipients. 𝒮 holds a secret message 𝗆 of length l bits and the goal is to allow 𝒮 to create a secret sharing of 𝗆 with privacy threshold t among the recipients, by broadcasting a single message 𝖼 to the recipients. Our goal is to do this with information theoretic security in a model with a simple form of correlated randomness. Namely, for each subset 𝒜 of recipients of size q, 𝒮 may share a random key with all recipients in 𝒜. (The keys shared with different subsets 𝒜 must be independent.) We call this Broadcast Secret-Sharing (BSS) with parameters l, n, t and q.
Our main question is: how large must 𝖼 be, as a function of the parameters? We show that (n-t)/q l is a lower bound, and we show an upper bound of ((n(t+1)/(q+t)) -t)l, matching the lower bound whenever t = 0, or when q = 1 or n-t.
When q = n-t, the size of 𝖼 is exactly l which is clearly minimal. The protocol demonstrating the upper bound in this case requires 𝒮 to share a key with every subset of size n-t. We show that this overhead cannot be avoided when 𝖼 has minimal size.
We also show that if access is additionally given to an idealized PRG, the lower bound on ciphertext size becomes (n-t)/q λ + l - negl(λ) (where λ is the length of the input to the PRG). The upper bound becomes ((n(t+1))/(q+t) -t)λ + l.
BSS can be applied directly to secret-key threshold encryption. We can also consider a setting where the correlated randomness is generated using computationally secure and non-interactive key exchange, where we assume that each recipient has an (independently generated) public key for this purpose. In this model, any protocol for non-interactive secret sharing becomes an ad hoc threshold encryption (ATE) scheme, which is a threshold encryption scheme with no trusted setup beyond a PKI. Our upper bounds imply new ATE schemes, and our lower bound becomes a lower bound on the ciphertext size in any ATE scheme that uses a key exchange functionality and no other cryptographic primitives.

BibTeX - Entry

  author =	{Damg\r{a}rd, Ivan Bjerre and Larsen, Kasper Green and Yakoubov, Sophia},
  title =	{{Broadcast Secret-Sharing, Bounds and Applications}},
  booktitle =	{2nd Conference on Information-Theoretic Cryptography (ITC 2021)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-197-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{199},
  editor =	{Tessaro, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-143299},
  doi =		{10.4230/LIPIcs.ITC.2021.10},
  annote =	{Keywords: Secret-Sharing, Ad-hoc Threshold Encryption}

Keywords: Secret-Sharing, Ad-hoc Threshold Encryption
Collection: 2nd Conference on Information-Theoretic Cryptography (ITC 2021)
Issue Date: 2021
Date of publication: 19.07.2021

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