Abstract

We consider multiparty information-theoretic private protocols, and specifically their randomness complexity. The randomness complexity of private protocols is of interest both because random bits are considered a scarce resource and because of the relation between that complexity measure and other complexity measures of boolean functions such as the circuit size or the sensitivity of the function being computed [Kushilevitz, Ostrovsky, and Rosén, J. Comput. Syst. Sci., 58 (1999), pp. 129--136] and [Gál and Rosén, SIAM J. Comput., 31 (2002), pp. 1424--1437]. More concretely, we consider the randomness complexity of the basic Boolean function \tt and, that serves as a building block in the design of many private protocols. We show that \tt and cannot be privately computed using a single random bit, thus giving the first nontrivial lower bound on the 1-private randomness complexity of an explicit Boolean function, $f: \{0,1\}^n \rightarrow \{0,1\}$. We further show that and, on any number of inputs $n$ (one input bit per player), can be privately computed using 8 random bits (and 7 random bits in the special case of $n=3$ players), improving the upper bound of 73 random bits implicit in [Kushilevitz, Ostrovsky, and Rosén, J. Comput. Syst. Sci., 58 (1999), pp. 129--136]. Together with our lower bound, we thus approach the exact determination of the randomness complexity of \tt and. To the best of our knowledge, the exact randomness complexity of private computation is not known for any explicit function (except for \tt xor, which is 1-random, and for several degenerate functions).

Keywords

  1. multiparty private computation
  2. randomness
  3. lower bounds

MSC codes

  1. 68Q17
  2. 68Q87

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 465 - 484
ISSN (online): 1095-7146

History

Submitted: 21 January 2020
Accepted: 7 December 2020
Published online: 23 March 2021

Keywords

  1. multiparty private computation
  2. randomness
  3. lower bounds

MSC codes

  1. 68Q17
  2. 68Q87

Authors

Affiliations

Funding Information

Defense Advanced Research Projects Agency https://doi.org/10.13039/100000185 : HR0011-20-2-0025
Department of Science and Technology, Ministry of Science and Technology https://doi.org/10.13039/501100001409
Israel Science Foundation https://doi.org/10.13039/501100003977 : 1709/14
United States - Israel Binational Science Foundation https://doi.org/10.13039/100006221 : 2018393, 2015782, 1619348, 2012366

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