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computational complexity

, Volume 8, Issue 2, pp 145–168 | Cite as

Randomness complexity of private computation

  • C. Blundo
  • A. De Santis
  • G. Persiano
  • U. Vaccaro

Abstract.

We consider the problem of n honest but curious players with private inputs \( x_1,\ldots,x_n, \) who wish to compute the value of a fixed function \( {\cal F}(x_1,\ldots,x_n) \) in such way that at the end of the protocol every player knows the value \( {\cal F}(x_1,\ldots,x_n) \). Each pair of players is connected by a secure point-to-point communication channel. The players have unbounded computational resources and they intend to compute \( {\cal F} \) in a t-private way. That is, after the execution of the protocol, no coalition of size at most \( t \le n - 1 \) can get any information about the inputs of the remaining players other than what can be deduced from their own inputs and the value of \( \cal F \).¶ We study the amount of randomness needed in t-private protocols. We prove a lower bound on the randomness complexity of any t-private protocol to compute a function with sensitivity n. As a corollary, we obtain that when the private inputs are uniformly distributed, at least k(n—1)(n—2)/2 random bits are needed to compute the sum modulo 2 k of n k-bit integers in an (n—2)-private way. This result is tight as there are protocols for this problem that use exactly this number of random bits.

Keywords. Cryptography, private protocols, randomness, information theory, entropy. 

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Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • C. Blundo
    • 1
  • A. De Santis
    • 2
  • G. Persiano
    • 3
  • U. Vaccaro
    • 4
  1. 1.Dip. di Informatica ed Appl., Università di Salerno, 84081 Baronissi SA, Italy, e-mail: carblu@dia.unisa.itIT
  2. 2.Dip. di Informatica ed Appl., Università di Salerno, 84081 Baronissi SA, Italy, e-mail: ads@dia.unisa.itIT
  3. 3.Dip. di Informatica ed Appl., Università di Salerno, 84081 Baronissi SA, Italy, e-mail: giuper@dia.unisa.itIT
  4. 4.Dip. di Informatica ed Appl., Università di Salerno, 84081 Baronissi SA, Italy, e-mail: uv@dia.unisa.itIT

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