Efficient Bit-Decomposition and Modulus-Conversion Protocols with an Honest Majority

  • Ryo KikuchiEmail author
  • Dai Ikarashi
  • Takahiro Matsuda
  • Koki Hamada
  • Koji Chida
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10946)


In this paper, we propose secret-sharing-based bit-decomposition and modulus-conversion protocols for a prime order ring \(\mathbb {Z}_p\) with an honest majority: an adversary can corrupt \(k-1\) parties of n parties and \(2k-1 \le n\). Our protocols are secure against passive and active adversaries depending on the components of our protocols. We assume a secret is an \(\ell \)-bit element and \(2^{\ell +\lceil \log m \rceil } < p\), where \(m= k\) in the passive security and \(m= \left( {\begin{array}{c}n\\ k-1\end{array}}\right) \) in the active security. The outputs of our bit-decomposition and modulus-conversion protocols are \(\ell \) tuple of shares in \(\mathbb {Z}_2\) and a share in \(\mathbb {Z}_{p'}\), respectively, where \(p'\) is the modulus after the conversion. If k and n are small, the communication complexity of our passively secure bit-decomposition and modulus-conversion protocols are \(O(\ell )\) bits and \(O(\lceil \log p' \rceil )\) bits, respectively. Our key observation is that a quotient of additive shares can be computed from the least significant \(\lceil \log m \rceil \) bits. If a secret a is “shifted” and additively shared as \(x_i\)s so that \(2^{\lceil \log m \rceil }a = {\sum _{i=0}^{m-1}}x_i = 2^{ \lceil \log m \rceil } a + qp\), the least significant \(\lceil \log m \rceil \) bits of \(\sum _{i=0}^{m-1} x_i\) determine q since p is an odd prime and the least significant \(\lceil \log m \rceil \) bits of \(2^{\lceil \log m \rceil } a\) are 0s.


Bit decomposition Modulus conversion Secure computation Secret sharing Honest majority 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ryo Kikuchi
    • 1
    Email author
  • Dai Ikarashi
    • 1
  • Takahiro Matsuda
    • 2
  • Koki Hamada
    • 1
  • Koji Chida
    • 1
  1. 1.NTT CorporationTokyoJapan
  2. 2.National Institute of Advanced Industrial Science and Technology (AIST)TokyoJapan

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