Constrained PRFs for \(\mathrm{NC}^1\) in Traditional Groups

  • Nuttapong Attrapadung
  • Takahiro Matsuda
  • Ryo NishimakiEmail author
  • Shota Yamada
  • Takashi Yamakawa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10992)


We propose new constrained pseudorandom functions (CPRFs) in traditional groups. Traditional groups mean cyclic and multiplicative groups of prime order that were widely used in the 1980s and 1990s (sometimes called “pairing free” groups). Our main constructions are as follows.

  • We propose a selectively single-key secure CPRF for circuits with depth \(O(\log n)\) (that is, NC\(^1\) circuits) in traditional groups where n is the input size. It is secure under the L-decisional Diffie-Hellman inversion (L-DDHI) assumption in the group of quadratic residues \(\mathbb {QR}_q\) and the decisional Diffie-Hellman (DDH) assumption in a traditional group of order q in the standard model.

  • We propose a selectively single-key private bit-fixing CPRF in traditional groups. It is secure under the DDH assumption in any prime-order cyclic group in the standard model.

  • We propose adaptively single-key secure CPRF for NC\(^1\) and private bit-fixing CPRF in the random oracle model.

To achieve the security in the standard model, we develop a new technique using correlated-input secure hash functions.



We thank Keita Xagawa for letting us know the relation between CIH and RKA-PRG. The first, second, and fourth authors were supported by JST CREST Grant No. JPMJCR1688. The fourth author was supported by JSPS KAKENHI Grant Number 16K16068.


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Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  • Nuttapong Attrapadung
    • 1
  • Takahiro Matsuda
    • 1
  • Ryo Nishimaki
    • 2
    Email author
  • Shota Yamada
    • 1
  • Takashi Yamakawa
    • 2
  1. 1.National Institute of Advanced Industrial Science and Technology (AIST)TokyoJapan
  2. 2.Secure Platform LaboratoriesNTT CorporationTokyoJapan

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