## Cryptology ePrint Archive: Report 2018/154

Constrained PRFs for NC1 in Traditional Groups

Nuttapong Attrapadung and Takahiro Matsuda and Ryo Nishimaki and Shota Yamada and Takashi Yamakawa

Abstract: 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, $\textbf{NC}^1$ 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 $\textbf{NC}^1$ 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.

Category / Keywords: foundations / pseudo-randomness, constrained PRF, pairing free group, correlated-input hash