**Multi-property-preserving Domain Extension Using Polynomial-based Modes of Operation**

*Jooyoung Lee and John Steinberger*

**Abstract: **In this paper, we propose a new double-piped mode of operation for multi-property-preserving domain extension of MACs~(message authentication codes), PRFs~(pseudorandom functions) and PROs~(pseudorandom oracles). Our mode of operation performs twice as fast as the original double-piped mode of operation of Lucks while providing comparable security. Our construction, which uses a class of polynomial-based compression functions proposed by Stam, makes a single call to a $3n$-bit to $n$-bit primitive at each iteration and uses a finalization function $f_2$ at the last iteration, producing an $n$-bit hash function $H[f_1,f_2]$ satisfying the following properties.
\begin{enumerate}
\item $H[f_1,f_2]$ is unforgeable up to $O(2^n/n)$ query complexity as long as $f_1$ and $f_2$ are unforgeable.
\item $H[f_1,f_2]$ is pseudorandom up to $O(2^n/n)$ query complexity as long as $f_1$ is unforgeable and $f_2$ is pseudorandom.
\item $H[f_1,f_2]$ is indifferentiable from a random oracle up to $O(2^{2n/3})$ query complexity as long as $f_1$ and $f_2$ are public random functions.
\end{enumerate}
To our knowledge, our result constitutes the first time $O(2^n/n)$ unforgeability has been achieved using only an unforgeable primitive of $n$-bit output length. (Yasuda showed unforgeability of $O(2^{5n/6})$ for Lucks' construction assuming an unforgeable primitive, but the analysis is sub-optimal; in this paper, we show how Yasuda's bound can be improved to $O(2^n)$.)

In related work, we strengthen Stam's collision resistance analysis of polynomial-based compression functions (showing that unforgeability of the primitive suffices) and discuss how to implement our mode by replacing $f_1$ with a $2n$-bit key blockcipher in Davies-Meyer mode or by replacing $f_1$ with the cascade of two $2n$-bit to $n$-bit compression functions.

**Category / Keywords: **secret-key cryptography / hash functions, message authentication codes

**Publication Info: **An extended abstract of this work was accepted for publication in Eurocrypt 2010.

**Date: **received 8 Mar 2010, last revised 10 Mar 2010

**Contact author: **jlee05 at ensec re kr

**Available format(s): **PDF | BibTeX Citation

**Version: **20100311:010223 (All versions of this report)

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