**PRF Domain Extension Using DAGs**

*Charanjit Jutla*

**Abstract: **We prove a general domain extension theorem for
pseudo-random functions (PRFs).
Given a PRF $F$ from $n$ bits to $n$ bits,
it is well known that
employing $F$ in a chaining mode (CBC-MAC)
yields a PRF on the bigger domain. Viewing each application of $F$
in this chaining mode to be a node in a graph, and the chaining as the edges
between the nodes, the resulting graph is just a line graph.
In this paper, we show that the underlying graph can be an arbitrary
directed acyclic graph (DAG), and
the resulting function on the larger domain is still a PRF.
The only requirement on the graph is that it have unique
source and sink nodes, and no two nodes have the same set of incident nodes.
A new highly parallelizable MAC construction follows
which has a critical path of only $3+\log ^{*} n $ applications of $F$.\\
\hspace*{0.1in}
If we allow Galois field arithmetic, we can consider edge colored DAGs,
where the colors represent multiplication in the field by the color number.
We prove an even more general theorem, where the only restriction on
the colored DAGs is that if two nodes ($u$ and $v$)
have the same set of incident nodes $W$,
then at least one $w$ in $W$ is incident on $u$ and $v$
with a different colored edge.
PMAC (parallelizable message authentication \cite{black})
is a simple example of such graphs. The general thoerem allows one to have
further optimizations over PMAC, and many modes which deal with variable
lengths.

**Category / Keywords: **foundations /

**Date: **received 25 Mar 2005, last revised 9 May 2005

**Contact author: **csjutla at watson ibm com

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**Version: **20050509:234818 (All versions of this report)

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