# Directed Acyclic Graphs, One-way Functions and Digital Signatures

## Daniel Bleichenbacher and Ueli Maurer

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```The goals of this paper are to formalize and investigate the general
concept of a digital signature scheme based on a general one-way function
without trapdoor for signing a predetermined number of messages.
It generalizes and unifies previous work of Lamport, Winternitz, Merkle,
Even et al. and Vaudenay. The structure of the computation yielding a public
key from a secret key corresponds to a directed acyclic graph $\cg$.
A signature scheme for $\cg$ can be defined as an antichain in
the poset of minimal verifyable sets of vertices of $\cg$ with the
naturally defined computability relation as the order relation
and where a set is verifyable if and only if the public key can be computed
from the set. Several types of graphs are analyzed, results on the number of
signatures of these schemes are presented (with and without restriction
on the size of signatures), and several open research problems are proposed.
In particular, a tree is shown which allows to sign $0.4162$ bits per
one-way function evaluation and it is proved that this is also an upper
bound for all trees. In contrast a general graph is exhibited which allows
to sign 0.4327 bits per one-way function evaluation.