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Essentially all known one-time signature schemes can be described as special instances of a general scheme suggested by Bleichenbacher and Maurer based on "graphs of one-way functions". Bleichenbacher and Maurer thoroughly analyze graph based signatures from a combinatorial point of view, studying the graphs that result in the most efficient schemes (with respect to various efficiency measures, but focusing mostly on key generation time). However, they do not give a proof of security of their generic construction, and they leave open the problem of determining under what assumption security can be formally proved. In this paper we analyze graph based signatures from a security point of view and give sufficient conditions that allow to prove the security of the signature scheme in the standard complexity model (no random oracles). The techniques used to prove the security of graph based one-time signatures are then applied to the construction of a new class of algebraic signature schemes, i.e., schemes where signatures can be combined with a restricted set of operations.

The Provable Security of Graph-Based One-Time Signatures and Extensions to Algebraic Signature Schemes