# Sufficient conditions for sound tree and sequential hashing modes

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## Abstract

Hash functions are usually composed of a mode of operation on top of a concrete primitive with fixed input-length and fixed output-length, such as a block cipher or a permutation. In practice, the mode is often sequential, although parallel (or tree) hashing modes are also possible. The former requires less memory, while the latter has several advantages such as its inherent parallelism and a lower cost of hash value recomputation when only a small part of the input changes. In this paper, we consider the general case of (tree or sequential) hashing modes that make use of an underlying hash function, which may in turn be sequential. We formulate a set of three simple conditions for such a (tree or sequential) hashing mode to be *sound*. By sound, we mean that the advantage in differentiating a hash function obtained by applying a tree hashing mode to an ideal underlying hash function from an ideal monolithic hash function is upper bounded by \(q^2/2^{n+1}\) with \(q\) the number of queries to the underlying hash function and \(n\) the length of the chaining values. We provide a proof of soundness in the indifferentiability framework. The conditions we formulate are easy to implement and to verify and can be used by the practitioner to build a tree hashing mode on top of an existing hash function. We show how to apply tree hashing modes to sequential hash functions in an optimal way, demonstrate the applicability of our conditions with two efficient and simple tree hashing modes and provide a simple method to take the union of tree hashing modes that preserves soundness. It turns out that sequential hashing modes using a compression function (i.e., a hash function with fixed input-length) can be considered as particular cases and, as a by-product, our results also apply to them. We discuss the different techniques for satisfying the three conditions, thereby shedding a new light on several published modes.

## Keywords

Hash functions Tree hashing modes Sequential hashing modes Indifferentiability## References

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