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The Wikipedia article about the Merkle signature scheme claims that it is

very adjustable and resistant against quantum computing.

What proof is there of this?

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The Merkle signature scheme is based on a hash tree over one-time signatures, e.g. using Lamport's scheme.

Roughly speaking, both Lamport's scheme and the hash tree are secure as long as the hash function is resistant to preimages and second-preimages. There can be details because when using hash trees, preimage attacks are often multi-target, i.e. there are several values for which the attacker would like a preimage, and a preimage for any one of them would yield a success. Yet, preimage resistance still leads the scheme overall robustness.

Against quantum computing, a "perfect" hash function of output size n bits still offers resistance 2n/2. E.g. with SHA-256 (a 256-bit output), the best quantum computer would still need 2128 operations (i.e. way too many to be feasible, with a huge margin) to break preimage resistance.

(This also means that against classical computers, a 256-bit hash function is total overkill when used in Merkle scheme. But, against classical computers, we can also use RSA or ECDSA, which are way more efficient than Merkle's scheme.)

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