Most real-world signature schemes like RSA and DSA can be broken by a quantum computer, due to Shor's algorithm . One could use lattice-based signatures but their security is not well understood: the authors of  note that "their quantitative security levels are highly unclear" even in the pre-quantum setting.
On the other hand, hash-based signature schemes using 'quantum-resistant' one-way hash functions are secure in the quantum setting , although the symmetric security parameter needs to be doubled for quantum resistance due to Grover's algorithm . The trouble is that such schemes tend to be inefficient or stateful, where the latter means one cannot have a signing key shared across multiple devices and the scheme may be vulnerable to 'restart attacks' where the scheme is forced to re-use a secret key, compromising security. The paper of Peter's talk seeks to address this by constructing a (relatively) efficient, stateless, hash-based signature scheme called SPHINCS. The construction is proved secure “based on weak standard-model assumptions, avoiding collision resistance and the random-oracle model”.
One Time Signature Schemes
From 'One Time' to 'Many Times'
- Given the path from a leaf to the root, let the sequence of siblings of each node on the path be called the authentication path of the leaf.
- In a signature, supply the index of the leaf node used and its authentication path.
- With the leaf and its sibling, the verifier can construct the parent.
- Then with the sibling of the parent, the verifier can construct the grandparent.
- The verifier continues like this, using the siblings given in the authentication path to recover the root node i.e. the public key.
From 'One Time' to 'A Few Times'
The main idea of the paper is to use a 'few times signature' scheme (instead of an OTS) at the bottom of the tree to reduce the number of leaves needed for security and hence the overall height of the tree, thus shortening the signature. Their choice of scheme for the bottom of the tree is called HORS: Hash to Obtain Random Subset. In HORS, the secret key is the tuple $(s_1, ..., s_t)$, the public key is the hashes $(H(s_1), ..., H(s_t))$ and a message $m$ determines a (“random”) subset $S$ of $\lbrace1, 2, ..., t\rbrace$ with fixed size $k$ (much smaller than $t$). Then the signature for $m$ is the set of secret key components corresponding to $S$, i.e. $\lbrace s_i | i \in S\rbrace$. Now we can use the secret key 'a few times' before security is compromised as only a small number of the components $s_i$ of the secret key are revealed in each signature. But again we have the problem that the public key needs to be very large and so a Merkle tree is once again employed: the new public key becomes the root node, recoverable from the index of a leaf node and the corresponding authentication path. Notice that this means the bottom of the tree in SPHINCS, the construction proposed in , is itself a tree. So SPHINCS consists of a hyper-tree (a tree of trees).
The SPHINCS Tree of Trees
The SPHINCS hyper-tree has total height $h$ consisting of $d$ layers, each of height $h/d$. The index from the hash of the message determines a HORS tree on the bottom layer of the hyper-tree and a leaf of this HORS tree from which we compute the HORS signature of the message. Then this signature is signed according to the OTS scheme on the next layer (where the initial index in some way determines the tree and the leaf to use). We repeat this OTS signing on each layer and finally output the SPHINCS signature consisting of the index, the HORS signature (which contains all the information needed to recover the HORS public key) and each OTS signature and each authentication path needed to recover the root at each layer.
Real World Considerations (in the Quantum Computing World!)