By Daniel J. Bernstein

This publication introduces the reader to the subsequent new release of cryptographic algorithms, the structures that face up to quantum-computer assaults: particularly, post-quantum public-key encryption platforms and post-quantum public-key signature systems.

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In particular, there is never a desired subtree at level L. The left part of Figure 6 depicts the adjacent existing and desired subtrees. As the name suggests, we need to compute the pebbles in the desired subtrees. 1) to the root of Desirei . For these purposes, the treehash algorithm is altered to save the pebbles needed for Desirei , rather than discarding them, and secondly to terminate one round early, never actually computing the root. Using this variant of treehash, we see that each desired subtree being computed has a tail of saved intermediate pebbles.

We also prove that this complexity is optimal in the sense that there can be no Merkle Tree traversal algorithm which requires both less than O(H) time and less than O(H) space. In the analysis of the ﬁrst two algorithms, the computation of a leaf and an inner node are each counted as a single elementary operation1 . The third Merkle tree-traversal algorithm has the same space and time complexity as the second. However it has a signiﬁcant constant factor improvement and was designed for practical implementation.

The checksum c is c = (4 − 1) + (4 − 0) = 7. Prepending one zero to the binary representation of c and splitting the extended string into blocks of length 2 yields c = 01||11. The signature is ⎞ ⎛ 0011 σ = (σ3 , σ2 , σ1 , σ0 ) = f (x3 ), x2 , f (x1 ), f 3 (x0 ) = ⎝ 0 0 0 1 ⎠ ∈ {0, 1}(3,4) . 0001 The signature is veriﬁed by computing f 2 (σ3 ), f 3 (σ2 ), f 2 (σ1 ), σ0 ⎛ ⎞ 0010 = ⎝ 1 1 1 0 ⎠ ∈ {0, 1}(3,4) 0101 and comparing it with the veriﬁcation key Y . Example 4. We give an example to illustrate why the signature keys of the WOTS must be used only once.