Editing Wedderburn–Etherington number (section)

==Applications==
{{harvtxt|Young|Yung|2003}} use the Wedderburn–Etherington numbers as part of a design for an [[encryption]] system containing a hidden [[Backdoor (computing)|backdoor]]. When an input to be encrypted by their system can be sufficiently [[data compression|compressed]] by [[Huffman coding]], it is replaced by the compressed form together with additional information that leaks key data to the attacker. In this system, the shape of the Huffman coding tree is described as an Otter tree and encoded as a binary number in the interval from 0 to the  Wedderburn–Etherington number for the number of symbols in the code. In this way, the encoding uses a very small number of bits, the base-2 logarithm of the Wedderburn–Etherington number.<ref>{{citation
 | last1 = Young | first1 = Adam
 | last2 = Yung | first2 = Moti | author2-link = Moti Yung
 | contribution = Backdoor attacks on black-box ciphers exploiting low-entropy plaintexts
 | doi = 10.1007/3-540-45067-X_26
 | isbn = 978-3-540-40515-3
 | pages = 297–311
 | publisher = Springer
 | series = [[Lecture Notes in Computer Science]]
 | title = Proceedings of the 8th Australasian Conference on Information Security and Privacy (ACISP'03)
 | volume = 2727
 | year = 2003}}.</ref>

{{harvtxt|Farzan|Munro|2008}} describe a similar encoding technique for rooted unordered binary trees, based on partitioning the trees into small subtrees and encoding each subtree as a number bounded by the Wedderburn–Etherington number for its size. Their scheme allows these trees to be encoded in a number of bits that is close to the information-theoretic lower bound (the base-2 logarithm of the Wedderburn–Etherington number) while still allowing constant-time navigation operations within the tree.<ref>{{citation
 | last1 = Farzan | first1 = Arash
 | last2 = Munro | first2 = J. Ian | author2-link = Ian Munro (computer scientist)
 | contribution = A uniform approach towards succinct representation of trees
 | doi = 10.1007/978-3-540-69903-3_17
 | mr = 2497008
 | pages = 173–184
 | publisher = Springer
 | series = Lecture Notes in Computer Science
 | title = Algorithm theory—SWAT 2008
 | volume = 5124
 | year = 2008| isbn = 978-3-540-69900-2
 }}.</ref>

{{harvtxt|Iserles|Nørsett|1999}} use unordered binary trees, and the fact that the Wedderburn–Etherington numbers are significantly smaller than the numbers that count ordered binary trees, to significantly reduce the number of terms in a series representation of the solution to certain [[differential equation]]s.<ref>{{citation
 | last1 = Iserles | first1 = A.
 | last2 = Nørsett | first2 = S. P.
 | doi = 10.1098/rsta.1999.0362
 | issue = 1754
 | journal =  Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
 | mr = 1694700
 | pages = 983–1019
 | title = On the solution of linear differential equations in Lie groups
 | volume = 357
 | year = 1999| bibcode = 1999RSPTA.357..983I
 | s2cid = 90949835
 | url = https://cds.cern.ch/record/323789
 }}.</ref>