Abstract
This chapter explores ways of performing a kind of commutative task by N parties, of which a particular scenario of contract signing is a canonical example. Tasks are defined as commutative if the order in which parties perform them can be freely changed without affecting the final result. It is easy to see that arbitrary N-party commutative tasks cannot be completed in less than N − 1 basic time units.
We conjecture that arbitrary N-party commutative tasks cannot be performed in N − 1 time units by exchanging less than 4N − 6 messages and provide computational evidence in favor of this conjecture. We also explore the most equitable commutative task protocols.
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Notes
- 1.
\(\mathcal{A}_{i}\) gets a container, decrypts it, and examines its contents. \(\mathcal{A}_{i}\) extracts any messages sent to him and erases these messages from the container. \(\mathcal{A}_{i}\) potentially inserts into the container new messages for other parties and re-encrypts the container for the next receiving party without changing the container’s size.
- 2.
Of \(\mathbb{S}_{N}\).
- 3.
Or isomorphic when there is no ambiguity as is the case until the other relation is presented.
- 4.
That is, given \(\mathfrak{P},\mathfrak{P}' \in \mathfrak{S}_{N-1}^{N}\), decide if \(\mathfrak{P} \equiv \mathfrak{P}'\).
- 5.
Our Ocaml code is available at http://www.eleves.ens.fr/home/xia/posting.
- 6.
- 7.
The reader is referred to Fig. 10 for a clarifying example.
Fig. 10 
Protocol from Fig. 6 where edges representing default envelopes are drawn in thick lines
An edge \((d,\alpha ) - (d + 1,\beta )\) means that \(\mathcal{A}_{\alpha }\) sends an envelope to \(\mathcal{A}_{\beta }\) on day d.
The path (1, 5, 6, 4, 7, 2, 3) is associated to the set \(\{(0, 1) - (1, 5); (2, 6) - (3, 4); (5, 2) - (6, 3)\}\) (the final destination of the corresponding contract is \(\mathcal{A}_{0}\)).
Note that all available paths on this graph are not necessarily taken by a contract, e.g., (4, 5, 6, 4, 7, 2, 3) is associated to \(\{(0, 4) - (1, 5); (5, 2) - (6, 3)\}\).
- 8.
That is, renumbering the parties by increasing workload.
- 9.
For example, if we launch seven shifted instances of Fig. 6 we get a very uneven split of cost where \(\mathcal{A}_{0}\) and \(\mathcal{A}_{3}\) pay $4 every day (i.e., a total of $28 each) whereas the other six parties pay $3 every day (i.e., a total of $21 each).
References
N. Ferguson, B. Schneier, Practical Cryptography (Wiley, Indianapolis, 2003)
K. McCurley, Language modeling and encryption on packet switched networks, in Advances in Cryptology - Eurocrypt 2006. Lecture Notes in Computer Science, vol. 4004 (Springer, Berlin, 2006), pp. 359–372
V. Voydoc, S. Kent, Security mechanisms in high-level network protocols. ACM Comput. Surv. 15, 135–171 (1983)
Acknowledgements
The authors thank Oğuzhan Külekci for interesting discussions and useful remarks notably concerning the variant of the traveling salesmen problem.
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© 2014 Springer International Publishing Switzerland
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Brier, E., Naccache, D., Xia, Ly. (2014). How to Sign Paper Contracts? Conjectures and Evidence Related to Equitable and Efficient Collaborative Task Scheduling. In: Koç, Ç. (eds) Open Problems in Mathematics and Computational Science. Springer, Cham. https://doi.org/10.1007/978-3-319-10683-0_13
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DOI: https://doi.org/10.1007/978-3-319-10683-0_13
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