Weighted threshold secret sharing schemes
Information Processing Letters
Enumeration of Threshold Functions of Eight Variables
IEEE Transactions on Computers
Truth functions realizable by single threshold organs
FOCS '61 Proceedings of the 2nd Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 1961)
Monotone circuits for monotone weighted threshold functions
Information Processing Letters
Characterizing ideal weighted threshold secret sharing
TCC'05 Proceedings of the Second international conference on Theory of Cryptography
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Possible characterizations of which positive boolean functions are weighted threshold were studied in the 60s and 70s. It is known that a boolean function is weighted threshold if and only if it is k-asummable for every value of k. Furthermore, for some particular subfamilies of functions (those with up to eight variables, and graph functions), it is known that a function is weighted threshold if and only if it is 2-asummable. In this work we prove that bipartite functions also satisfy this property: a bipartite function is weighted threshold if and only if it is 2-asummable. In a bipartite function the set of variables can be partitioned in two classes, such that all the variables in the same class play exactly the same role in the function.