Parallel computation with threshold functions
Journal of Computer and System Sciences - Structure in Complexity Theory Conference, June 2-5, 1986
On the Size of Weights for Threshold Gates
SIAM Journal on Discrete Mathematics
Simulating Threshold Circuits by Majority Circuits
SIAM Journal on Computing
Weighted threshold secret sharing schemes
Information Processing Letters
Communications of the ACM
Generalized Secret Sharing and Monotone Functions
CRYPTO '88 Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Characterizing ideal weighted threshold secret sharing
TCC'05 Proceedings of the Second international conference on Theory of Cryptography
On the optimization of bipartite secret sharing schemes
ICITS'09 Proceedings of the 4th international conference on Information theoretic security
Any 2-asummable bipartite function is weighted threshold
Discrete Applied Mathematics
The complexity of game isomorphism
Theoretical Computer Science
Ideal hierarchical secret sharing schemes
TCC'10 Proceedings of the 7th international conference on Theory of Cryptography
On the optimization of bipartite secret sharing schemes
Designs, Codes and Cryptography
Improved Approximation of Linear Threshold Functions
Computational Complexity
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Weighted threshold functions with positive weights are a natural generalization of unweighted threshold functions. These functions are clearly monotone. However, the naive way of computing them is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; this algorithm is inherently non-monotone since addition is a non-monotone function. In this work we by-pass this addition step and construct a polynomial size logarithmic depth unbounded fan-in monotone circuit for every weighted threshold function, i.e., we show that weighted threshold functions are in mAC^1. (To the best of our knowledge, prior to our work no polynomial monotone circuits were known for weighted threshold functions.) Our monotone circuits are applicable for the cryptographic tool of secret sharing schemes. Using general results for compiling monotone circuits (Yao, 1989) and monotone formulae (Benaloh and Leichter, 1990) into secret sharing schemes, we get secret sharing schemes for every weighted threshold access structure. Specifically, we get: (1) information-theoretic secret sharing schemes where the size of each share is quasi-polynomial in the number of users, and (2) computational secret sharing schemes where the size of each share is polynomial in the number of users.