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by Amos Beimel, Enav Weinrebý
In Proc. of the 20th ACM Symp. on the Theory of Computing
http://www.cs.bgu.ac.il/~weinrebe/Papers/wtas_poly.ps
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Abstract:
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 it 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 inÑ��. (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 weightsÛ�����ÛÒ where the size of each share is quasi-polynomial in the number of users, and (2) computation secret sharing schemes where the size of each share is polynomial in the number of users. 1
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