TY - GEN

T1 - Monotone circuits for weighted threshold functions

AU - Beimel, Amos

AU - Weinreb, Enav

PY - 2005/11/16

Y1 - 2005/11/16

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=27644460973&partnerID=8YFLogxK

U2 - 10.1109/CCC.2005.12

DO - 10.1109/CCC.2005.12

M3 - Conference contribution

AN - SCOPUS:27644460973

SN - 0769523641

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 67

EP - 75

BT - Proceedings of the 20th Annual IEEE Conference on Computational Complexity

T2 - 20th Annual IEEE Conference on Computational Complexity

Y2 - 11 June 2005 through 15 June 2005

ER -