TY - JOUR

T1 - Monotone circuits for monotone weighted threshold functions

AU - Beimel, Amos

AU - Weinreb, Enav

N1 - Funding Information:
✩ A preliminary version of this paper appeared in [A. Beimel, E. Weinreb, Monotone circuits for weighted threshold functions, in: Proc. of 20th Annual IEEE Conf. on Computational Complexity, 2005, pp. 67–75]. * Corresponding author. E-mail addresses: beimel@cs.bgu.ac.il (A. Beimel), weinrebe@cs.bgu.ac.il (E. Weinreb). URL: http://www.cs.bgu.ac.il/~beimel. 1 The work of the second author was partially supported by a Kreit-man Foundation Fellowship and by the Frankel Center for Computer Science.

PY - 2006/1/16

Y1 - 2006/1/16

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 mAC1. (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 mAC1. (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.

KW - Computational complexity

KW - Cryptography

KW - Parallel algorithms

KW - Theory of computation

UR - http://www.scopus.com/inward/record.url?scp=27844491900&partnerID=8YFLogxK

U2 - 10.1016/j.ipl.2005.09.008

DO - 10.1016/j.ipl.2005.09.008

M3 - Article

AN - SCOPUS:27844491900

SN - 0020-0190

VL - 97

SP - 12

EP - 18

JO - Information Processing Letters

JF - Information Processing Letters

IS - 1

ER -