Abstract
We study the relationship between the multiparty communication complexity of functions over certain communication topologies and the complexity of inverting those functions. We show that if a function ofn variables has aring-protocol or atree-protocol of communication complexity bounded by ϕ, then there is a circuit of size\(2^{0(\phi )} n\) that computes an inverse of the function. Consequently, we prove that although invertingNC 0 Boolean circuits isNP-hard, planarNC 1 Boolean circuits can be inverted inNC, and hence in polynomial time. From the ring-protocol theorem, we derive an ω(n logn) lower bound on the VLSI area required to lay out any one-way function. Our results on inverting boolean circuits can be extended to algebraic circuits over finite rings. We prove that on certain topologies no one-way function can be computed with low communication complexity.
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Communicated by Joan Feigenbaum
The preliminary version of this paper appeared inCRYPTO 91. This work was supported in part by National Science Foundation Grant DCR-8713489. Part of this work was done while the author was at the School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. Current address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
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Teng, SH. Functional inversion and communication complexity. J. Cryptology 7, 153–170 (1994). https://doi.org/10.1007/BF02318547
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DOI: https://doi.org/10.1007/BF02318547