On the Communication Complexity of Secure Computation

Information theoretically secure multi-party computation (MPC) is a central primitive of modern cryptography. However, relatively little is known about the communication complexity of this primitive. In this work, we develop powerful information theoretic tools to prove lower bounds on the communication complexity of MPC. We restrict ourselves to a 3-party setting in order to bring out the power of these tools without introducing too many complications. Our techniques include the use of a data processing inequality for residual information - i.e., the gap between mutual information and G\'acs-K\"orner common information, a new information inequality for 3-party protocols, and the idea of distribution switching by which lower bounds computed under certain worst-case scenarios can be shown to apply for the general case. Using these techniques we obtain tight bounds on communication complexity by MPC protocols for various interesting functions. In particular, we show concrete functions that have "communication-ideal" protocols, which achieve the minimum communication simultaneously on all links in the network. Also, we obtain the first explicit example of a function that incurs a higher communication cost than the input length in the secure computation model of Feige, Kilian and Naor (1994), who had shown that such functions exist. We also show that our communication bounds imply tight lower bounds on the amount of randomness required by MPC protocols for many interesting functions.Comment: 37 page

Branching Programs for Tree Evaluation

Abstract. The problem FT h d (k) consists in computing the value in [k] = {1,..., k} taken by the root of a balanced d-ary tree of height h whose internal nodes are labelled with d-ary functions on [k] and whose leaves are labelled with elements of [k]. We propose FT h d (k) as a good candidate for witnessing L � LogDCFL. We observe that the latter would follow from a proof that k-way branching programs solving FT h d (k) require Ω(k unbounded function(h) ) size. We introduce a “state sequence ” method that can match the size lower bounds on FT h d (k) obtained by the Ne˘ciporuk method and can yield slightly better (yet still subquadratic) bounds for some nonboolean functions. Both methods yield the tight bounds Θ(k 3) and Θ(k 5/2) for deterministic and nondeterministic branching programs solving FT 3 2 (k) respectively. We propose as a challenge to break the quadratic barrier inherent in the Ne˘ciporuk method by adapting the state sequence method to handle FT 4 d (k).

NA49 results on hadron production: indications of the onset of deconfinement ?

The NA49 experiment at the CERN SPS measured the energy and system size dependence of particle production in A+A collisions. A change of the energy dependence of several hadron production properties at low SPS energies is observed which suggests a scenario requiring the onset of deconfinement.Comment: XXXV International Symposium on Multiparticle Dynamics 200

Upper Limit of D0 Production in Central Pb-Pb Collisions at 158A GeV

Results are presented from a search for the decays D0 -> Kmin piplus and D0bar -> Kplus pimin in a sample of 3.8x10^6 central Pb-Pb events collected with a beam energy of 158A GeV by NA49 at the CERN SPS. No signal is observed. An upper limit on D0 production is derived and compared to predictions from several models.Comment: REVTEX 5 pages, 4 figure

Event-by-event fluctuations of the kaon to pion ratio in central Pb+Pb collisions at 158 GeV per Nucleon

We present the first measurement of fluctuations from event to event in the production of strange particles in collisions of heavy nuclei. The ratio of charged kaons to charged pions is determined for individual central Pb+Pb collisions. After accounting for the fluctuations due to detector resolution and finite number statistics we derive an upper limit on genuine non-statistical fluctuations, perhaps related to a first or second order QCD phase transition. Such fluctuations are shown to be very small.Comment: 4 pages, 2 figure

Strangeness from 20 AGeV to 158 AGeV

New results from the energy scan programme of NA49, in particular kaon production at 30 AGeV and phi production at 40 and 80 AGeV are presented. The K+/pi+ ratio shows a pronounced maximum at 30 AGeV; the kaon slope parameters are constant at SPS energies. Both findings support the scenario of a phase transition at about 30 AGeV beam energy. The phi/pi ratio increases smoothly with beam energy, showing an energy dependence similar to K-/pi-. The measured particle yields can be reproduced by a hadron gas model, with chemical freeze-out parameters on a smooth curve in the T-muB plane. The transverse spectra can be understood as resulting from a rapidly expanding, locally equilibrated source. No evidence for an earlier kinetic decoupling of heavy hyperons is found.Comment: Contribution to the proceedings of "Strangeness in Quark Matter 2003" (March 2003, Atlantic Beach NC, USA), to be published in Journal of Physics G. 11 pages, 14 figure

Energy dependence of particle ratio fluctuations in central Pb+Pb collisions from sNN=\sqrt{s_{_{NN}}} =~6.3 to 17.3 GeV

We present measurements of the energy dependence of event-by-event fluctuations in the K/pi and (p + \bar{p})/pi multiplicity ratios in heavy ion collisions at the CERN SPS. The particle ratio fluctuations were obtained for central Pb+Pb collisions at five collision energies, \sqrt{s_{_{NN}}}, between 6.3 and 17.3 GeV. After accounting for the effects of finite-number statistics and detector resolution, we extract the strength of non-statistical fluctuations at each energy. For the K/pi ratio, larger fluctuations than expected for independent particle production are found at all collision energies. The fluctuations in the (p + \bar{p})/pi ratio are smaller than expectations from independent particle production, indicating correlated pion and proton production from resonance decays. For both ratios, the deviation from purely statistical fluctuations shows an increase towards lower collision energies. The results are compared to transport model calculations, which fail to describe the energy dependence of the K/pi ratio fluctuations.Comment: 11 pages, 10 figures, update to published versio

Report from NA49

The most recent data of NA49 on hadron production in nuclear collisions at CERN SPS energies are presented. Anomalies in the energy dependence of pion and kaon production in central Pb+Pb collisions are observed. They suggest that the onset of deconfinement is located at about 30 AGeV. Large multiplicity and transverse momentum fluctuations are measured for collisions of intermediate mass systems at 158 AGeV. The need for a new experimental programme at the CERN SPS is underlined.Comment: invited talk presented at Quark Matter 2004, 10 page