Private Collaborative Forecasting and Benchmarking

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1 Prvate Collaboratve Forecastg ad Becharkg Mkhal Atallah Mara Bykova Jagtao L Keth Frkke Merca Topkara Coputer Sceces Departet ad CERIAS Purdue Uversty {ja,bykova,jtl,kbf,karaha}@cs.purdue.edu ABSTRACT Suppose a uber of hosptals a geographc area wat to lear how ther ow heart-surgery ut s dog copared wth the others ters of ortalty rates, subsequet coplcatos, or ay other qualty etrc. Slarly, a uber of sall busesses ght wat to use ther recet potof-sales data to cooperatvely forecast future dead ad thus ake ore fored decsos about vetory, capacty, eployet, etc. These are sple exaples of cooperatve becharkg ad (respectvely) forecastg that would beeft all partcpats as well as the publc at large, as they would ake t possble for partcpats to aval theselves of ore precse ad relable data collected fro ay sources, to assess ther ow local perforace coparso to global treds, ao avod ay of the effceces that curretly arse because of havg less forato avalable for ther decso-akg. Ad yet, spte of all these advatages, cooperatve becharkg ad forecastg typcally do ot take place, because of the partcpats uwllgess to share ther forato wth others. Ther reluctace to share s qute ratoal, ad s due to fears of ebarrasset, lawsuts, weakeg ther egotatg posto (e.g., case of over-capacty), revealg corporate perforace ad strateges, etc. The developet ad deployet of prvate becharkg ad forecastg techologes would allow such collaboratos to take place wthout revealg ay partcpat s data to the others, reapg the beefts of collaborato whle avodg the drawbacks. Moreover, ths kd of techology would epower saller orgazatos who could the cooperatvely base ther decsos o a uch broader forato base, a way that s today restrcteo oly the largest corporatos. Ths paper s a step towards ths Portos of ths work were supported by Grats IIS , IIS , IIS , ad IIS fro the Natoal Scece Foudato, Cotract N fro the Offce of Naval Research, by sposors of the Ceter for Educato ad Research Iforato Assurace ad Securty, ad by Purdue Dscovery Park s e- eterprse Ceter. Persso to ake dgtal or hard copes of all or part of ths work for persoal or classroo use s grated wthout fee provdehat copes are ot ade or dstrbuted for proft or coercal advatage ahat copes bear ths otce ahe full ctato o the frst page. To copy otherwse, to republsh, to post o servers or to redstrbute to lsts, requres pror specfc persso ad/or a fee. WPES 04, October 28, 2004, Washgto, DC, USA. Copyrght 2004 ACM /04/ $5.00. goal, as t gves protocols for forecastg ad becharkg that reveal to the partcpats the desred aswers yet do ot reveal to ay partcpat ay other partcpat s prvate data. We cosder several forecastg ethods, cludg lear regresso ae seres techques such as ovg average ad expoetal soothg. Oe of the ovel parts of ths work, that further dstgushes t fro prevous work secure ult-party coputato, s that t volves floatg pot arthetc, partcular t provdes protocols to securely ad effcetly perfor dvso. Categores ad Subject Descrptors K.4.4 [Coputers ad Socety]: Electroc Coerce securty; F.2. [Aalyss of Algorth ad Proble Coplexty]: Mscellaeous Geeral Ters Desg, Securty Keywords Prvacy, secure ult-party coputato, forecastg, becharkg, e-coerce, secure protocol 1. INTRODUCTION Oe drawback that saller ettes (e.g., dvduals, chartes, sall busesses, etc.) have copetg wth large ettes (gat corporatos ad ult-atoals) s that the latter s sze ad resources eable the to ake decsos usg ore accurate forato (e.g., about future dead). Ths better forecastg ablty ca, over te, drve the saller players out ad leave the feld uder the cotrol of the largest ettes. Prvacy-preservg cooperatve coputato, whch s of obvous beeft to the prvacy of dvduals, s also a valuable techology for eablg saller ettes to cooperate ad ake as hgh-qualty decsos as larger ettes (decsos about plag, producto, qualty cotrol, etc.). Ths paper s focus s o the two specfc areas of forecastg of custoer dead ad secure becharkg, whch are descrbed below. Before we do so, we rehe reader that the broad fraework for ths work s the usual prvacy-preservg coputato odel, whch two or ore partes egage a collaboratve coputato order to produce results that are sgfcat to both partes wthout revealg the prvate forato of ay of the partes, eve though the jotly-coputed results deped o the forato of all the partes.

2 The frst proble we are explorg s secure ad prvate collaboratve forecastg, whch a uber of retalers jo ther efforts to geerate ore accurate forecasts of custoer dead. We assue that each of the partcpats has ts ow propretary data gathered over soe perod of te the past ad ca produce a local forecast. They decde to partcpate jot coputato to obta ore relable results. Cosder the followg busess scearos: A uber of sall retalers the area whch sell slar products caot copete wth gat stores ther forecastg capabltes. Thus they decde to collaborate wth each other order to better estate future cosuer dead. Revealg data about the past volues of sales s uacceptable to ay of the, as they are copetg the sae arket. The retalers, however, are wllg to share the data a secure fasho f all that ay party lears fro the collaborato s the geeral tred the custoer dead (.e., crease or decrease sales ad by what aout). After partcpatg the protocol, each retaler ca copare ts ow locally geerated forecast wth the large-scale tred, draw coclusos about the accuracy of the local forecast ad dffereces, f ay, the behavor of the sales fucto at the local ad global scopes. Aother settg where slar collaborato s useful s stuatos whe o sgle retaler ca accurately estate future dead. Cosder a product that has bee troduceo the arket recetly such that o sgle (eve very large) retaler ca accurately predct cosuer dead for t. Ths happes whe dfferet retalers target dfferg groups of custoers, for whch shoppg patters ad adaptablty to ew products vary. The t s beefcal to all such stores to egage to jot forecastg, whle stll preservg the prvacy of the data o whch the forecast s bult. We ca also odel a scearo where there s oe suppler ad ay retalers, ahe cycle of producto s very log. For exaple, order for a overseas copay to aufacture clothes, t ay eeo start producto 7 oths advace cludg shppg te. The suppler wats to kow the custoer deads,.e., the sze of the arket. Each retaler s reluctat to provde ts ow hstorcal data. However, t ay beeft the whole supply cha, f the retalers together ca collaboratvely provde a forecast o custoer deads to the suppler. Or, as a alteratve, the suppler ght provde a dscout to all retalers who partcpate the jot coputato of custoer dead, ad uses the results for aufacturg ore precse quattes. All of the above scearos produce forecasts based o te seres. Aother type of forecastg that we also explore ths work s based o regresso techques. A otvatg busess odel ca be as follows. A hosptal perfors a certa type of surgeres that result a rather hgh ortalty rate copareo other types of surgeres. The hosptal would lke to vestgate the correlato of the ortalty rate to the age of the patets, ther health codtos, ad possbly other paraeters, to be able to exclude the rskest category of patets fro beg cosdered for such surgeres. The hosptal, however, does ot have eough cases to draw a relable correlato betwee the ortalty rate ad other paraeters. The hosptal also would lke to kow how t perfors o ths kd of surgeres copareo other slar sttutos. Thus, the hosptal would lke to egage collaborato wth other sttutos to be able to draw coclusos o the aggregate data, but for prvacy reasos caot share ts data wth other partcpats. The soluto ths case s to use secure ult-party coputato (SMC) techques that apply regresso to aggregate data ad dstrbute the results to all partcpatg partes. Havg the results, the hosptal the ca lear the overall correlato o the large scale, as well as ake coclusos about ts perforace copareo other hosptals. To address these two probles, we cosder forecastg based o te seres ovg average, weghted ovg average, ad expoetal soothg ad regresso-based techques lear regresso. Sce the fuctos used the coputato are lear, soe copaes ght decde that the output of the coputato reveals forato about ther puts f the uber of partcpats s low (e.g., two). Cosequetly, they ght decde to partcpate the coputato oly f the uber of partcpats s suffcetly large. Therefore, we provde solutos to the probles for a geeral case of players. I ths work, we preset effcet protocols for coductg secure collaboratve forecastg ad becharkg for all statstcal ethods lsted above. Before provdg our fal protocols, we gve sub-protocols, or buldg blocks, whch ake presetato of the fal protocols crsper ad at the sae te add flexblty to the protocols theselves by allowg the partcpats to choose the ost approprate buldg blocks. I cases whe we gve ore tha oe protocol for perforg the sae task, the protocols dffer ther coplexty, coucato overhead, ad robustess agast colludg players. A ovel part of ths work s that we troduce floatg pot coputato secure ult-party coputato. We preset several dvso protocols that for the core of our forecastg ad becharkg solutos, ao the best of our kowledge are the frst attepts to perfor dvso wthout buldg a geerc crcut, as well as the frst attepts to operate o floatg pot ubers. Our dvso protocols splfy prvacy-preservg busess forecastg, ad ca also be appleo other forecastg ethods as well as other SMC applcatos. A suary of our results s gve table 1. For each protocol descrbed ths work, we lst ts uber of coucato rouds, total coucato easured essages exchaged betwee the players, aotal coputatoal coplexty (sued over all players). Each essage for ost protocols s of l bts log, where l s the legth of ubers we operate (wth the -key dvso protocol beg a excepto). I the table, refers to the uber of players, k s a colluso threshold descrbed secto 4.1 such that 1 k 1, ad s a (costat) uber of data pots used the lear regresso. All of these protocols are later evaluated wth respect to the a odel of the adversary used ths paper: That of colludg players,.e., they exhbt the behavor of se-hoest players but ca also collude together order to dscover soe addtoal forato about other players data (ore o ths later). We aalyze the colluso-resstace characterstcs of each protocol edately followg ts descrpto.

3 Protocol Coucato Total Total Rouds Coucato Coputato Splt O(1) O(k) O(k) Dvso wth a Appotee O(1) O(k) O(k) 2-party Dvso wth Scalg O(log l) O(log l) O(log l) ecryptos 2-party 2-key Dvso O(1) O(1) O(1) ecryptos -key Dvso O(1) O( 2 ) O( 2 ) ecryptos Movg Average sae as dvso sae as dvso sae as dvso Expoetal Soothg sae as dvso sae as dvso sae as dvso Lear Regresso splt + dvsos splt + dvsos splt + dvsos Table 1: Suary of protocols: s the uber of players, k s a colluso threshold where 1 k 1, l s the legth of ubers bts, ad s a uber of data pots the lear regresso. The rest of ths paper s orgazed as follows. Secto 2 revews related work. I secto 3 we brefly provde backgroud forato such as dfferet forecastg ethods ahe provde a ore precse defto of our protocols. Secto 4 descrbes buldg blocks that we developeo ad desgg our a forecastg protocols. The buldg blocks clude a secure algorth for bldg dvdual prvate puts ad the ost terestg ad dffcult secure dvso protocols. Sectos 5 ad 6 descrbe our a protocols, where Secto 5 covers forecastg based o te seres ad Secto 6 cotas regresso-based becharkg. Lastly, Secto 7 cocludes the paper ad provdes drectos for future work. 2. RELATED WORK Forecastg s creasgly beg appleo busess decso akg. May forecastg ethods (for exaple, see [14, 26]) have bee developed, such as te-seres techques ad regresso techques. Collaboratve forecastg allows dfferet ettes to jotly perfor busess forecastg where each etty cotrbutes ts ow data. As poted out [2], collaboratve forecastg, coparso to tradtoal forecastg, gves better productvty ad proftablty throughout the supply cha. Collaboratve forecastg has bee extesvely studed by ay copaes [25, 19], orgazatos [10], ad acadea [15]. Most of the solutos ether assue exstece of a cetral plaer who has all the forato about the syste, or assue that each partcpat of the coputato shares all of her forato wth other partcpats. These solutos, however, are probleatc whe the data s sestve ahe partcpats are reluctat to share ther prvate, propretary forato. Our approach s to perfor collaboratve forecastg a prvacypreservg aer, therefore elates the above cocer. The proble of secure forecastg ad becharkg s closely relateo secure ult-party coputato [27]. The SMC proble was troduced by Yao [27] ad exteded by Goldrech, Mcal, Wgderso [17] ad others ([23, 18], to lst a few). Goldrech states [16] that although the geeral secure ult-party coputato proble s solvable theory, usg the solutos derved by these geeral results for specal cases ca be practcal. I other words, effcecy dctates developet of specal solutos for specal cases. Ad as we ca see, ay other exaples of cooperatve prvacy-preservg coputatos have bee cosdered the lterature: electroc auctos [7], card playg [17], dgtal certfed al, data g [20], etc. Du ad Atallah recetly have developed effcet protocols for ay secure two-party coputato probles [11], cludg scetfc coputato [12], geoetrc coputato [5], ad statstcs aalyss [13]. Atallah et al. [6] have proposed Secure Supply-Cha Collaborato (SSCC) proble, ad developed SSCC protocols for sple e-aucto scearos ad sple capacty-allocato proble. Our secure collaboratve forecastg ad becharkg ca be vewed as a brach of the SSCC proble. I ths paper, we propose ovel protocols for coputg a rato floatg pot ubers securely, a portat copoet used ay forecastg techques. To the best of our kowledge, o oe has studehs before. 3. PROBLEM DESCRIPTION 3.1 Backgroud Before presetg our results, we brefly revew several forecastg ethods (see [14, 26]) that are the bass of our protocols. Te-Seres Techques. A te seres s a teordered sequece of observatos take at regular tervals over a perod of te (daly, weekly, othly, aually, etc.). A exaple of such data s a othly estate of custoer dead. Let us here cosder a sgle user evroet, where oly a local forecast s geerated. We use to deote th te perod, ad d to deote data te perod. Let t be the curret te perod. Usg ths otato, the three ethods that we cosder are as follows: 1. Movg Average: Let deote the uber of perods used calculato of the average. For te pero, the ovg average forecast s: F t = 1 =0 / where F t dcates the forecasted value for te terval t Weghted Movg Average: Let w = {w 0, w 1,..., w 1} be a weght vector such 1 that =0 w = 1. For te pero, the weghted ovg average forecast s: F t = 1 =0 w

4 3. Expoetal Soothg: Let F be the forecasted value te perod, ad α be a soothg costat. For te pero, the expoetal soothg techque coputes: F t = F t 1 + α( 1 F t 1) where F t s also the predcted value for the ext te perod. Regresso Techques. As etoed above, our regresso solutos are bult o the ost wdely used regresso ethod lear regresso: Gve two varables wth lear correlato, the goal s to copute a lear fucto such that the su of the devatos of all the pots fro the fucto s zed. Cosder a lear fucto y = ax+b where all data pots x are kow. If there are hstorcal data pars of (x, y), the after applyg regresso to the, we wll be able to estate the coeffcets a ad b. The coeffcets a ad b ca be coputed usg the followg equatos: ( xy) ( x)( y) a = ( x 2 ) (, b = y a x. (1) x) Protocol Defto Now we defe the terfaces of our forecastg protocols. I the deftos below ad the rest of the paper we use the followg otato. We assue that there are players P 1, P 2,..., P egaged the coputato, where 2. Notato 1. Ay te superscrpted wth (j) s held by ad kow oly to player P j. The sae te wthout a superscrpt ark correspods to the su of the tes held by all players, whch s assueo be addtvely splt aog the players. For exaple, f we have that player P j has x (j), the x s equal to j=1 x(j). I the frst two protocols, whch are based o te seres, t s udesrable to lear the absolute result: F t (the forecasted value) ght be cosdereo be revealg too uch forato because a player ca lear hs share of the value ad possbly soe addtoal forato about other players data. Therefore, stead of provdg ts absolute value, we output oly the slope Ft dt,.e., the percetage by whch the value s expecteo crease or decrease the ext te terval. Defto 1 correspods to forecastg based o ovg average techques, ad defto 2 s for expoetal soothg forecasts. The detaled protocols are gve secto 5. Defto 1. Secure Collaboratve Forecastg Movg Average Techques Usg Iput Player P j, 1 j, provdes put data d (j) t for te tervals, where 0 1. I case of coputg the weghted ovg average, the weght vector w s publc. Output Player P j, 1 j, lears Ft dt wthout aythg else, where F t s coputed usg the ovg average or the weghted ovg average techque. Defto 2. Secure Collaboratve Forecastg Soothg Techques Usg Iput Player P j, 1 j, supples d (j) t 1, d(j) t, ad F (j) t 1, where the value of F t 1 fro the prevous te terval coputato s kept addtvely splt aog all players. The value of α s publc. Output Player P j, 1 j, lears Ft dt wthout aythg else, where F t s coputed usg the expoetal soothg techque. For the lear regresso protocols, we assue that the x- axs s publc, ahe set of possble x values s fte. We use x 1, x 2,..., x to deote possble x-values. I our odel, each player supples the y-axs data ahey jotly copute the result the oralzed for. Ths eas that the data, for stace, s gve as the average uber of accdets per custoer case of car surace data, or as the ortalty rate for surgcal cases. I ths case each data pot y s gve as two tegers c ad d where y = c /d. The aggregate values for each data pot coputed durg the executo of the protocol s the foud as y = j=1 c(j) / j=1 d(j). The protocol that correspods to the defto below s provded secto 6. Defto 3. Secure Collaboratve Becharkg Usg Lear Regresso Techques Iput Player P j, 1 j, provdes data pots y (j) where 1 ad each y (j) s suppled the for of (c (j), d (j) ) wth y (j) = c (j) /d (j). If P j does ot have data for x, the he sets both c (j) ad d (j) to 0., Output Player P j, 1 j, lears the coeffcets a ad b, such that y = a x + b where a ad b are the cooperatvely coputed lear regresso paraeters. I ters of the hosptal exaple, suppose hosptals wat to jotly bechark ther ortalty rates of surgery operatos o heart dseases. x the could be the categorzed health codtos of the patets, whereas y s the overall ortalty rates for patets wth catalog x. We assue there s a lear relato betwee x (health codto) ad y (ortalty rate). For each x, hosptal P j provdes ts ow ortalty rate y (j) = c (j) /d (j), where c (j) ad d (j) are the uber of deaths ahe uber of patets, respectvely. y s the overall ortalty rate for x -codtoed patets, therefore t s the su of deaths dvded by the su of patets x catalog,.e., j=1 c(j) / j=1 d(j). All protocols preseted ths work are evaluated ters of ther coputatoal ad coucato coplexty, as well as the uber of coucato rouds they requre. For the purposes of our evaluato, a coucato roud s defed as a exchage of essages betwee the players durg whch a sgle player ca () receve oe or ore essages fro other players, perfor calculatos, ad sed oe or ore essages, or () sed oe of ore essages ad the receve oe or ore essages fro other players. Every player partcpates a sgle roud at ost oce, wth the total uber of essages set over the etwork durg oe roud beg up to BUILDING BLOCKS Gvg the fully-developed protocols would ake the too log ad rather haro coprehed. Ths secto as at

5 akg the later presetato of the protocols crsper by presetg parts of our solutos ahead of te. The buldg blocks that we descrbe ths secto are a portat part of ths work, because they lay the groud for solvg the forecastg probles a secure fasho ad provde a desg choce for the fal protocols. Ths secto presets splt ad dvso protocols, where the later protocols operate o floatg pot ubers ahus are ew to SMC. We have also developed secure suato ad coparso protocols that provde alteratve solutos to forecastg based o ovg average ad weghted ovg average techques. They are ot cluded ths paper due to space costrats but ca be foud [4]. We cosder three dfferet types of players wth respect to alcous behavor: 1. Se-hoest players: Se-hoest players (also kow as hoest but curous ) wll follow the protocol as prescrbed, but ght also attept to dscover ore forato based o the data they receve at varous steps of the protocol. 2. Colludg players: Colludg players exhbt the behavor of se-hoest players but ca also collude together order to dscover soe addtoal forato about other players data. 3. Malcous players: Malcous players ay arbtrarly sbehave: they ca collude agast other players ad ca devate fro the correct steps of the protocol. Dfferet types of devato fro the protocol clude supplyg correct data, odfyg data at teredate steps of the protocol (possbly collaborato wth other alcous players), preaturely quttg the protocol, or perforg correct coputatos at certa steps of the protocol. I our solutos, we focus o the frst two types of players. Cosderg oly se-hoest players s ot suffcet because our case the players ght be copetg busesses, whch does ot allow us to exclude colludg behavor fro cosderato. We do ot cosder the thrype of sbehavg players o the grouds that all the players are terested the outcoe of the coputato ad wll ot attept to dsrupt t. Soe of our solutos ca be tued to provde a trade-off betwee coplexty ad robustess agast colludg behavor, ad should be setup to accout for the expected behavor of the players wth respect to collusos. I other words, f durg the coputato t s ot expectehat a sgfcat uber of players wll collude, the protocol ca be ade ore effcet by settg the colluso threshold low. 4.1 Secure splt protocol The frst protocol that we preset s a secure splt protocol that s used as a buldg block the fal protocols as well as other buldg blocks. Pror to executo of the splt protocol all players addtvely share a te where the dvdual shares are prvate forato. The goal of ths protocol s to bld dvdual shares such a way that o share reveals prvate forato, but the total su of all shares stays the sae as before. At the ed of the protocol each player holds a large rado uber, ahe tal prvate put stays hdde. The detals of ths protocol are rescet of the techques used the dg cryptographers [9]. Protocol 1. Secure Splt Protocol Iput Player P j, 1 j, provdes prvate put x (j). Output Player P j, 1 j, obtas z (j) j=1 x(j) = j=1 z(j). such that Protocol Steps: 1. All players jotly agree o a colluso threshold k, such that 1 k Each player P j splts x (j) betwee k + 1 players the followg way: Player P j geerates k large rado values r (j) 1,..., r(j) k (both postve ad egatve) ad seds the to radoly chose k players fro the reag 1 players. He sets hs share of x (j) to be r (j) 0 = x (j) r (j) 1... r (j) k. 3. After recevg k essages fro other players (o average k = k), each player P j coputes z (j) = r (j) 0 + j r() l for soe 1 l k. such that player P set r () l to P j Aalyss Ths ethod of hdg data s secure but ot a forato-theoretc sese. As ca be see fro the protocol, a sgle prvate put s dstrbuted aog k + 1 players. Whe ths protocol s used as a part of aother protocol, dvdual shares z (j) s are revealed. I order for a dvdual x (j) to be revealed, however, all of the k players to who player P j set essages step (2), ad all players who set essages to player P j step (3), ust collude. Assue that < s the uber of players that collude agast player P j. The the probablty of coprosg x (j) s 0 whe < k. Whe k, the probablty of coprose s less tha: 1 k 1 k 1 1 whch expoetally decreases as k creases. For stace, whe the uber of colluders s less tha /2 ad s eve, the probablty of a successful coprose s less tha: 2 k 1 k 2 1 whch s upper-bouded by 2 k. If we use k = c log, where c s a costat, the probablty s upper-bouded by c. Ths eas that a sub-lear colluso threshold k results a perforace that s probablstcally colluso-resstat agast a lear uber of colluders. The protocol s eve better wth respect to collusve behavor tha ths probablty aalyss ples, because the colluders have o way of kowg whether they succeeded or ot. Eve f the colluders kew fro step (2) that all of the k essages that player P j set wet to the colludg players, they stll would ot kow whether soe o-colludg player step (3) chose player P j to be a recpet of oe of hs essages. Thus the colluders have o success dcator to tell the whether they succeeded the coprose or ot.

6 Ths protocol s perfored 1 roud (assue all the players execute the protocol sultaeously); the total coucato s O(k) essages; ad coputatoal coplexty at each player s O(k). Whe k takes the hghest value avalable k = 1, a colluso of ay uber of players less tha 1 has zero probablty to succeed. Coucato coplexty ths case reaches O( 2 ). 4.2 Secure dvso protocols It s possble to use the geeral crcut sulato results to carry out secure dvso, ad we eeo copare ths approach to our approaches. The practcal crcuts for 2-party l-bt dvso have sze O(l 2 ), ad sulato of ths crcut requres O(l) 1-out-of-2 Oblvous Trasfers ad O(l 2 ) evaluatos of pseudorado fuctos (such as AES). Although there are asyptotc proveets to these crcuts, they coe at the cost of huge costat factors; the asyptotcally best of the (ahe worst ters of havg practcally large costat factors) s a crcut of sze O(l log l log log l) [8, 3] derved fro the textbook Schoehage-Strasse teger ultplcato algorth [24] (whch s tself of aly theoretcal terest, ad ot used practce). Covertg a 2-party dvso protocol to a geeral -party dvso protocol adds a addtoal factor of O( 2 ). The protocols we have developeo hadle secure ultparty dvso use a secure ultplcato protocol ther varous steps, ad each such ultplcato protocol s easly carred out usg O(1) hooorphc ecrypto coputatos; we cout the uber of expesve cryptographc operatos (e.g., hooorphc ecrypto, oblvous trasfer) rather tha bts coucated because, for both crcut sulatos ad our protocols, the latter ca be obtaed fro the forer by ultplyg by l, hece for relatve coparsos we ca detere whch s better usg the forer. The protocols we gve dffer ther costrats o the uber of players, coucato ad coputatoal coplexty of the protocols, aher robustess agast colludg players. We preset the the order of ther splcty (splest frst). The frst protocol operates o floatg pot ubers. The other three protocols utlze hooorphc ecrypto ad operate o tegers, but the result s stll coputed as a floatg pot uber. Gve two ubers x ad y addtvely splt aog players, all of our protocols wll output x/y f y 0 ad a specal sybol whch dcates that dvso s ot possble whe y = 0. Ths eas that all players wll lear whether y = 0, whch s uavodable as t s a heret part of the aswer (hece t s ot a forato leak). The dea behhe frst protocol s that oe of the players s radoly selecteo perfor dvso o tes that all other players prevously bld. Ths player perfors fuctoalty slar to that of a utrusted server ad returs the result to all other players, who the ubl. A verso of ths protocol that uses a exteral utrusted server ca be foud [4]. Protocol 2. Secure Dvso Protocol DIV1 Iput Player P j, 1 j, provdes x (j) ad y (j). Output Player P j, 1 j, lears x y. Protocol Steps: 1. All players radoly select oe player aog all of the who wll perfor dvso. Wthout loss of geeralty, assue that player P s chose. 2. Player P splts hs tes x () ad y () to 1 rado ubers each,.e., x () 1 = j=1 r(j) x ad y () 1 = j=1 r(j) y. Player P seds each par r x (j) ad r y (j) to player P j. 3. Player P j, 1 j 1, receves r (j) x fro player P ad sets x (j) = x (j) + r x (j) y (j) = y (j) + r (j) y. ad r y (j) ad 4. Players P 1 through P 1 egage the secure splt protocol two tes provdg 0 as ther put. Player P j stores the results of the protocol vocatos as ρ (j) 1 ad ρ (j) 2, respectvely. 5. Players P 1 through P 1 jotly agree o two rado floatg pot ubers α ad β. 6. Player P j, 1 j 1, coputes a par of values a (j) = α(x (j) + ρ (j) 1 ), b(j) = β(y (j) + ρ (j) 2 ) ad seds the par to player P. 7. Player P receves 1 pars a (j), b (j), ad 1 coputes a = j=1 a(j) 1, b = j=1 b(j), ad the δ = a. Player P seds δ to each of players b P 1 through P Player P j, 1 j 1, recovers the value of x y by coputg β δ. The value of x s the set to α y player P. Note: If t s desreo have the result splt aog all partes at the ed of the protocol, player P ca splt δ to 1 rado floatg pot ubers δ (j),.e., 1 δ = j=1 δ(j) ad sed each δ (j) to the correspodg player P j. Player P j, 1 j 1, the recovers the result by coputg β α δ(j), ad splts t to two parts, oe of whch s kept locally, whle the secod oe (purely rado) s set to player P. Player P the collects these 1 ubers ad sets hs share of the result to be ther su. Aalyss Ths protocol works whe the total uber of players 3. It s reslet to collusos f the player who perfors dvso (P ) does ot collude wth other players, but f that player colludes wth ay other player the the aggregate x ad y ca be revealed. The probablty of learg dvdual x (j) or y (j) depeds o the colluso threshold k used the splt protocol (see the aalyss of protocol 1) ad s further lowered by the fact that the player who perfors dvso ust be aog the colludg players. Ths protocol ca be perfored 5 rouds, wth total coucato of O(k), where k s the colluso threshold for the splt protocol. The coputatoal coplexty for the player who perfors dvso s O(), a s O(k) for every other player. Istead of dedcatg a sgle player for perforg dvso, all players ca be dvded to two groups where the frst group perfors dvso for the secod oe ahe secod group perfors dvso for the frst group; the result s recovered jotly by both groups. Such protocol ca be further geeralzeo a larger uber of groups. See [4] for ore detals o the protocol.

7 As the prevous protocol oly works for 3, ext we preset solutos to two-party dvso. I the two-party dvso protocols ahe oe followg the, all of whch use hooorphc ecrypto, we assue that all players pror to protocol tato agree o a rage of possble values. That s, they defe MAXIN T to be a large uber, such that all possble (aggregate) values of x ad y wll be less tha MAXINT, but soe radoly geerated ubers ay exceed MAXINT ( whch case ths s explctly stated whe they are defed). Also, we cosder 1/MAXIN T to be a eglgble error. Aother assupto that we ake these protocols s that both x ad y are o-egatve ubers. Ths s a acceptable ltato because all the forecastg ethods that we solve operate o postve quattes. Lastly, all ecrypto arthetc s teger-based. If players wat to provde ther puts as floatg pot ubers, they eeo agree to covert the to teger represetato by gorg decal pots up to a certa precso. We ow gve a two-party dvso a protocol that s provably secure ad operates O(log l) rouds for l-bt ubers, ad requres O(log l) hooorphc ecryptos ad O(log l) oblvous trasfers. We suarze what t acheves the theore that follows, where by secure we ea provably secure a forato-theoretc sese. Theore 1. If two l-bt ubers x ad y are gve odularly addtvely splt betwee two partes P 1 ad P 2, the t s possble for the two partes to securely copute the rato θ = x/y to wth l bts of precso O(log l) rouds, where each roud s doe wth O(1) hooorphc ecryptos ad oe oblvous trasfer. Proof Sketch: If we could soehow copute the teger z = 2 2l 1 /y splt fasho wth the claed bouds, the t would be easy to copute x z splt fasho by dog oe addtoal splt ultplcato of x ad z; recall that such a splt ultplcato of two tegers ca be doe O(1) rouds ad usg O(1) hooorphc ecrypto coputatos. After ths, the two partes would exchage ther halves of x z ahereby lear θ to wth the desred accuracy. So the a proble s how the two partes ca securely copute, splt fasho ao wth l bts of accuracy, the teger z. (Note that z has to be coputed splt fasho, otherwse both partes ca deduce y fro t.) Before turg our atteto to ths coputato of z, we observe that we ca choose, wthout loss of securty, the odulus of the addtve-splttg to be the sae as the oe for the hooorphc ecrypto syste. That s, f arthetc s odulo T for the hooorphc ecrypto, the we ca assue that P 1 (resp., P 2) tally has x (1) ad y (1) (x (2) ad y (2) ), such that x (1) + x (2) od T = x, ad y (1) + y (2) od T = y. For reasos that wll becoe clear later, we also assue that T s chose such that T > 2 3l+1. Because z s the recprocal of y, the frst thought that coes to d s to use the recprocal-coputato techque based o the cetures-old Newto s ethod (as was doe [1] ad [3]). Ths, however, rus to a subtle dffculty due to the fact that y s splt, ahe aswer z (ad all teredate aswers of the teratve process) ust also be splt for: Each terato, whe used o tegers ad producg a teger aswer, volves scalg (dvso by a quatty C kow to both partes). But because the values to be dvded by that C are addtvely splt odulo T, the two partes caot sply dvde ther respectve shares by ths C as t ay troduce a error due to the odulo T wraparoud. The protocol we gve below overcoes ths dffculty. Before gvg the protocol, we recall that the basc dea of the teratve coputato of z s to use a sequece of approxatos to z that are progressvely ore accurate: Specfcally, f z approxates z to wth α correct bts, the: z +1 = 2z (z ) 2 y/2 2l 1 approxates z to wth 2α correct bts (the earleretoed scalg factor C s therefore 2 2l 1 ). It wll so happe (as wll becoe clear the splt scalg protocol below) that because we operate o splt values our protocol s z +1 approxates z to wth 2α 2 bts. Therefore, as log as we start wth a splt z 0 that approxates z to wth a few correct bts, each terato approxately doubles the uber of correct bts the approxato. Ths ples that O(log l) teratos suffce (actually, log l f z 0 starts out wth 3 correct bts). The above teratve forula actually always coverges as log as z 0 {2, 2 l 1}: It just coverges faster f z 0 starts out wth at least a few (e.g., 3) correct ost sgfcat bts. There are ay ways of coputg a good eough tal z 0, ahese wll be descrbed the full verso of the paper. Each terato requres carryg out two splt ultplcatos (hece O(1) hooorphc ecryptos) ad oe splt subtracto. Each terato also requres scalg w = (z ) 2 y by the factor C to obta w/c splt fasho, ahe protocol for dog so s gve below (we droppehe celg otato t to avod uecessarly clutterg the presetato). Protocol 3. Secure Two-Party Protocol for Scalg by a Costat Iput Player P 1 has (1), P 2 has (2), such that (1) + (2) od T = w, ad w < 2 3l (because w s the product of three l-bt ters). Output Player P 1 receves updated (1) ad P 2 receves (2), such that (1) + (2) od T = w/c. Protocol Steps: 1. P 1 chooses a rado r [0, T 1] ad updates (1) by dog (1) = r + (1) /C od T. 2. The two players egage a oblvous trasfer protocol whch P 1 prepares a par (a 0, a 1) ad P 2 obtas a x {a 0, a 1} where (a 0, a 1) = (r, r T/C) f (1) < 2 3l (a 0, a 1) = (r T/C, r T/C) f (1) 2 3l x = a { (2) 2 3l },.e., x s a1 f (2) 2 3l ad a 0 otherwse. 3. P 2 updates (2) by dog (2) = x + (2) /C. Correctess of the splt scalg protocol follows fro the two possble cases for (1) + (2) pror to the update doe by the protocol: If, before the protocol s updates, (1) + (2) < 2 3l the both (1) ad (2) are < 2 3l, ahe protocol updates by settg (1) = r + (1) /C ad (2) = r + (2) /C, whch s correct because the ew values (1) ad (2) result (1) + (2) od T = w/c, as requred.

8 If, before the protocol s updates, (1) + (2) 2 3k the, because w = (1) + (2) od T = w < 2 3l, t ust be the case that (1) + (2) [T, T + 2 3l ]. Ths eas that dvdg each of (1) ad (2) by C ths case would troduce a addtve T/C error that ust be subtracted out by the protocol. To see that t s subtracted out, ote that ths case at least oe of { (1), (2) } s 2 3l, aherefore P 2 s guarateed to obta x = r T/C ahe T/C ter x cacels out the above-etoed addtve error of +T/C. Ths copletes the proof. The followg s a hgh-level suary of the protocol descrbed the above proof. Protocol 4. Secure Two-party Dvso Protocol DIV2 Iput Player P 1 provdes x (1) ad y (1), ad player P 2 provdes x (2) ad y (2). Every oe of x (1), y (1), x (2), y (2) s [0, T ) where T s the odulus of the hooorphc ecrypto syste used. Both x = x (1) + x (2) od T ad y = y (1) + y (2) od T are assueo be l bts log, hece saller tha 2 l. T s chose such that T > 2 3l+1. Output Players P 1 ad P 2 lear x y bts. to wth l sgfcat Protocol Steps: 1. Startg wth splt z 0, for = 1, 2,..., 2 log l, P 1 ad P 2 securely copute z splt fasho accordg to the terato equato: z +1 = 2z (z ) 2 y/2 2l 1 by usg secure splt ultplcato twce, ad usg oce the splt scalg protocol descrbed the proof of Theore 1. A kow techque for perforg secure two-party ultplcato s gve Appedx A. As oted earler, z = z 2 log l s a l bt teger that equals 2 2l 1 /y. 2. P 1 ad P 2 securely ultply z by x splt fasho. The resultg x z s x/y scaled up so t s teger for. They exchage ther shares of x z ad lear x/y to wth l sgfcat bts. Aalyss See the proof of Theore 1 for coputatoal ad roud coplexty. Coucato overhead for each user s O(1) essages per roud, aherefore O(log l) essages total. Although the above protocol s uch better tha a crcut sulato approach to solvg the secure dvso proble, t s stll expesve. Because the quattes used forecastg are ofte prvate oly a approxate sese, t s worthwhle to cosder protocols that leak soe forato but are ore effcet (for exaple, oe ay ot d f others kow the sales fgures creased by betwee 5 ad 10 percet, as log as they do ot kow the exacts percetage). Protocol 5. Secure Two-party Dvso Protocol DIV3 Iput Player P 1 provdes x (1) ad y (1), ad player P 2 provdes x (2) ad y (2). Output Both players lear x y, where x = x(1) + x (2) y = y (1) + y (2). ad Protocol Steps: 1. Player P 1 geerates a (publc, prvate) key par a hooorphc seatcally secure ecrypto syste [21, 22] where arthetc s odulo N, wth N 2 MAXIN T 2 (Recall that such a syste E(a) E(b) = E(a + b), ad othg ca be leared about c fro E(c).) 2. Player P 1 coputes E(x (1) ) ad E(y (1) ), ad seds the to P 2 alog wth the publc key of the hooorphc ecrypto syste. 3. Player P 2 coputes E(x) = E(x (1) ) E(x (2) ) ad E(y) = E(y (1) ) E(y (2) ). 4. Player P 2 chooses four rados α 1, α 2, β 1, β 2 fro a rage [u, MAXINT u] where u s kow to P 2 but ot to P 1, ahe coputes: p 1 = E(x) α 1 od N = E(α 1 x) q 1 = E(y) β 1 od N = E(β 1 y) p 2 = E(x) α 2 od N = E(α 2 x) q 2 = E(y) β 2 od N = E(β 2 y) Player P 2 the coputes v = p 1 q 1 = E(α 1x + β 1y) ad w = p 2 q 2 = E(α 2x + β 2y) ad seds the to player P 1. Note that what s sde the ecrypto s less tha N so there s o wraparoud due to the odulo N arthetc. 5. Player P 1 decrypts v ad w ad gets D(v) = α 1x + β 1y ad D(w) = α 2x + β 2y. He the coputes ther (floatg pot) rato δ = (α 1x + β 1y)/(α 2x + β 2y) ad seds t to P Player P 2 coputes the rato x as (β1 δ β2)/(δ y α 2 α 1) ad forwards the aswer to P 1. Aalyss Ths protocol has a ltato: whe x = 0, the protocol reveals β 1y ad β 2y to player P 1. Player P 1 the ca detere possble values of y (usg a gcd coputato, etc.). Thus, ths protocol should ot be used whe x ca take the value of 0, or, alteratvely, the coeffcets β 1 ad β 2 could be costructed a way to ze the probablty of a successful attack whe x = 0 (see [4] for ore forato). As ths protocol s desged for two-party coputato, there s o eed to cosder colluso. The protocol cossts of 2 rouds. Each player eeds to perfor two ecryptos odular arthetc. Player P 1 addtoally creates a key par ad perfors two decrypt operatos. Player P 2 also perfors a sall (costat) uber of ultplcato ad expoetato operatos odular arthetc. Thus the overall coputatoal ad coucato coplexty s O(1). Lastly, we gve a protocol that s secure agast collusos of up to 1 players. I what follows, the ultplcatve coeffcets α ad β are plctly costructed as a product of dvdual α j s ad β j s,.e., α = j=1 αj ad β = j=1 βj where α j ad β j are kow oly to player P j. Protocol 6. Secure Dvso Protocol DIV4 Iput Player P j, 1 j, provdes x (j) ad y (j).

9 Output Player P j, 1 j, lears x y. Protocol Steps: 1. Each player P j geerates a (publc, prvate) key par E j ad D j a hooorphc seatcally secure syste odulo N j wth N j MAXINT +1. P j seds to P 1 the publc key E j ahe tes p j = E j(x (j) ) ad q j = E j(y (j) ). Throughout what follows, eve as they get updated, p j ad q j should be thought of as the ecryptos of the curret (.e., updated) x (j) ad (respectvely) y (j). 2. For = 1,..., tur, the followg steps are repeated: (a) I ths step player P updates the p j ad q j other tha hs ow (.e., wth j ) that he receved ( step 1 f = 1, otherwse fro P 1 the prevous terato of (a) (c)). He does so as follows. Frst, P creates two rado ubers α ad β the rage [MAXINT/2, MAXINT ]. Next, player P geerates 1 pars of rado ubers (oe par a,j, b,j for each other P j), where each such rado s less tha MAXINT +1. For each p j ad q j, j, P the coputes: p j q j = p α j E j(a,j) = E j(x (j) α ) E j(a,j) = E j(x (j) α + a,j) = q β j E j(b,j) = E j(y (j) β ) E j(b,j) = E j(y (j) β + b,j) whch plctly ultples x (j) (resp., y (j) ) by α (β ) ahe adds a rado to t. (b) Player P ow updates hs ow p ad q by dog p = E (α D (p ) j a,j) q = E (β D (q ) j b,j) whch plctly ultples x () (resp., y () ) by α (β ) ahe subtracts fro t a rado that cacels out the rado ubers plctly added (a) to the other x (j) s (resp., y (j) s). Note: The above decrypto ad reecrypto of p ad q are ot ecessary, the sese that the coputato (b) could have bee perfored o ecryptees just lke the coputato of (a) was, but we chose to do the arthetc o u-ecrypted values for effcecy reasos. (c) If < the player P seds all of the p j ad q j (cludg hs ow p ad q ), as well as all ecrypto keys E j, to P +1. Otherwse = ad P seds every p j, q j par to the correspodg player P j who the decrypts the wth hs prvate key D j ad obtas hs fal x (j), y (j), that s, x (j) = D j(p j) ad y (j) = D j(q j) for all j (cludg j = ). At the ed of the kth terato of (a) (e) the su of the tes x (j) s (α 1 α k x) ahe su of the tes y (j) s (β 1 β k y). Therefore at the ed of step (1) the su of the resultg x (j) s αx. Slarly, the su of the resultg y (j) s βy. Note that o player kows (or wll kow) α or β. 3. Every player P j geerates two rado ubers r x (j) ad r y (j) less tha (MAXINT 1 )/(2 1 ), ad sets x (j) = x (j) + r x (j) ad y (j) = y (j) + r y (j). Now the su of all x (j) s wll gve αx+r x ahe su of y (j) s s βy + r y, where r x ad r y are eglgble copareo αx ad βy (ore dscusso of ths follows). 4. All players egage the secure splt protocol provdg ther y (j) as put ad obtag y (j). The every player P j publshes hs share ρ (j). 5. After recevg all the ρ () s, each player P j coputes the su of all the ρ () s, whch equals βy + r y. Sce everyoe ow kows βy+r y, each player P j ca copute δ (j) = x (j) /(βy + r y). 6. Every player P j reveals to all others the (floatg pot) rato t j = β j/α j. 7. Every player P j coputes δ (j) t 1t 2 t = δ (j) β/α, whch results the approxato of x y addtvely splt for aog the players (wth P j s share beg δ (j) β/α). To recover the aswer as the su of these shares, they ru a secure splt protocol, the copute (ad reveal to all) the su of all shares, whch s x wth the ecessary precso. y Aalyss The aggregate rado ubers r x ad r y are addeo αx ad βy to ze the possblty of factorg αx ad βy. For stace, step (3) all players receve the su of y (j) s, ad wthout protectg βy wth r y soe players ght attept to factor the value. Whle t s very coputatoally expesve to factor ths uber ad furtherore, gve ts factors, ot possble to deterstcally dfferetate betwee factors of β ad y, we stll would lke to lower the possblty of success as uch as possble. Thus, we requre that α j ad β j are at least as large as M A XIN T/2, whch gves us αx/r x M A XIN T ad βy/r y MAXIN T ad s acceptable (recall that we cosder 1/MAXINT to be a eglgble error). Furtherore, we copute: αx + r x βy + r y = αx βy 1 + r x/αx 1 + r y/βy αx βy 1 + rx αx βy ry whch coverges to αx/βy whe r x αx, r y βy. Now order to successfully factor βy, a attacker ust try all possble r y, whch s a prohbtvely large uber o the order of MAXINT 1. Ths protocol does ot scale well to large s because the legth of the ubers that players operate s lear the uber of players. The protocol s coducted 5 rouds, wth the total coucato of O( 2 ) tes (or O() essages). The coputatoal coplexty at each player s bouded by key geerato (whch ca be precoputed) ad O() ecryptos. The above protocol provdes a hgh degree of protecto where a colluso of ay uber of players caot succeed. Ths ght ot be requred certa settgs, ahe protocol ca be tueo lower ts robustess ad at the sae

10 te lower ts coucato ad coputatoal cost. Slar to protocol 1, we ca radoly select a subset of k players (2 k ) who wll coduct the above protocol after all other players dstrbute ther dvdual shares aog those k. By tug the value of k, the players ca fd a balace betwee the acceptable reslece ad coplexty of the protocol. We do ot provde detaled aalyss of ths protocol here due to space ltatos. Other tradeoffs are possble, e.g., decreasg the roud coplexty at the expese of a hgher coputato coplexty by usg crcut sulato. Aga, we ot the detals. 5. SECURE TIME-SERIES FORECASTING Ths secto gves fal protocols for perforg collaboratve forecastg based o te seres. We start wth ovg average, the proceed wth weghted ovg average, ad lastly cover expoetal soothg. 5.1 Movg average The goal of ovg average forecastg s to fhe behavor of the fucto at te t + 1 relatve to the curret te t. Ths value ca be coputed as: x = = Ft dt = 1 / =0 (2) dt dt 1 ( 1)dt Below we provde a protocol for solvg the ovg average proble based o our dvso protocols. For ovg average ad weghted ovg average, we have developed alteratve solutos that use bary search ad a secure coparso protocol as ther buldg blocks ad ca be foud [4]. We do ot gve ther detals ths paper. Protocol 7. Secure Movg Average Protocol Iput Player P j, 1 j, has put data d (j) t te tervals, where 0 1. for Output Player P j, 1 j, lears Ft dt, F t s coputed as the ovg average. Protocol Steps: 1. Each player P j sets x (j) = d (j) t d (j) t 1 ( 1)d(j) t ad y (j) = d (j) t. 2. All players jotly coduct a secure dvso protocol, wth each player P j provdg put x (j) ad y (j). The output of the dvso protocol s the output of ths protocol,.e., Ft dt. Aalyss Both coplexty ad robustess of ths protocol deped o the uderlyg secure dvso protocol. Coucato ad coplexty requreets are also those of the dvso protocol because there s o coucato step (1) ad oly O(1) coputato ( s costat). 5.2 Weghted ovg average Coputato of the weghted ovg average s very slar to the prevous case of the ovg average coputato. The dfferece s that all players agree o a weght vector w = {w 0, w 1,..., w 1}, whch s publc. Accordg to the forula for coputg the weghted ovg average, equato (2) for ths case becoes: x = = Ft dt = 1 w =0 w0dt w 2dt 1 (1 w 1)dt Protocol 8. Secure Weghted Movg Average Protocol Iput Player P j, 1 j, supples data pots d (j) t, where 0 1. Output Player P j, 1 j, obtas Ft dt, where F t correspods to jot coputato of the weghted ovg average. Protocol Steps Very slar to Protocol 7 s steps: 1. Each player P j sets x (j) = w 0d (j) t w 2d (j) t 1 (1 w 1)d(j) t ad y (j) = d (j) t. 2. All players jotly coduct a secure dvso protocol, where each player P j supples put x (j) ad y (j). The coputato results the desred value. Aalyss See aalyss of protocol Expoetal Soothg The forula for expoetal soothg ca also be rewrtte to splfy jot coputato. I the forula below assue α s publc, F t 1 s calculated durg the prevous executo of the protocol ad s addtvely splt betwee players. The goal s the to copute: x = = Ft dt = Ft 1 + α(dt 1 Ft 1) dt (1 α)ft 1 + αdt 1 dt Protocol 9. Secure Expoetal Soothg Protocol Usg Dvso Protocol Iput Player P j, 1 j, provdes put data d (j) d (j) t t 1 ad, as well as the result of the prevous executo of the protocol F (j) t 1. Output Player P j, 1 j, lears Ft dt, where F t s the result of expoetal soothg coputato, ad also gets a share F (j) t of F t. Protocol Steps: 1. Each player P j sets x (j) = (1 α)f (j) t 1 + αd(j) t 1 d(j) t ad y (j) = d (j) t. 2. All players jotly execute a secure dvso protocol, where each player P j provdes x (j) ad y (j) as hs put. The output of the dvso protocol s the output of ths protocol. 3. Each player P j sets F (j) t as (1 α)f (j) t 1 + αd(j) t 1. Aalyss The core of ths protocol s the uderlyg dvso protocol, therefore all coplexty ad coucato aalyss, as well as robustess agast colludg players s the sae as for the dvso protocol used.

11 6. SECURE LINEAR REGRESSION BENCHMARKING As was etoed earler, we apply the lear regresso techque to a set of x, y values, where the uber of pots s set advace. The each y s gve the for of two ubers c ad d, where y = c /d to ake t possble to operate o oralzed values ad guaratee correct outcoe. We cosder ths scearo to be ore geeral tha the oe where each player provdes oly hs y s values. Ths s because every protocol that solves a proble wth y values provded the for of c ad d pars ca be useo solve that proble where y s provded as a sgle value. I our case, f players decde that dvso s ot ecessary, they ca follow ether of the paths below: (a) They ca agree o the values of d (j) s such that j=1 d(j) = 1. (b) They ca ot the step of the protocol where the y s are coputed usg the dvso protocol ad use ther orgal shares of y s stead. Aother assupto that we ake ths odel s that all values of x are kow to all players ad are agreed upo pror to protocol tato. Ths eas that all of the x s values wll be used coputato of the regresso coeffcets eve f a player does ot have data for all of the pots. If, however, oe of the players have data for a specfc value of x, that pot ust be excluded fro the coputato. Ths eas that the players lear what data pot s beg excluded, whch s vewed as addtoal forato about other players put that should be kept secret. Chagg the protocol so that t ca hadle cases where o data s avalable for a certa pot ad o player lears ths forato wll result sgfcatly ore coplex solutos both ters of coputato ad coucato. Therefore, we decde to solve ths ssue the followg way. The protocol starts as usual, ad for each data pot we copute y = /. If t s detected that j=1 c(j) j=1 d(j) ths dvso s ot possble to perfor because all c (j), d (j) pars for a specfc data pot x are zero, the the executo s suspeded. Each player wll be otfehat coputato caot be carred out, ahey have two optos: they ca abort the protocol or cotue ts executo, but the latter case forato about the ssg values wll be revealed to all players. If all of the players decde to cotue, the value of x that causehe proble s excluded fro the set of possble pots ahe protocol s restarted. If at least oe of the players decdes to abort, executo terates. To copute the regresso coeffcets theselves, we use the forulas gve equato (1). Here the value of s publc ad ca be coputed by each player. equatos becoes: a = A =1 x y =1 x =1 y, b = =1 =1 x The the y / B where A ad B are kow to all players ad ca be precoputed, such that A = 1/ =1 x2 =1 x 2 ad a =1 B = x / (otce that B ca be coputed oly after a s kow as a result of jot coputato). Protocol 10. Secure Lear Regresso Protocol Iput Player P j, 1 j, provdes a set of pars c (j) d (j),, where correspods to data pots x 1, x 2,..., x. Output Player P j, 1 j, lears the coeffcets a ad b such that y = a x + b. Protocol Steps: 1. All players egage a secure dvso protocol tes to copute y for 1, where y s rea addtvely splt aog all players. 2. Each player P j locally coputes a (j) = A =1 xy(j) x =1 =1 y(j). 3. All players egage the secure splt protocol wth a (j), publsh ther outputs, ad copute the su a = j=1 a(j). 4. Each player P j locally coputes b (j) = /). =1 (y(j) 5. All players egage the secure splt protocol wth b (j), publsh ther outputs ad each player coputes b = j=1 b(j) B, where B s subtracted by each player separately fro coputg the su of b (j) s. Aalyss Ths protocol s as secure agast colludg players as ts uderlyg blocks are (.e., dvso ad splt). Coucato ad coputatoal coplexty of the protocol are also bouded by the dvso ad splt protocols, where both of the are voked a costat uber of tes (the dvso protocol s executed tes, where the uber of pots s costat). 7. CONCLUSIONS AND FUTURE WORK I ths work, we provded prvacy-preservg solutos to collaboratve forecastg ad becharkg that ca be used to crease the relablty of local forecasts ad data correlatos, ao coduct the evaluato of local perforace copareo global treds. We gave both buldg blocks aher use protocols for a uber of dfferet forecastg ethods based o te-seres ad regresso techques. The buldg blocks are geeral eough to be used other protocols for forecastg ad becharkg, as well as other applcatos. I partcular, the dvso protocols preseted ths work, to the best of our kowledge, are the frst attept to perfor dvso secure ult-party coputato as well as to perfor coputatos o floatg pot ubers. Ths work ca be exteded a uber of ways. Future drectos clude: The odel ca be extedeo other te-seres forecastg techques. Alog wth provdg short-rage forecastg, we would lke to be able to perfor log-rage forecasts. Log-rage forecasts take to accout seasoal chages ad other log-rage patters. We also would lke to desg protocols to cover other types of regressos for becharkg collaborato. Ths wll allow us to draw relable coclusos for dfferet types of data dstrbutos.

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