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A New Bth FHE Shee oe the Iteges Kw W Sog K Cho U Shoo of Mthets K I Sg Uesty yogyg Deot eoe s e of Koe Astt The FHE (fy hoooh eyto) shees [7 3] sed o the odfed AGCD oe (osefee AGCD oe) e ee to t tts ese ts sety ees ty o the hdess of ftog d soe FHE shees sed o the deso AGCD wthot the ose-fee ssto fo ee [] hs dw tht ts hetets e ey ge I ths e we ostt ew th FHE shee sed o the deso AGCD oe to oeoe these weesses d oe ts sety Keywods: FHE AGCD oe Bth Eyto Chese ede Theoe Itodto I 978 est Ade d Detoos [8] fsty toded the oet of the FHE shee The go of the de s to ow ottos o eyted dt wthot oss of the dt sety Thee dedes te 9 C Gety [ ] oosed the fst FHE shee sed o the de tte The sety of ths FHE shee s sed o the Boded Dste Deodg (D) oe d the Sse Sset S (SSS) oe te Gety s FHE shee ws oed y C Gety & S He [] N St & F Vetee [9] D Stehѐ & Stefed [] et I D Gety He d Vtth [] oosed the tete FHE shee The sety of the FHE shee ees o the SSS d the Aote Getest Coo Dso (AGCD) oe The effey of the DGHV shee hs ee oed y D Beoh et [] H Cheo et [7] S Coo et [9] H Cheo & D Stehѐ [8] et Nowdys My FHE shees towds the esste to t tts t the sety o two otto oes: () the eg Wth Eos (WE) oe defed y ege[7] d () Howge- Gh s AGCD oe[] U to eset thee e o oyo-te t goths to soe these oes Z Bes & V Vtth [ 5] deeoed the fst WE-sed FHE shees These shees wee oed to oes wth ette effey d sety y sseet wos sh s Z Bes [] Z Bes d C Gety et [3] Z Bes d V Vtth [] C Gety d S He et [3 ] C Gety d A Sh et [5] et At Cyt 5 Cheo d Stehѐ [8] oght ot edto fo the WE oe to the AGCD oe Theefoe we osde tht the AGCD oe whh fds ot the hdde oo dso fo y ote tes of e e o odt of es e hde th the WE oe whose oyo te t goth s ow yet [] The AGCD-sed FHE shees fsty oosed y D et [] e gettg teests of es of esehes fo ts dtge of deedey o etey esy tege oetos [ 7 9 8] d the AGCD-sed FHE shees e ofte ed the FHE oe the teges (FHE-OI) The FHE shee sggested [] hs dw tht t hs to eyt/deyt oy oe t t te Cheo et [7] d Coo et [9] eseted the th FHE shees tht eyt/deyt see ts t oe t the sety of the th eso s sed o the ose-fee t of the AGCD oe tht hs ght oo te og ge ote oo tes Ths odfed AGCD oe s ese th the gee oe d ttes gess soe of the hdde oo dso y ftog the ght oo te wth the he of t ote

I [8] the thos sggested ew AGCD-sed FHE shee ee to t tt ode to oeoe the defets of [7 9] Howee they od ot oeoe the defet of eyto/deyto e oe t Theefoe they sggested the ostto of th oe of the FHE shee s oe oe O the othe hd Beoh et [] oosed ew FHE-OI shee d ts th eso tht e esstt to t tts t they he oge hetets th the oe [8] O Cotto: I [8] the thos oy ostted the o-thed FHE shee d oted ot tht the shee y e eteded to th eso sy to [7 9] Howee thee s seos dffeee of the shees [7 9] d the oe [8] The dffeee s tht the sety of the shee [8] s sed o the og AGCD oe d the oes [7 9] s sed o the ose-fee t of the AGCD oe To e oe et the fst eeet of the -ey te of es t [7 9] hs dffeet oety [8] d [7 9] I [8] s ote oo s et oo te of the hdde e Theefoe thee e soe sses to ostt th eso of [8] I ths e we sggest ew th FHE shee wth h shote hetets th the oe [] whh e t soto of oe oe oe [8] d oe ts sety ees Notto We deote y foy t do fo fte set A Whe s dstto we deote y eest tege to e A seetg eeet seetg se odg to the dstto If odg wds f thee e two Ge d tht stsfes the odto od Z we se s e the s the to deote e We se the CT eesettos Fo ge wse oe teges we defe CT s e tege T T CT od fo Ety whee T : d : T whee ( ) Fo : sh tht od CT : CT CT CT Z we defe the fto Fo Z [ ) d y The we oe the foowg eesso: : [ ) Z d : wth y yy y : We so e the teso odt of two ge etos: y y y : T Moeoe t stsfes the foowg eto wth the s odt: : s foows:

Soe Dsttos d AGCD oes Fo -t es we defe soe dsttos s foows: : CT Z D D : Z : y CT Z Defto [7] (The e AGCD deso oe) et e the sety etes d -t es The deso oe s to dstgsh etwee the dstto D the fo dstto U [ ) Z Defto (The e -t es d e the hose de The deso oe s to dstgsh etwee the dstto D AGCD deso oe) et d e the sety etes d the fo dstto U [ ) Z The AGCD deso oe d the AGCD deso oe he the se degee of otto oety e f thee s oyo-te t goth to soe the AGCD deso oe the we ostt oyo-te t goth to soe the AGCD deso oe Moeoe the ese get s oet O the othe hd fo the ow edto of WE oe to AGCD oe [8] we sse tht the AGCD deso oe s hd: M ssto The AGCD deso oe s hd to soe y y t otes 3 O Bth Soewht Hoooh Eyto shee I ths seto we geee the FHE shee [8] to ew th Soewht HE shee d the oe ts sety To ostt ew th SHE(Soewht Hoooh Eyto) shee we se the CT eesetto 3 The Costto We defe soe etes I ths e s the sety ete s the t egth of the eo s the t egth of the seet e teges s the t egth of the hetets s the e of eytos of eo ey d s the e of dstt seet es We sse tht these etes stsfy the foowg ostts y dsssos [ 8]: og d KeyGe The seet ey s set of -t dstt odd es D ( : s : ) d ee the dees so tht d Choose et

X : Fo hoose do sset S of y : CT The et Y : y y Choose so tht the eto : s to e tege eto fo et y g CT : whee Eyt Choose do sset S S The hoose g D ( g s se th of Deyt s Ott : od Ete C Ge the y t C wth t d tto gtes of) C Add The ey s : X Y y d ott : od to t : s foows: Mt The t ets the estg tege 3 Coetess : y t ts d hetets e (Eyto ose) et s KeyGe fo The Q oof S y CT d S ) t y we y (ddto e hetet of tet fo soe Q Z d Z wth CT t S ts CT t S ts The oe eesso s e to the foowg oe fo soe CT O the othe hd ese of D F fo soe Z So ts t F Z d Z S G wth G ts Z t S G

t ts S G fo : t G S Theefoe Q fo soe ts To fd e od of S t G t S ts ts G Q od Z e (Deyto ose) et s e seet ey Fo ge eto ( tege Q f the s oof We he Theefoe f the Theefoe Tht s De Q ) d od Theoe (Coetess) et s KeyGe d et E ( ) The we he De whe og oof Asse og The og O the othe hd se we he Se s -t tege og d the we so he Fo e fo soe Q Z d soe Z deote the hetet of wth 8

whee Aodgy Theefoe fo e s Q De 33 Sety Z e 3 (eftoe Hsh e) [8] The sttst dste of the fo dstto d the dstto s ess th s ) Z s s ( U Theoe (Sety) O Bth FHE shee s CA-see de the ssto of hdess of AGCD oe oof Fo the ssto the ey d the sedo ey y whh ws de fo foy do seeto e ottoy dstgshe Fo the ft tht d e 3 the oty dste etwee dstto d fo dstto U ( s ) Z S Z e ess th - y Ths we ode tht the oty fo the tte to esey dstgsh the t hetet fo foy d doy seeted tege s ess th Bth eeed Hoooh Eyto shee Defto 3 ( hoooh shee) et s e KeyGe A shee s ed - hoooh shee f fo y y tege t C tht hs the t deth d -ts t hods tht De s E C C wth oty gete th fo the tets d the oesodg hetets E e (Addto ose) Fo ( ) Fo f the fo soe Q Z d Z wth et E deote the hetets of Q Add oof Aodg to the defto of Addto oeto Add Theefoe thee est soe tege sh tht O the othe hd Q fo soe Z D Q d Z ese of

The Q We ow the dffeee etwee d s oy whe othewse If the Theefoe thee s soe tege sh tht Q Q Ad the Add Q od Theefoe od Add fo : Fo ths we ode tht Add Q fo soe Z Q To fd e od of e 5 et e -t odd e tege d Z The fo soe Z wth d soe wth oof Fo ge If : the d od O the othe hd od Se

Theefoe thee s soe tege Z Se s -t odd e tege d the od sh tht - 3 3 Theefoe 3 d the e od of e (Mtto ose) Fo Fo fo soe Q s et E deote hetets of f the Q Z oof We he et The d Z Mt wth y CT fo y ooet of the g g CT A B CT : g B Moeoe se g we he : CT T -deso tege eto CT s ess th T A g whh es tht thee ests tege G [ ] sh tht d Fo the eto Mt y A B G A B G A G od B od od Mt s tege eto d fo Chese ede Theoe Ths we he CT od T

T B od od The thee ests -deso tege eto sh tht od F T od F T Fo e 5 fo thee est Z d we ote s foows od F T whee d et deote set of oeo ooet of the deso tege eto The B od F T od O the othe hd s te of Ths od whh es tht B od Fo D y thee est teges d sh tht y d sy thee ests teges Q d sh tht Q I ddto we he

G G y G y G A od Cog the oe two ests we he Mt G od et G : the we he od Mt whh es tht thee ests tege Q sh tht Q Mt Now et s fd the e od of G G Theefoe Theoe 3 O SHE shee s hoooh f the foowg ety hods: og og oof Fo eh ] [ d ] [ et e hetet wth fte the eto of the th ee gtes et e od of Fo e Fo e d e the foowg etosh hods etwee d The eee of the tye B A hs the soto A A B A A B A Theefoe we he eg wth

Fo e ths shee s hoooh f Fo o hyothess we he og og se s t tege Ths we he whh es tht Theefoe we ode tht t s hoooh efeees [] D Beoh Z Bes d T eot: FHE Oe the Iteges: Deoosed d Bthed the ost- Qt ege IAC Cytoogy et Ahe eot 7/5 (7) [] Z Bes: Fy hoooh eyto wthot ods swthg fo ss GSV I: Sf-N Cett (Eds) CYTO (NCS 77) 88 88 Sge-Veg () [3] Z Bes C Gety S He: ed hetets WE-sed hoooh eyto I: K Kosw G Ho (Eds) -Key Cytoghy-KC 3 (NCS 7778) 3 Sge-Veg (3) [] Z Bes V Vtth: Effet fy hoooh eyto fo (stdd) WE I o of FOCS 97 IEEE Cote Soety ess () [5] Z Bes V Vtth: Fy hoooh eyto fo g-we d sety fo ey deedet essges I: ogwy (Ed) CYTO (NCS 8) 55 5 Sge-Veg () [] Z Bes V Vtth: tte-sed FHE s see s KE I: o of ITCS ACM ess () [7] H Cheo S Coo T eot M Toh d A Y: Bth fy hoooh eyto oe the teges I: Thos ohsso hog Q Ngye (Ed) Ades Cytoogy EUOCYT 3 (NCS 788) 35 335 Sge-Veg (3) [8] H Cheo D Stehѐ: Fy hoooh eyto oe the teges ested I: E Oswd M Fsh (Eds) Ades Cytoogy EUOCYT 5 (NCS 95) 53 53 Sge-Veg (5) [9] S Coo T eot M Toh: Se-t fy hoooh eyto oe the teges I: H Kwy (Ed) -Key Cytoghy-KC (NCS 8383) 3 38 Sge-Veg () [] C Gety: A fy hoooh eyto shee hd thess Stfod Uesty (9) [] C Gety: Fy hoooh eyto sg de ttes I: M Mtehe (Ed) oeedgs of the st A ACM Syos o Theoy of Cotg STOC 9 9 78 ACM (9) [] C Gety S He: Ieetg Gety s fy-hoooh eyto shee I: K G teso (Ed)

Ades Cytoogy EUOCYT (NCS 3) 9 8 Sge-Veg () [3] C Gety S He N St: Fy hoooh eyto wth oyog oehed I: D othe T ohsso (Eds) Ades Cytoogy EUOCYT (NCS 737) 5 8 Sge- Veg () [] C Gety S He N St: Hoooh eto of the AES t I: Sf-N Cett (Eds) CYTO (NCS 77) 85 87 Sge-Veg () [5] C Gety A Sh B Wtes: Hoooh eyto fo eg wth eos: oety-se sytoty-fste ttte-sed I: Cett A Gy (Eds) CYTO 3 t I (NCS 8) 75 9 Sge-Veg (3) [] N Howge-Gh: Aote tege oo dsos I: H Se (Ed) CC (NCS ) 5 Sge-Veg () [7] O ege O ttes eg wth eos do e odes d ytoghy ACM 5(): 9 ey eso STOC 5 [8] est Ade d M Detoos: O dt s d y hooohss Fodtos of See Cotto () 9 8 (978) [9] N St F Vetee: Fy hoooh eyto wth etey s ey d hetet ses I: Q Ngye D othe (Eds) -Key Cytoghy-KC (NCS 5) 3 Sge-Veg () [] D Stehѐ Stefed: Fste fy hoooh eyto I: M Ae (ed) Ades Cytoogy ASIACYT (NCS 77) 377 39 Sge-Veg () [] M D C Gety S He V Vtth: Fy hoooh eyto oe the teges I: H Get (Ed) Ades Cytoogy EUOCYT (NCS ) 3 Sge-Veg ()

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