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Drawback: n communication bits (vs. logn+1 w/o privacy). Main research goal: minimize communication complexity. 2. User masks i with additional random indices. Drawback: gives a lot of information about i. 3. Enable anonymous access to database. Addresses a different concern: hides identity of user, not the fact that xi is retrieved. Fact: PIR as described so far requires W(n) communication bits. J!" Ps" P'PP f ! "" "   ^ f"x"  g 1      _Two Approaches (   Information-Theoretic PIR [CGKS95,Amb97,...] Replicate database among k servers. Unconditional privacy against t servers. Default: t=1$$(    QModel for I.T. PIR(    $%Information-Theoretic PIR for Dummies&&(%  #Computational PIR for Dummies ,Tool: homomorphic encryption Protocol: F-  cBounds for Computational PIR     ~Bounds for I.T. PIR    Upper bounds: O(log n / loglog n) servers, polylog(n) [BF90,BFKR91,CGKS95] 2 servers, O(n1/3); k servers, O(n1/k) [CGKS95] k servers, O(n1/(2k-1)) [Amb97,Ito99, IK99, BI01,WY05] t-private, O(nt/(2k-1)) [BI01,WY05] k servers, O(ncloglogk /(klogk)) [BIKR02]. Lower bounds: log n +1 (no privacy) 2 servers, ~5log n ; k servers, ck log n [Man98,WdW04] Better for restricted 2-server protocols [CGKS95, GKST02, BFG02, KdW03, WdW04]FF%F-$FFF$FF           f           *&                         aWhy Information-Theoretic PIR?*(((   Cons: Requires multiple servers Privacy against limited collusions Worse asymptotic complexity (with const. k): O(nc) vs. polylog(n) f Lu             bLocally Decodable Codes [KT00]"(   1Requirements: High fault-tolerance Local decoding*$$&From I.T. PIR to LDC"(   JConverse relation also holds. Binary LDC @ PIR with one answer bit per server Best known LDC are obtained from PIR protocols. const. q: m=exp(nc loglogq / qlogq) P%PP*& &     VP  =  A Question about MPC>Beaver, Micali, Rogaway, 1990 B., Feigenbaum, Kilian, R., 1990?Z?P  Question Reformulated Connecting MPC and LDC   .cPIR and the Crypto World"(     0From PIR to OT [DMO00]0(   Rabin s OT Sender holds a secret b. Following interaction with Receiver: w/prob , Receiver outputs b w/prob , Receiver outputs ? and cannot learn b Sender cannot tell which is the case Enough to consider  honest-but-curious parties [GMW87]  Z>ZuZ8ZZ 33%3333%0    ,I:From PIR to OT [DMO00]0(   Analysis Privacy of PIR Sender can t distinguish between two cases Sublinear comm. in Case 2, Receiver cannot learn b Privacy amplified using XOR lemma (cf. [Haitner]) Interesting corollary: Homomorphic symmetric encryption PKE  q3' ,#(       f       >E U 1From PIR to CRHF [IKO05]"(   Def. {hq} is a CRHF if: hq shrinks its input Given qGen(1n), hard to find distinct x,x s.t. hq(x)=hq(x ) CRHF from 1-round PIR: i [n]; q Query(i) hq(x)=Answer(x,q) Analysis: Sublinear communication shrinks input Collision resistance: hq(x)=hq(x ) xi=x i i diff(x,x ) Gives noticeable advantage in guessing i from q Interesting corollary: HE CRHF ZSZZ$ZZ Z?Z^ZZ ZZf   ffff                 ff':(  6  VPIR as a Building Block*Private storage [OS98] Sublinear-communication secure computation 1-out-of-n Oblivious Transfer (SPIR) [GIKM98,NP99,& ] Keyword search [CGN99,FIPR05] Statistical queries [CIKRRW01] Approximate distance [FIMNSW01, IW04] Communication-preserving secure function evaluation [NN01] BZZZ+%  " " "4~      =Time Complexity of PIR (Focus so far: communication complexity Obstacle: time complexity Server/s must spend at least linear time. Workarounds: Preprocessing [BIM00] Amortization [BIM00, IKOS04]  CZ+ZZ3ZZZ 8   Protocols    nHigh level structure of all known protocols User maps i into a point zFm User secret-shares z between servers using some t-threshold LSSS over F Server j responds with a linear function of x determined by its share of z. Two types of protocols: Polynomial-based [BF90,BFKR91,CGKS95,& ,WY05] LSSS = Shamir Scale well with k,t Replication-based [IK99,BI01,BIKR02] LSSS = CNF Do not scale well with k,t - involve (k choose t) replication overhead However, dominate over polynomial-based up to (k choose t) factor [CDI05] Best known protocols for constant k ,PPP.P#P&PPPPP,f"   f    $ PEJPolynomial-Based Protocols Step 1: Arithmetization Fix a degree parameter d (will be determined by k) Goal: Communication = O(n1/d) User maps i[n] into a weight-d vector z of length m=O(n1/d). 1 11100& .0 2 11010& 0 n 00& 0111 Servers view x as a degree-d m-variate polynomial P(Z1,& ,Zm)= x1Z1Z2Z3 + x2Z1Z2Z4 + & + xnZm-2Zm-1Zm Privately retrieving i-th bit of x privately evaluating P on z.24>,4"*" "  "+2ffffffffffffff$vS  y @,Basic Protocol: t=1 GGoal: user learns P(z) without revealing z. Step 2: Secret Sharing of z|H3Basic Protocol: General t GGoal: user learns P(z) without revealing z. Step 2: Secret Sharing of z|H3Improved Variant [WY05]   GGoal: user learns P(z) without revealing z. Step 2: Secret Sharing of zdH3P*Breaking the O(n1/(2k-1)) Barrier [BIKR02]8+( ( ( * Arithmetization As before, except that now F=GF(2) Fix a degree parameter d (will be determined by k) Goal: Communication = O(n1/d) User maps i[n] into a weight-d vector z of length m=O(n1/d). 1 11100& .0 2 11010& 0 n 00& 0111 Servers view x as a degree-d m-variate polynomial P(Z1,& ,Zm)= x1Z1Z2Z3 + x2Z1Z2Z4 + & + xnZm-2Zm-1Zm Privately retrieving i-th bit of x privately evaluating P on z.2#4>,"4"*" "  "+2ffffffffffffff$du  y @,TEffect of Degree Reduction"$     U4Degree Reduction Using Partial Information "54 4     V  W Back to PIR " (       XInitial Protocol "(     PUser picks random y1,& , yk s.t. y1+& + yk= z, and sends to Sj all y s except yj. Servers define an mk-variate degree-d polynomial Q(Y1,& ,Yk)= P(Y1+& +Yk) . Each Sj computes degree-(d/k) poly. Qj , such that Q(y)= Q1(y)+& +Qk(y). Sj sends a description of Qj to User. User computes SQj(y)=xi .)                    x                                                              Y "$    V" M $ Sj missing at most d/k variables. , &    #  Z "$     [  \  ]^_` For each block b, define: V(b) = set of servers j[k] not occurring in b d(b) = # of * s in b. Ex. k=3, b = Z1Z2Y3,1Z4Y5,1 (*,*,1,*,1) V(b) = {2,3} d(b) = 3 Want: V(b) large, d(b) small. ^8( *    a &A block set B is said to be spanning if it spans Z1Z2& Zd. For any spanning B, can write: where Qb is a deg-d(b) polynomial known to all servers in V(b). Retrieve each Qb(z) from servers in V(b) using 2d PIRs For every k, d=d(k) and spanning set B=B(k), 46      % 7    )    H    !    >    b Necessary condition for spanning: Blocks in B cover all kd monomials in Z1Z2& Zd. Observation: The following conditions are sufficient. (I) Above covering condition (II) Closure under intersection: Any nonempty intersection of b1,b2B is in B. In fact, (II) may be relaxed to: (II ) Any intersection of b1,b2B is spanned by B. v "4,      63 &=L8            c "Naive strategy: C =  optimal covering block set (e.g., all b with |V(b)| k , d(b) d ), B = closure of C under intersections. Problem: Intersection can make things worse. (1,1,1,1,2,*,*,*,*) (1,1,1,1,*,3,*,*,*) (1,1,1,1,2,3,*,*,*) N1V     d  B = intersections of blocks b such that |V(b)|=2, d(b)=4, e.g., (1,1,1,*,*,*,*). Case 1: (1,1,1,*,*,*,*) Case 2: (1,1,1,*,*,*,*) (*,*,1,1,1,*,*) (*,*,*,2,2,2,*) (1,1,1,1,1,*,*) (1,1,1,2,2,2,*) j(     e  Let l,k <k be parameters and d(l+1)k-(l-1)k +(l-2). Then the following B is spanning: All b such that d(b) 1 and |V(b)|>0 All b such that: (1) |V(b)| k (2) Each server index in b occurs there more than l times (3) d(b) l|V(b)| Y         Open Problems  Better upper bounds Known: O(ncloglogk /(klogk)) What is the true limit of our technique? 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E eb, B   EF"Q&UVWԔ? ~~]%%it  t-p^t %;)^-;^0w258c9:G;a<G<<#=|0>>\AvBHD4 E Gx _H7vI$JiHKG_K%`T%`T?S C1REPGwOIMKKaMvIOFiPvCQ@R'=(=)L=+==,j=^-=u.J>/>/e?/@G/RA.AG.OB-B@E`@`@`@0*`T%0*`T%&R;Lock  l     ",$D 0`   C 0AHH01518_[1]    0?   na+b ( B   EF"Q&UVWԔ? ~~]%%it  t-p^t %;)^-;^0w258c9:G;a<G<<#=|0>>\AvBHD4 E Gx _H7vI$JiHKG_K%`T%`T?S C1REPGwOIMKKaMvIOFiPvCQ@R'=(=)L=+==,j=^-=u.J>/>/e?/@G/RA.AG.OB-B@E`@`@`@0*`T%0*`T%&R;Lockl P+f   ;Pf ,$D  0   <?` 0f  Q=     <?P+ 8   <̣? = ,$  0 nn1/22    <Ш?\ V ,$  0 nn1/22    0?,$ 0 Qi  % <?: ]Z ,$  0 ZX=   & <$@,$ 0 & User sends E(ei) E(0) E(0) E(1) E(0) (=c1 c2 c3 c4) Server replies with E(Xei) c2c3 c1 c2c3 c1c2 c4 User recovers ith column of X(#( (8 F fffffffffffffffff          f  f f  f  f  f  f f  f  f f  f  f  f  f f  f  f  f  f f          J :+  hh ( < P` oJ,$ 0 F PIR with ~ O(n1/2) communication $ # H  0޽h ? ̙33_2W2___PPT1072+P/D[0' = @B D0' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* +%(D' =%(D@' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*  %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*% %(DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*  %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*& %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*& 5%(D' =%(DM' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*& 5R%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*& RX%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*& Xb%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*& bi%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*& il%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*& l%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*( %(++0+  ? ++0+ ? ++0+ ? ++0+ ? ++0+ ? ++0+% ? ++0+& ? ++0+( ? + h P  (   x  c $?  ? F  0A` ,$ 0  servers comm. assumption [CG97] 2 O(ne) one-way function [KO97] 1 O(ne) QRA / [CMS99] 1 polylog(n) Q-hiding & DCRA [Lipmaa] [KO00] 1 n-o(n) trapdoor permutation J 2@7DFB @@@@@BFDB@@@@@B@DB@@@@@BFD;D BFFB@@@@@@@FDbO8& U   0LB  {homomorphic encryption"     <XI 6  *H  0޽h ? 33̙33 y ___PPT10Y +&D ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 9%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 9s%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* s%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* G%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* GJ%(+8+0+  +7#     7 (   x  c $T     c $Ӗ0<$ 0    l m   m ,$D 0ZB   s *D ZB   s *D     0Ֆ|  ZCGKSg     <xٖl  YAMBg     0}  XIKg     0$ P p  XBIg     0, 0  ZBIKRg  `2   0  < `2   0 `2   0P `2   0   `2   0 0 `    <؇y   W efficient      <m   \ clean     <  \ dirty     <X  Y inefficient      0  X ]  XWYg  `2   0p @ p H  0޽h ? 33̙3380___PPT10+0OPD' = @B Dw' = @BA?%,( < +O%,( < +D^ ' =%(%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L|%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 0j%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* j%(+8+0+  + f  (   x  c $,   x  c $     0D pP,$ 0 NPros: Neat question Unconditional privacy Better  real-life efficiency Allows very short queries or very short answers (+apps [DIO98,BIM99] ) Closely related to a natural coding problem [KT00] 3333r,    H  0޽h ? 33̙33___PPT10+PQFDO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+  +t4 g 00M @(   ~  s *    - B  <$ 0    s *h| Yx((2   X  00 PXB  0DXB  0DPPXB  0DXB  0DXB   0DXB   0DPP   s *@  Yy((2   XB   0D XB   0D XB  0D XB  0D@@XB  0DXB  0DXB  0DXB  0D@@XB  0DXB  0D z      ,$D 0   BA J???0@ JZ   s * Z  B s * `   VfAGreen marble     VfAGreen marble     VfAGreen marble@   VfAGreen marblez Z   Z,$D  0   <`Z Ui$   Z    s *N 0 !  0xb " HGHEI 0 xb # HGHI 0`xb $ HGXHI 0X % 0@ 4 s * ,$D  0 4Question: how large should m(n) be in a k-query LDC?5   l 8  0A 3? ?C0a 3 : @ H 68 M G Mr 6 2 TA ? ?S   <  6M 6X D <f8`,$D  0 rk=2: 2Q(n) k=3: 2O(n^ 0.5) W(n2) *:  K 0 `  , 2 L <) @ D ,$ 0 v.Recover from dm faults& , ff ff M <. @  ,$ 0 t,& with +e probability, ffffH  0޽h ??`   "    #   % $  33̙33=5___PPT10+-@D' = @B D' = @BA?%,( < +O%,( < +DE' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*- %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*- #%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(DX' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*- #2%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*M %(D2' =%(D' =%(D' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*4 %(D' =-6B%slide(fromBottom)*<3<*4 D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D %(D' =-6B%slide(fromBottom)*<3<*D ++0+-  ++0+4  ++0+D  ++0+L  ++0+M  +  PH@6 0 (   x  c $`?    * 0Q   , 2p + s *S0p,$D  0 bk-server PIR with a-bit queries and b-bit answersd2(2 d - s *[ff ,$D  0 Pk-query LDC of length 2a over S={0,1}bj)(2  . B4^    <$ 0   1 s * } ,$D  0< 0 0d,$D 0 y[q]=Answer(x,q)P $ H  0޽h ? 33̙33___PPT10+uD' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+ %(D' =-s6Bwipe(left)*<3<*+ D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*1 %(D' =-s6Bwipe(left)*<3<*1 D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*- %(D' =-s6Bwipe(left)*<3<*- DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0 %(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*. %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*. P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*. P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*. %(++0++  ++0+-  ++0+.  ++0+0  +    ~ (  x  c $I `     c $# <$ 0   8z m P6   Pm 6 ,$@ 0  s *L m 6  bjCan k computationally unbounded players compute an arbitrary f with communication = poly(input-length)? k   T   8  < P7 ROpen question:(0*gf3z s f   s f ,$D 0x  s *ؘ s 0 f  & or with work = poly(formula-size) and constant rounds [BB89,& ]DA  $ 8   s *쫙 l s f  3Can this be done using a constant number of rounds? 43 8F DM v   MD v[   0p D  ABen-Or, Goldwasser, Wigderson, 1988 Chaum, Crpeau, Damgrd, 1988BZBb   8   s *dʙ 0 6  k3 players can compute any function f of their inputs with total work = poly(circuit-size) x^? I   8   <̙Pr 7 j&Information-theoretic MPC is feasible!('0*'g`B  0D) M vH  0޽h ? 33  ___PPT10 ./]+7ZD0 ' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+ +7  :$2$0""#(  ~  s *P     <H&  6"0 c  s */  (|Is the communication complexity of MPC strongly correlated with the computational complexity of the function being computed?}0 2c gcg c g,c  <2 *,  8$0*c  <Ԕ`V ,$@ 0 >*0*g2  <DԔ | ,$D 0 >*0*g  <t  ,$ 0 b efficiently computable functions&!0*!c    < g: ,$ 0 M All functions$0*cz E <   E<,$@ 0Z   s *v   <DE-? ]= communication-efficient MPC$0*cZ   s *  <B < ` = no communication-efficient MPC$!0*!c^z $   $ ,$D  0B  B^CgDEdF.&(.%8&> Y.^YPN(M0Z8V@RHFD>HgJL`"A:7()&&(@        `S"  p x   <BOCNDEFB(('9:< -(,)/.=3?0?*C*G*G0I3J6O;L<F=:6:6#<.HNE99630)     @             `S" Q A *  B8C1DEPF*$$68*&$")0/ 1-* % !@      `S" 5 ^ m B  BSC=DEdF.G<N7S7S-S@2,   ''47<:)=59A=;<G<@        `S" ] H  =BSC=DEdF.G<N7S7S-S@2,   ''47<:)=59A=;<G<@        `S" . k B  B<C3DEdF.    $$-4<60*'!#!#03 / )$! @        `S" $ ` B  BSC=DEdF.G<N7S7S-S@2,   ''47<:)=59A=;<G<@        `S" j  B  B^CgDEdF.&(.%8&> Y.^YPN(M0Z8V@RHFD>HgJL`"A:7()&&(@        `S" [  B B B^CgDEdF.&(.%8&> Y.^YPN(M0Z8V@RHFD>HgJL`"A:7()&&(@        `S"   % "  B8CFDELF&3( 34!8$6'4)5 6FA3@      `S"  9 2  B3C8DEXF*7"0*.3%/*$-4757877@       `S"   V "  B3C>DELF&8'3.,30**)00$ /< >8@      `S" ] "  B3C>DELF&8'3.,30**)00$ /< >8@      `S"  6   <BOCNDEFB(('9:< -(,)/.=3?0?*C*G*G0I3J6O;L<F=:6:6#<.HNE99630)     @             `S"  !V   <BOCNDEFB(('9:< -(,)/.=3?0?*C*G*G0I3J6O;L<F=:6:6#<.HNE99630)     @             `S" % es   <BOCNDEFB(('9:< -(,)/.=3?0?*C*G*G0I3J6O;L<F=:6:6#<.HNE99630)     @             `S" F  "   B8CFDELF&3( 34!8$6'4)5 6FA3@      `S" P " ! B8CFDELF&3( 34!8$6'4)5 6FA3@      `S"  $ " " B8CFDELF&3( 34!8$6'4)5 6FA3@      `S" S U H  0޽h ? 33h`___PPT10@..e+?D4' = @B D' = @BA?%,( < +O%,( < +D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*D' =1:B solid*a3>Bfill.type<*D' =1:B true*]3>Bfill.on<*D!' =4@BBBB%(D' =,54*3>Bfillcolor=@BPB<*D' =1:B solid*a3>Bfill.type<*D' =1:B true*]3>Bfill.on<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+ ++0+ ++0+ ++0+  +  @!(    xA  fire1Fire Safety*<$D 0 ~  s *)     bA* nuts-walnuts-inshellf  hf)xA* nuts-walnuts-inshell/j5 \  c 4Awalnut    p WG?MPCTimes New Romang   p WGg?LDCTimes New Romanr   p WG?PIRTimes New Roman J R   s *j0 AR  @ s *j0 R   s * BR @ s *   < /'  D[KT00]"0 cz s    a  ,$D  0  09    P1990"%Zf  0<  #  1995 F%Z$Zff8HD  03 `   Q2000 "%Zf`B  0D>s s ZB  s *D> s  ZB  s *D> s ZB  s *D>cs c   0P  e ,$ 0 The three problems are  essentially equivalent up to considerable deterioration of parameters*0/0/ 8  <PZ&)  D[IK04]"0 cH  0޽h ? 33  ___PPT10w ./]+08D ' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(Du' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+ +A    -( b (  ( x ( c $    ( 08   , 2  ( s *< E  hcPIR*0*g l p  "( p ,$D  0`B (  0Do p `B ( B 0Dop  (  6B   cCRHF*&0*c  (  s *O   `OT&0*c l @ J )(  @J,$D  0@ p J !( p J`B ( B 0Do@ @V  (  s *hSp 9 @ pSecure Computation&0*c  (  s *Tp J nNI UH Commitment&0*c `B ( B 0Do  ` @ @   (( @  `B &( B 0DjJ  V  '(  6X\C"?@   VKAc kl  &  -( &  ,$D  0@ &   ( &   (  s *`&  ~Homomorphic Encryption&0*c   `B ( B 0Do@ $@  +(  0de 04 sTrapdoor Permutation*&0*c lB ,(  <Do.H ( 0޽h ? 33̙33  ___PPT10 +Q=D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-( %(D' =-o6Bwipe(up)*<3<*-( D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"( %(D' =-o6Bwipe(up)*<3<*"( D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)( %(D' =-o6Bwipe(up)*<3<*)( +E K0 .A0 e(  0 x 0 c $0e    0 0ph   , 2 0 0kS M * 0 <Pj    0 6n` ph ,$@ 0 SPIR l @ p T  0   T ,$@ 0  0  <\s@ p  XSender   0  <w@   ZReceiver    0  <n Z T  Sb33 l 8 `  .0  l ` ,$D 0  0  <H    Sx   0  <lz8   Si ZB 0 B s *D  ` ZB 0  s *D ` l   /0  ,$D 0ZB 0 B s *Dh   0  <   fxi, l H 0  00  p,$D  0ZB 0  s *DH0  0  <   Si l H0  10 p,$D   0ZB 0  s *DH0  0  <  bxiJ33  0 6t,$D  0 Sb33  @ p T  0 #  @pT ,$D 0  0  <@ p  XSender  !0  <@   ZReceiver   "0  <̤ Z T  Sb33  l   A0  ,$D  0 0  <p ` h  SPIR  #0  B   Sx  $0  Bl8   Si `B %0 B 0D ` `B &0  0D ` `B '0 B 0Dh   (0  B  fxi, l H 0 30  ,$D  0ZB )0  s *DH0 *0  < U Vi  l H0 40 ,$D  0ZB +0  s *DH0  ,0  <   bxi J33  -0 6HĜ,$D 0 S?  50 6Ԕ 0 ,$@ 0 60 6Ԕ P ,$@ 0H 0 0޽h ? 33̙33''___PPT10a'+|aD&' = @B DX&' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0 %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*50 %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*60 %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0 %(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0 %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.0 %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/0 %(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*00 %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*00 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*00 Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*10 %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*10 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*10 DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0 %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*A0 %(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*30 %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*30 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*30 Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*40 %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*40 D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*40 DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-0 %(++0+0  ++0+0  ++0+-0  +&- K0 '/2l (  l x l c $4    l 0(   , 2 l 0S M *8 0  1l  0 l  6h` ph  SPIR N @ p T  l   T  l  <@ p  XSender   l  <@   ZReceiver    l  <H Z T  Sb33 N 8 `   l   l `   l  <@    Sx   l  <, 8   Si ZB l B s *D  ` ZB l  s *D ` N   l   ZB l B s *Dh   l  <   fxi, N H 0  l   pZB l  s *DH0  l  <  Si N H0  l  pZB l  s *DH0  l  <<  bxiJ33  l  6Tt Sb33 T @ p T  l # @ pT  l  < @ p  XSender  l  <$@   ZReceiver   l  <( Z T  Sb33 N   l    l  6, ` h  SPIR   l  <'   Sx  !l  <48   Si ZB "l B s *D ` ZB #l  s *D ` ZB $l B s *Dh   %l  <L/  fxi, N H 0 &l   ZB 'l  s *DH0 (l  << U Vi  N H0 )l  ZB *l  s *DH0  +l  <@   bxi J33  ,l  6G S? f -l  6Ԕ0 f .l  6ԔP  2l <J  <$ 0  H l 0޽h ? 33̙33___PPT10g+ D' = @B D' = @BA?%,( < +O%,( < +D ' =%(%(D' =%(D7' =4@BB@BB%()?)?D' =.S7 BBBBBeM -3.33333E-6 -2.22222E-6 L -3.33333E-6 -0.39444 *3>*B ppt_xB ppt_y=@0BBAApBBBI<*1l D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2l %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2l E%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2l Ez%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2l z%(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2l %(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2l %(+8+0+2l  + K0 xp8 (  8 x 8 c $4    8 0@b   , 2 8 <\` <$ 0  H 8 0޽h ? 33̙33'___PPT10+tRsD' = @B Df' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 .%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 .l%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 l%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =%(DM' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 #%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 #S%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 Sk%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8 ku%(+8+0+8  +y   0(   x  c $D   x  c $   H  0޽h ? ̙33y___PPT10Y+D=' = @B +  %  ~(     3 rgֳgֳ ?     3 rxgֳgֳ ?@<$ 0  ~l     s,$D  0   60  ] Single user      6h `Multiple users    6"   ZAdaptive     6r  L   ^ Non-adaptive      6   :f   6   9?   6 9?   60  9?H  0޽h ? ̙33___PPT10+i:DA' = @B D' = @BA?%,( < +O%,( < +D ' =%(%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* '%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (C%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* Cn%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* o|%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+  +  `T(  Tr T S ע    H T 0޽h ? ̙33___PPT10i.l@v+D=' = @B +v  % @ L*(  L L 3 rܢgֳgֳ ?<$ 0  H L 0޽h ? ̙33ph___PPT10H++D' = @B D' = @BA?%,( < +O%,( < +D,' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L,J%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*LJ%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L'%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L'5%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L5I%(D' =%(DM' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*LJp%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Lp{%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L{%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L 1%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L37%(+8+0+L +  pXP(  Xr X S `    X S P0p<$ 0  H X 0޽h ? ̙33y___PPT10Y.qv +FD' = @B D' = @BA?%,( < +O%,( < +Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*XL%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*XLk%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Xk%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X;%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X;%(+8+0+X +-  7/%\(  \x \ c $    \ c $<$ 0   l  O  %\`8  ,$D 0 \ 6  O  , \ B 9  ^Fm$  ZB  \ s *D?2 zT r   \#  = `2  \ 0   \ 0| r  Fz+a$T r  \#  E `2 \ 0  \ 0 r  dz+2a$ T r  \#  7| `2 \ 0  \ 0$r  dz+3a$ T r  \#  = `2 \ 0  \ 0)r  dz+4a$ }T r   \# R O_ Z2 \ s *  \ 0x.r  Oz   \ BX2e _  Rt =1 J !\ 0\6pp p,$ 0 rt=1: Pick random  direction aFm zj = z+ja goes to Sj0K250K2 >$  "\ 0E  3$ 0nf___PPT9H@ "Step 3: Sj responds with P(zj) User can extrapolate P(z) from P(z1),& ,p(zk) if k>d. Define deg-d univariate poly Q(W)=P(z+Wa) Q(0)=P(z) can be extrapolated from d+1 distinct values Q(j)=P(zj) Query length m=O(n1/d), answer length 1 Using k servers, O(n1/k-1) communication 5l(*        f ?          /TH \ 0޽h ? ̙33?7___PPT10.qv +D-DC' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\,H%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%\%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"\%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"\T%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"\T~%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"\~%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"\%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"\%(++0+\ ++0+!\ ++0+"\ +  "` M(  `x ` c $R    ` c $G<$ 0  8 ` 04 $ 0`X___PPT9:2 < Step 3: Sj responds with P(zj) User can extrapolate P(z) from P(z1),& ,P(zk) if k>dt. Define deg-dt univariate poly Q(W)=P(z+Wa1+W2a2+& +Wtat) P(z)=Q(0) can be extrapolated from dt+1 distinct values q(j) O(n1/d) communication using k=dt+1 servers. n 6u-       f  #f   '     5/r ` 0pp ,$ 0 FGeneral t: zj = z+ja1+j2a2+& +jtat H 0K20K20K2 , /l " s  "`s"  ,$D 0 ` 6" s  , ` B,  ^Fm$   ` 6pD  Oz  ` BC4DEdF,AA4$Y|oq@@Cy,}c\ @        0$ `2 ` 0 Z2 ` s *`@pZ2 ` s *(XZ2 ` s *@0pZ2  ` s *H@xZ2 !` s *@0p`H ` 0޽h ? ̙33  ___PPT10 .qv + ̚D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`,H%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(++0+` ++0+` ++0+` +     d (  dx d c $|   x d c $P   d 0`  h`X___PPT9:2 Step 3: Sj responds with P(zj) along with all m partial derivatives of P evaluated at zj User can extrapolate P(z) if k>dt/2. Define deg-dt univariate poly Q(W)=P(z+Wa1+W2a2+& +Wtat) P(z)=Q(0) can be extrapolated from 2k>dt distinct values Q(j),Q (j) Complexity: O(m) communication both ways Same communication using half as many servers! tY%|)" 0  7ffff #f     0    9 # Kp d 0,pp  FGeneral t: zj = z+ja1+j2a2+& +jtat z 0K20K20K2 , H d 0޽h ? ̙33___PPT10i.qv +D=' = @B +_  ~` (     6/fP 0 ,R  s *xH R  s *|` 0` X  0xX ` r  S 3`P`     s *40  o 0    `   c $A C??P8 Ce  09h`  k = 2 k = 3 k = 4 k = 5 k = 60K2   H  0޽h ? 33̙33y___PPT10Y+D=' = @B +  p0(  px p c $`   x p c $P0p  H p 0޽h ? ̙33___PPT10i.qv +D=' = @B +      P (   x  c $\0Tt   F A"    q@ 2   >.C:SENGvJ:SQo `T `T:S`T `T:S:S `T:S:SA"s z D2  B  >.^C:SENGvJ:SQo `T `T:S`T `T:S:S `T:S:Sn (  @   ^B  6Do) a!  <@c degree d, m variablesd       0dk !  nsize n,    z T    T ,$ 0M   <oT  degree d/c, m variables     fB   6Do  F   0$z   size O(n1/c)      H  0޽h ?/    ̙33y___PPT10Y+D=' = @B +~  %% Q(     <4V >     B      <<@P h [BKL95,BI01] $        z  M   M ,$D 0   s *  1 Each entry of y is known to all but one server.B2  2  ? N  Mg     Mg    0l $M  lS1:$$      0@   ZUser&$      <У M{ uQ QB      < eg  Uy$      0D $s lS2:$$      0l$ zSk:$$       <иM UQ$      <Ծ ^,$$   N `     j`    0¦M  `  Q(y)N   T Pd   # NlB  <D> d lB  <D> P 4 lB  <D>\ @ 7 F  Mg    V p    0\ʦ $M  lS1:$$      0Ϧ@   ZUser&$      <Ӧ Mb uP PB      <ڦ Xg  Uz$      0Pަ $s lS2:$$      0l$ zSk:$$       <M UP$      <X ^,$$   N `     j`    0M  `  P(z)N   T Pd  ! # NlB " <D> d lB # <D> P 4 lB $ <D>\ @  % s *  vz is hidden from servers"    H  0޽h ? ̙33e]___PPT10=+iD!' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +G  ,, VV d,(   FF u    u N u    u `   0  i    0 D  @   Z   s * _  Z   s * : `   0  Z   s * Z   s *  ` `   0 { ; Z   s * V  Z   s * 1 `   0 Z   s * Z   s *` Z   s *{ ; Z   s *V  Z   s *1  Z   s *  Z   s *     < p  WQ =$        <$ a ;  S+       <$ X #x  S+       0 e  @   Z   s *  `   0     <u   S1 S2 S3d      Z   s *` `   0{ ; Z   s *V  `   01  ` !  0  Z "  s *  Z #  s *` Z $  s *{ ; Z %  s *V  Z &  s *1  Z '  s *  Z (  s *   )  <&a   S+     *  <|*X x  S+     +  <.  k=3,d=6L   z  _ 0 ,   _ ,$@ 0 -  <t5 _ 0 \S1*    Z .  s *  ` Z /  s * : Z 0  s *  Z 1  s * z  o0 2   o ,$D 0 3  <(; o0 \S3*    Z 4  s * : Z 5  s * a ! Z 6  s * Z 7  s *  `  P0 8 # O ,$D 0 9  <tAP0 \S1*    Z :  s *i Z ;  s *{ ; Z <  s *X  Z =  s *  J /0 > # / ,$D 0 ?  <HH/0 \S2*    Z @  s *` Z A  s *a ! Z B  s *{ ; Z C  s *:  Z D  s *  J 0 E #  ,$D 0 F  <N0 \S3*    Z G  s *` Z H  s *a ! Z I  s *{ ; Z J  s *:  Z K  s *   L 08Y j,$  0 LQ(y)=Q1(y)+Q2(y)+Q3(y) degQj d/k = 2n'    &       Hz  c M   C ,$D  0fB N B 6D #fB O  6DfB P  6D ; fB Q  6D fB R B 6D S  <o imc dQ12     T  <tD`Z dQ32     U  <y`Z dQ22     V s * q,$D 0H  0޽h ? ̙33c[___PPT10;+yD' = @B D' = @BA?%,( < +O%,( < +D~' =%(D&' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*, %(D' =-o6Bdissolve*<3<*, Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*V %(D' =-o6Bdissolve*<3<*V D' =%(D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*2 %(D' =-o6Bdissolve*<3<*2 D' =%(D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*8 %(D' =-o6Bdissolve*<3<*8 D' =%(D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*> %(D' =-o6Bdissolve*<3<*> D' =%(D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*E %(D' =-o6Bdissolve*<3<*E D' =%(DF' =%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*M %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*M D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*M DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L %(+8+0+L  +  91@"" (     <P<V6 >     Bp  @   n  s *̊ j o  D Each entry of y is known to all but one server. O(n1/k) comm. bitsE fff f f f E    0< E  jS18      0$ ZUser&     <(I) 0 ` Q Q&      <Xe! Uy$     0s  jS28      0į4` xSk8       <@E!' UQ$     <` 9m* Z(   L .   # K    0нN  .  Q(y)N   T Pd   # NlB  <D> d lB  <D> P 4 lB  <D>\ @   0Pĩ)H jS18      0ɩR< ZUser&     <Tͩp W b P P&      <ѩ Uz$     0թ)@|H jS28      0ک)#H xSk8       <ߩy` UP$     <ha Z(   L 5   # eH   0M  5  P(z)N   T Pd   # NlB  <D> d lB  <D> P 4 lB  <D>\ @  ! s *0   h`X___PPT9:2@ %z is hidden from servers n comm. bitsb&!f f % f0           " <  Let z=y1+& + yk , where the yj are otherwise random Q(Y1,& ,Yk)= P(Y1+& +Yk) hM         |                     H  0޽h ? ̙33y___PPT10Y+D=' = @B +9  ` E(   x  c $  @p     < 0p  r  < o`p,$D  0 O(m) = O(n1/d)    Pr  <'o p ,$D  0  O(n1/k) v    H  0޽h ? ̙33, $ ___PPT10 .+2 Dl ' = @B D' ' = @BA?%,( < +O%,( < +D' =%(DS' =%(D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(DS' =%(D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* +p+0+  ++0+  +b  ME (   x  c $8: @p     <:     6A ` lA Closer Look "$      <$Fg J'  >   e z  P     P ,$D 0   0L    :FUseful parameters: d=k-1 query length O(n1/(k-1)) d/k =0 answer length 1 d=2k-1 query length O(n1/(2k-1)) d/k =1 answer length O(n1/(2k-1))dl:        >        "  1N P    P N      r   68sp   @      0O@  zBest previous binary PIR2   N BT P    BT P r   6D}pT P  @      0<B pP  YBest previous PIR   V  0Ѓd `  deg Qj d/k2        6z H`   H`,$D 0   0( ` j 0      0TH@  j 0   H  0޽h ? ̙33___PPT10.+;Dm' = @B D(' = @BA?%,( < +O%,( < +D#' =%(D' =%(Ds' =4@BB BB%())))?D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+J>   11TT 0(   x  c $  @p     6T ` 'Boosting the Integer Truncation Effect "('$'      0  ,$ 0 DfIdea: apply multiple  partial degree reduction steps. Generalized degree reduction: Assign each monomial to the k servers V which jointly miss the least number of variables. k %   z u    p ,$D 0N u    u `   0  i    0ķ D  @   Z   s * _  Z   s * : `   0  Z   s * Z   s *  ` `   0 { ; Z   s * V  Z   s * 1 `   0 Z   s * Z   s *` Z   s *{ ; Z   s *V  Z   s *1  Z   s *  Z   s *     < Ep  WQ =$        <d a ;  S+       <ƪ X #x  S+       0Pʪ e  @   Z   s *  `   0     <,Ϊu   S1 S2 S3d      Z   s *` ` !  0{ ; Z "  s *V  ` #  01  ` $  0  Z %  s *  Z &  s *` Z '  s *{ ; Z (  s *V  Z )  s *1  Z *  s *  Z +  s *   ,  <٪a   S+     -  <ݪX x  S+    $ .  <|ܪ  k=3,d=6, k =2f   +  T l / # " Tl,$D 0T 0  c $X99?T l 1  6v w- ]&G% 2  6@ i C=&G 3  6,'   C=&G 4  6 l ?'" 5  6 # ll ?|" 6  6T l ?|" 7  6v  ?)" 8  6 U  ?(" 9  6 K  ?)" :  6d   ?(" ;  6@ i Ak$ <  6 Fi AV$ =  6!Q=   AV$ >  6   AQ$ ?  6t$l N  AQ$ @  6X.    Ay$ A  62J   Ay$. z  p B  p ,$D 0 C  <p6  tS1S2@     Z D  s *[ p 0 Z E  s *[   F  <9  tS2S3@     Z G  s *l , Z H  s *l p,0  I  <PD  tS1S2@     Z J  s *L/   K  <pK  tS1S2@     Z L  s *M  Z M  s *MK  Z N  s *M  rT (  O # (  P  <S  tS1S3@     Z Q  s *-  Z R  s *-(  Z S  s *-  Z T  s *K  H  0޽h ? ̙33  ___PPT10 +GDk ' = @B D& ' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 9%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 9%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*B %(D' =-o6Bdissolve*<3<*B D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/ %(+8+0+  +  YQ (     6e ` @   OF r    r    <ir ,$ 0 replication degree size k d n k d n =O(n d /d) k d n =O(n d  /d ) & & & '            '       <n S reduction       <0 S reduction       <ē`Z S reduction    (  <ܚs' ,$D 0 T 1 d/k n d/k /d +ffffffff fffffffff                          |z  J    J,$D 0  @    P ,$D 0  T     @,$D 0   < T ,$D 0 t k d n k d n N      M     <   hThe missing operator: "      <`T @ V4 #vars m m =O(md/d )     L                <ϫ   T conversion    $   <0ԫ*J VAdditional cost: re-distribute new point y *,) ,  H  0޽h ? ̙33___PPT10+UPD' = @B DF' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+  +X   w(   -z        ,$D 0;N  `     ` Z2   s *fPJ   < A (  cQueries,   `2   0  z`2   0 V P   0H>A `  dAnswers     `   <4   2 O(n4/21) communicationr    $        < Iwreplication degree #vars size 3 7 O(n1/7) n x3(   &    U    6 ` @     6 `  Example: k=3 8 $$$     z    ,$D 0   <t S reduction       <<p 9 2 4 O(n1/7) O(n4/7):'  0        z b   b,$D 0   <d T conversion    x   <`b , 2 3 O(n4/21) O(n4/7)-   0        z p   p,$D 0   <) S reduction       <,Pp #3 1 1 O(n4/21) O(n4/21)4    0      &  H  0޽h ? ̙33  ___PPT10 +*D ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +    (     6@ p "In Search of the Missing Operator "#"$"      0I0 0*,$ 0 `dMust have m =W(md/d ). Question: For which d <d can get m =O(md/d )? Possible when d |d . Open otherwise. Positive result Better PIR Simplest open case: d=3, d =2, m =O(m3/2)En   CX    (    :    7  2  6+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* E%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* EZ%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* Z%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+  +B ia (    6@ p k A Workaround " $     6  0x|,$ 0  Can t solve the polynomial conversion problem in general. & but easy to solve given the promise weight(y)=const. Stronger degree reduction:Vb     c $A G??F8 G$ 0  c $A H?? hQ 8 H$ 0X2  0P "  0`  &,$ 0 F Main technical lemma: good parameters for strong degree reduction.GG G  H  0޽h ? ̙33y___PPT10Y+D=' = @B +-K0 <4@ (      68 Ppp tAn Abstract Framework "$       0P­,$ 0 j  L = linear space of polynomials in Yj,h , j[k], h[d], spanned by the kd monomials Let Zh=Sj Yj,h . Block = poly in L expressible as a product of Y s and Z s. `.R&!%$                ?        j  `   c $A I?? '. IF @@        0C"? <    0C"?x  2    0C"?t . `    0C"?R l     0C"?H  `    0@@  >TY1,1 Y1,2 Y1,3 Y1,4 Y1,5 Y2,1 Y2,2 Y2,3 Y2,4 Y2,5 Y3,1 Y3,2 Y3,3 Y3,4 Y3,5 U               T    F          0C"? 8 6   0C"?92 0   0C"?/2 0   0C"?+2     0C"?2     0   >TY1,1 Y1,2 Y1,3 Y1,4 Y1,5 Y2,1 Y2,2 Y2,3 Y2,4 Y2,5 Y3,1 Y3,2 Y3,3 Y3,4 Y3,5 U               T       < Or 0U XZY3,1Y3,2Y1,3Y2,4 Y1,5 (3,3,1,2,1) . -        <4`   DZ1Z2Y3,1Z4Y5,1 (*,*,1,*,1)@#  #  H   0޽h ? ̙33y___PPT10Y+D=' = @B + K0 `(  x  c $P@     6$D Ppp wCost Measures for Blocks "$    H  0޽h ? ̙33___PPT10i.N+D=' = @B + K0 (  x  c $VP@     6E Ppp j Back to PIR " $       c $A J??r8 J$ 0`  c $A K?? x KH  0޽h ? ̙33___PPT10i.N+D=' = @B + K0 $(  $x $ c $jP@    $ 6 Ppp eSpanning vs. Covering$     $ c $A L?? X 8 L$ 0H $ 0޽h ? ̙33___PPT10i.N+D=' = @B +_ K0 ^V,(  ,x , c $|P@    , 6lF Ppp jFinding Good Spanning Sets$    ^B , 6Do@  @ H , 0޽h ? ̙33___PPT10i.N+D=' = @B +R K0 QI4(  4x 4 c $P@    4 6` Ppp Example: k =3, d =7@ $$$$$    ` 4 c $A M?? p  M^B 4 6Do p ^B 4 6Do p H 4 0޽h ? ̙33___PPT10i.N+D=' = @B + K0 <N(  <x < c $P@    < 6죰 Ppp `The General Case$    ` < c $A N?? +  N` < c $A O??e   OH < 0޽h ? ̙33___PPT10i.N+D=' = @B +t  tP(  tr t S `0    t S 0<$ 0  H t 0޽h ? ̙33\T___PPT104+&D' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t2%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t2[%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t[~%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t~%(D' =%(DM' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(+8+0+t +\ 0  6(   d  c $09     s *0 ?      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