Yehuda Lindell Bar-Ilan University
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1 Wintr Shool on Sur Computtion n iiny Br-Iln Unirsity, Isrl 3//2-/2/2 Br Iln Unirsity Dpt. o Computr Sin Yhu Linll Br-Iln Unirsity
2 Br Iln Unirsity Dpt. o Computr Sin Protool or nrl sur to-prty omputtion Constnt numr o rouns Sur or smi-honst rsris Mny pplitions o th mthooloy yon sur omputtion Gnrl sur omputtion Cn us to surly omput ny untionlity Bs on th Booln iruit or omputin th untion Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 2
3 Br Iln Unirsity Dpt. o Computr Sin Grl iruit An nrypt iruit tothr ith pir o ys (, ) or ry input ir so tht in on y on ry ir: It is possil to omput th output (s on th input trmin y th y proi on ry ir) It is not possil to lrn nythin ls Oliious trnsr Snr hs x,x ; rir hs Rir otins x only Snr lrns nothin Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 3
4 Br Iln Unirsity Dpt. o Computr Sin Yo s protool Prty P onstruts rl iruit P sns P 2 th ys ssoit ith its input on its on input irs P sns only th ys so P 2 osn t no ht th tul input is P n P 2 us oliious trnsr so tht or ry on o P 2 s input irs: P 2 otins th orrt y ssoit ith its input P lrns nothin out P 2 s input P 2 omputs th iruit n ris th output, n sns it to P Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 4
5 Br Iln Unirsity Dpt. o Computr Sin Trpoor prmuttion (I,D,F,F - ) I: smpls untion n trpoor t in th mily D(): uniormly smpls lu in th omin o F(,x): omputs (x) F - (t,y): omputs - (y) Hr to inrt rnom y, in (ut not t) nhn trpoor prmuttions Hr to inrt y, n in th rnom oins us to smpl y (usin D) Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 5
6 Hr-or prit B Gin y=(x), n uss B(x) ith proility only nliily rtr thn ½ quilntly, in y=(x), th it B(x) is psuornom Br Iln Unirsity Dpt. o Computr Sin Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 6
7 Snr s input: (z,z ); rir s input Snr s irst mss: Snr hooss (,t) usin smplin lorithm I Snr sns to rir Rir s irst mss: Rir hooss x n omputs y =(x ) Rir hooss rnom y - Rir sns (y,y ) to snr Snr s son mss: Snr omputs (x,x ) y inrtin Snr omputs i = z i B(x i ) Snr sns (, ) to rir Rir outputs z = x Br Iln Unirsity Dpt. o Computr Sin Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 7
8 Br Iln Unirsity Dpt. o Computr Sin S (z,z ) Choos (,t) x = - (y ) = z B(x ) y,y R () Choos x, omput y =(x ) Choos y - x = - (y ) = z B(x ), Output z = B(x ) Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 8
9 Br Iln Unirsity Dpt. o Computr Sin Simultor is in (z,z ); thr is no output SIM nrts (,t) SIM hooss rnom y,y usin D() SIM omputs, s in snr s instrutions Th trnsript is xtly li rl protool xution Choosin x usin D() n omputin y =(x ) is intil to hoosin y usin D() Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 9
10 Simultor is in (,z ) SIM nrts (,t) SIM hooss rnom x,y - usin D() SIM omputs y =(x ) SIM omputs = B(x ) z SIM hooss - t rnom Br Iln Unirsity Dpt. o Computr Sin Th trnsript is inistinuishl rom rl xution By th hr-or proprty o B n th nhnmnt proprty o TDP, B(x - ) is inistinuishl rom rnom Sur Computtion n iiny Br-Iln Unirsity, Isrl 2
11 Br Iln Unirsity Dpt. o Computr Sin For th ntir iruit, ssin rnom lus/ys to h ir (y or, y or ) nrypt h t, so tht in on y or h input ir, n omput th pproprit y on th output ir Sur Computtion n iiny Br-Iln Unirsity, Isrl 2
12 Br Iln Unirsity Dpt. o Computr Sin u u
13 Br Iln Unirsity Dpt. o Computr Sin u u u u u u u u
14 Br Iln Unirsity Dpt. o Computr Sin u u u u u u u u u ( ( ( ( ( u ( u ( u ( u u u u
15 Th tul rl t u ( ( u ( ( u u u ( ( ( ( Gin n n otin only Furthrmor, sin th tl is prmut, th prty hs no i i it otin th or y u u u Br Iln Unirsity Dpt. o Computr Sin
16 I th t is n output t, n to proi th ryption o th output ir Output trnsltion tl Br Iln Unirsity Dpt. o Computr Sin,,, u u u Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 6
17 Br Iln Unirsity Dpt. o Computr Sin Gin Booln iruit Assin rl lus to ll irs Construt rl ts usin th rl lus Cntrl proprty: Gin st o rl lus, on or h input ir, n omput th ntir iruit, n otin rl lus or th output irs Gin trnsltion tl or th output irs, n otin output But, nothin ut th output is lrn!
18 ,,,,,, Br Iln Unirsity Dpt. o Computr Sin ( ( ( ( AND OR ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( AND r Computtion n iiny Iln Unirsity, Isrl 2 x x 2 y y 2 8
19 Br Iln Unirsity Dpt. o Computr Sin Ho os th prty omputin th iruit no hih is th orrt ntry It hs on y on h ir, ut symmtri nryption my rypt orrtly n ith inorrt ys To possiilitis (tully mny ) Us nryption s on PRF ith runnt zros; only orrt ys i runnt lo A it to sinl hih iphrtxt to rypt Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 9
20 Br Iln Unirsity Dpt. o Computr Sin Option : nryption: K (m) = [r, F K (r) (m n )] By psuornomnss o F, proility o otinin n ith n inorrt K is nliil Option 2: For ry ir, hoos rnom sinl it tothr ith th ys u u u Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 u 2
21 Th tul rl t (,) (,) (,) (,) Ant u u u u ( ( ( ( u = Computin th iruit rquirs just to ryptions pr t (rthr thn n r o 5) ( ( ( ( u u u = Br Iln Unirsity Dpt. o Computr Sin = Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 2
22 Br Iln Unirsity Dpt. o Computr Sin N to ormlly pro tht in 4 nryptions o rl t n only 2 ys Nothin is lrn yon on output Atully, in orr to simult th protool, n somthin stronr Nottion: Doul nryption: Orl: (,, ) ( u,, m) ( u ( m Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 22
23 Br Iln Unirsity Dpt. o Computr Sin Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 23
24 Br Iln Unirsity Dpt. o Computr Sin Input: x n y o lnth n P nrts rl iruit G(C) L, L r th ys on ir L Lt,, n th input irs o P n n+,, 2n th input irs o P 2 P sns P 2 th strins x,, n xn P n P 2 run n OTs in prlll P inputs n+i, n+i P 2 inputs y i Gin ll ys, P 2 omputs G(C) n otins C(x,y) P 2 sns rsult to P Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 24
25 ,,,,,, Br Iln Unirsity Dpt. o Computr Sin ( ( ( ( AND OR ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( AND OT Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 25
26 Br Iln Unirsity Dpt. o Computr Sin Prty P s i onsists only o th msss it ris in th oliious trnsrs In th OT-hyri mol, P ris no msss in th oliious trnsrs Simultion: Gnrt n mpty trnsript Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 26
27 Br Iln Unirsity Dpt. o Computr Sin Mor iiult s N to onstrut rl iruit G(C ) tht loos inistinuishl to G(C) Simult i ontins ys to input irs n G(C ) G(C ) tothr ith th ys omputs to (x,y) Simultor osn t no x, so nnot nrt rl rl iruit Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 27
28 Br Iln Unirsity Dpt. o Computr Sin Simultor Gin y n z = (x,y), onstrut rl iruit G (C) tht lys outputs z Do this y hoosin ir ys s usul, ut nryptin th sm output y in ll iphrtxts ( ( u u ( ( ( ( u ( This nsurs tht no mttr th input, th sm non rl lus on th output irs r ri u ( Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 28
29 Simultor (ontinu) Simultion o output trnsltion tls Br Iln Unirsity Dpt. o Computr Sin Lt, th ys on th i th output ir; lt th y nrypt in th prin t I z i =, rit [(,),(, )] I z i =, rit [(, ),(,)] Simultion o input ys phs Input irs ssoit ith P s input: sn ny on o th to ys on th ir Input irs ssoit ith P 2 s input: simult output o OT to ny on o th to ys on th ir Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 29
30 Br Iln Unirsity Dpt. o Computr Sin N to pro tht th simultion is inistinuishl rom th rl First stp moiy simultor s ollos Gin x n y (just or th s o th proo), ll ll ys on th irs s ti or inti ti: y is otin on this ir upon inputs (x,y) inti: y is not otin on ir upon inputs (x,y) Th sinl y to nrypt in h t is th ti on This simultion is intil Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 3
31 Br Iln Unirsity Dpt. o Computr Sin Pron y hyri rumnt Consir rl iruit G L (C) or hih: Th irst L ts r nrt s in th (ltrnti) simultion Th rst o th ts r nrt honstly Clim: G L- (C) is inistinuishl rom G L (C) Proo: Dirn is in L th t Intuition: us inistinuishility o nryptions to sy tht nnot istinuish rl rl t rom on hr sm y is nrypt Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 3
32 Br Iln Unirsity Dpt. o Computr Sin Osrtion L th t Th nryption unr oth ti ys is intil in oth ss Th irn is ht th inti ys nrypt (only th nxt ti y, or lso th inti) Th tripl in th xprimnt r ll nryptions unr inti ys Th prolm Th inti ys in this t my ppr in othr ts s ll Us orls to nrt rst Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 32
33 ,,,,,, Br Iln Unirsity Dpt. o Computr Sin ( ( ( ( AND OR ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( AND Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 33
34 ,,,,,, Br Iln Unirsity Dpt. o Computr Sin ( ( ( ( AND OR ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( AND Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 34
35 ,,,,,, Br Iln Unirsity Dpt. o Computr Sin ( ( ( ( AND OR ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( AND Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 35
36 ,,,,,, Br Iln Unirsity Dpt. o Computr Sin ( ( ( ( AND OR ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( AND Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 36
37 ,,,,,, Br Iln Unirsity Dpt. o Computr Sin ( ( ( ( AND OR ( ( ( ( ( ( ( ( ( ( ( ( Not hn in nrypt y ( ( ( ( ( ( ( ( AND Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 37
38 RAL,,,,,, Br Iln Unirsity Dpt. o Computr Sin SIM ( ( ( ( AND OR ( ( ( ( ( ( ( ( ( ( ( ( SIM ( ( ( ( ( ( ( ( AND Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 38
39 RAL,,,,,, Br Iln Unirsity Dpt. o Computr Sin RAL ( ( ( ( AND OR ( ( ( ( ( ( ( ( ( ( ( ( SIM ( ( ( ( ( ( ( ( AND Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 39
40 Br Iln Unirsity Dpt. o Computr Sin In th simult OR s, th inti y nrypts th y In th rl OR s, th inti y nrypts th y Inistinuishility ollos rom th inistinushility o nryptions unr th inti y Sur Computtion n iiny Br-Iln Unirsity, Isrl 2 4
41 Br Iln Unirsity Dpt. o Computr Sin ollos rom th inistinushility o nryptions unr th inti y h oo ns Ky is not nrypt nyhr (s t) us prior ts r simult h ns Th y ns to us to onstrut th rl AND t or th hyri h solution Th spil oul-nryption
42 , (, ) r ti ys, (, ) r inti ys Cn us orl to nrt th RAL AND t Br Iln Unirsity Dpt. o Computr Sin
43 in h t-rplmnt is nistinuishl, usin hyri rumnt h tht th istriutions r nistinuishl Br Iln Unirsity Dpt. o Computr Sin D
44 Br Iln Unirsity Dpt. o Computr Sin -4 rouns (pnin on OT n i oth or n prty ris output) y oliious trnsrs C symmtri nryptions to nrt iruit n 2 C to omput it (usin th inl it) or iruit o 33, ts: Btn 7 n 4 sons Btn 53 n 362 Kyts (pns on nryption us)
45 Br Iln Unirsity Dpt. o Computr Sin ssum tht th OT is sur or mliious : A orrupt P nnot lrn nythin (it ris no msss in th protool, in th hyri-ot mol) Thus, h priy W n pro ull surity or th s o orrupt P 2 his n usul, ut B rn tht this osn t ompos ith nythin.., onsir P tht uils iruit so tht i P 2 s irst it is, th iruit osn t rypt I P n tt this in th rl orl, priy is lost
46 n omput ny untionlity surly in rsn o smi-honst rsris Br Iln Unirsity Dpt. o Computr Sin rotool is iint nouh or us, or iruits tht r not too lr ommntion: r ull proo
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