Practical multi-candidate election system

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Submitted on 11 Feb 2011

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Practical multi-candidate election system

Olivier Baudron, Pierre-Alain Fouque, David Pointcheval, Jacques Stern, Guillaume Poupard

To cite this version:

Olivier Baudron, Pierre-Alain Fouque, David Pointcheval, Jacques Stern, Guillaume Poupard. Prac- tical multi-candidate election system. Proceedings of the Twentieth Annual ACM Symposium on Principles of Distributed Computing : PODC 2001, 2001, Newport, Rhode Island, United States.

pp.274-283, �10.1145/383962.384044�. �inria-00565273�

Practical Multi-Candidate Election System

Olivier Baudron and Pierre-Alain Fouque

École Normale Supérieure Laboratoire d’informatique

45, rue d’Ulm F-75230 Paris Cedex 05

David Pointcheval and Jacques Stern

École Normale Supérieure Laboratoire d’informatique

45, rue d’Ulm F-75230 Paris Cedex 05

Guillaume Poupard

SGDN/DCSSI DCSSI Crypto Lab 18, rue du Docteur Zamenhof

F-92131 Issy-les-moulineaux

[email protected]

ABSTRACT

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1. INTRODUCTION 1.1 Motivation

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Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.

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1.2 Related Work

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1.3 Achievements

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1.4 Outline of the paper

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2. ELECTION ORGANIZATION 2.1 Architecture

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2.3 Players

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H&!+%&1ÿPNN!4+8;-C80-'4!+]N!/.44ó

"Q'4*S#] "—™¹'G-(aò#6!6D:9#6^,6!J—8˜<;'E>

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˜>‘ÿàC,9+ -( BCC -(DEF > m

" D

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ë

!( " "8 (z>

pk

2

pk

3

pk

1

Regional Level

pk1,1 pk1,2 pk1,3 pk2,1 pk2,2 pk2,3 pk3,1 pk3,2 pk3,3

Local Level

National Level pk

â4‚

~

yzi eJ

lLK)i

~

{!uG‚NM{x/‚ƒhBubhPO1x/×

e

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e

k|

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šN>4ÿZ "%6VU9ìWXQR/ó_ "N(ý`HZY…ìD 7

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õ

^]X

"N(

ýŠ!(a!(! _A8>

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[ Y

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3.3.2 Proof that an encrypted message lies in a given set of messages.

†…!\!C=%&!`: "H‘8*"%B3!@:%!%B-(\

%&,-z!„ "„O91%6!3/&ZìTÐ'ó

´

ÑÒÒ/ÒrÑó

² Ó

µ@ "4.48!6&‘/*,=*•6&!4ƒ *'w>

3.3.3 Proof of equality of plaintexts.

†…!\!C=%&!`: "H‘8*"%B3!@:%!%B-(\

%&,O!‘A.„=&%/(5 "L+/^,6->

4. THE VOTING SCHEME

?oI!$wH.)(3B)9 "%!J%,<6&

! "HG6a!)=&% %!,L%N(`K

šN>5–F%B8=,HG.).4:a "%!!$!6<`+) D

7-(5!+ " )6!/a3BA.„a$+C !/

H#-'-!Hq-w!H!(*#,/w!>

4.1 Setup

†:9!(&J7˜CDE6DEƒDµ`7.4!!J/!((!')

!Z "!"µY!>I?o:!"[/'\%,HF/C

68A=,H!#,=r q8q%6!3!+]=-(aC;!

8L.48K:-(&%B-(&,+8;!/K68A=,>J†K)6!)

ƒ/.4@! 'C/O w®<µBðƒ$L--<6D

8o=,Hwµ-ð.1•!*-468"-(IµBð.+ab•!*,/

68>

p4ƒq!L% "-'=a%-H/ 76&!o=@%6!3!!#

8‘/.4@36!3B (5#1%!63!#]C=®Oƒ4¦ "%!H

3ced+f+!z68A="#]C=zµBðÑ•µBð . H!-(*!GgId^f9,/

68A=9BGcd^f !B68A=J!‘]C=Bµ-ðÑµ-ð.ѕµBð.eabN>

†:5(*3,=`±9!*N6! )3BCL1C)-(:.*"æ

,a'5-R±>TM!5¦ "%!H„.I/…C,`æ ì

™ihkjl#mTn2oqp'H$%B/.q<™9 " "/(=*'1-±>

4.2 The Voting Phase

„!(CO‘CrQ T‘,N‘.48+%6!3!<]=&NµBðBяµBð.2яµBð.+ab

.4$.4!F+ƒ„‘ó d+f /!((!'>

m

‘6$3!

a#•/.4!"./=®

˜>

m

(/.4(&wƒ T36!+3B ($„š‘%!63!<]=&

µBðBяµBð .яµBð .+ab O4[!>

o

­#L˜ è

_wv+xy:xz

.+ab

è

_wvexy:xz

.

è2_wvexy:xz

­##™ è _

v+xy:x

n

.+ab

è _ vexy:x

n

. è

_wvexy:x

n

­#qð è _

v+xy:x{

.+ab

è _ vexy:x{

. è _ vexy:x{

­#4± è _ v+xy:x|

.+ab

è _vexy:x|

. è _ vexy:x|

ù&6 à1S.eab } ñ U .eab~a

ñ  ñ è _ v+xy:x{

.  ñ è _ v+xy:x{

â4‚

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yzi e€

l4‚9yG•

e

x/‚•u‚Jh-{!ijRhiOqx/×

e

QhwÖ{-{-ywx/×zh-i‚ƒx f |

™N>­#!Y/Cb%6!37]=,H4I!C=%&5INIæßõ

.4*WO„C "4!() !‘%!C¦&ƒrtuBH

r s !(Cr

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Ì>

m

< G%&,&1NN,=$C;!r-w!

+%!¦N‘=&%‘+ "$>„!4q(!

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9%,G46-(*3B/6#!q%N6‘‘6! ,

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4.3 Computation of the result

?AJ!O6!39.‘%,O! "%!6& 9-„ 8=,>

MG82/HN.$(&3Bqq36!*3B ("z!1,/r6!8

.4! @/@3BL/(@3,=w64•1¦ "%!#6!a!

.3@8+O!#,G68A=ES.+ab,>

MG6&"™"!/.4+6! 536!`3B (I.4!J!ð d+f 6!C

/Z(\-(K.‘8aZ!9'.4Z/. 3!6&)Z=Y(K!

4/.4>

!C.‘!8"! "*C!".48V8"8;D

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!J( "@!L%N6q6! "qF8q/.P-(aL

4!/(3N=9 "$2 "%a/>

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-/IZ8Q=¬$!7%&,•@-(…-6&@)6@-(

‘ ""2 "%:C/>a!rHG!",/‘6&!8LS.+ab

"%!64!q%N(&6!„zq456 "!‘™+9Ì&>

¥#C¦&/H&!C=@$q+ 93!!1aL&!(@(C=%&

WO<=&%2=2 ߃G<;2G%&N(6!Cz.4! L%B-(&

7!8"%6!3!5]=,>Z@(=&%Z/V3B5 "/(Z3N=

6&(4!++(C=%&a%&,11%6!38=58;-3!>

!4,/- ‘4%!63!!/(136!"3B' (!

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!C=.‘8q!q%N(&6!„r!1 ",„*!8‘/.453!6!

3B' (r>!"(&ŽC,L6&!8S.+ab.4! :(&%B-(7!

6&!o=…S

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6 "a˜><?o6 "*™NH,=J.‘4!4%&N(6!C!8,

6&!>

ü

,/zpq6&!8 8 b~a

ñ 

b~a

ñ r s  b~a

ñ r u

ü

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ü

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ñ  ñ è _ v+xn x{

.  ñ è _ vex n x{

ü

,/zpq6& g } ñ U$.eab~a

ñ  ñ è _ v+xy:x{

.  ñ

è2_

vexyˆx{

ü

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.  ñ è _ vex|‡x{

ùN6! à%S. }

b~a

ñ U .+ab~a ñ 

b~a

ñ è _ vexyˆx{

.

‰ (&=&%

}

b~a

ñ U .eab~a ñ 

b~a

ñ è _ vexy:x{

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n1!Bp168 8P.eab~a

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.eab~a

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b~a

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b~a

ñ è _ z‡xy:x{

n1!zpq6a™ }

b~a

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b~a

ñ è _ nxy:x{

n1!zpq6†c }

b~a

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b~a

ñ

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n1!zpq6*± }

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!!46!8A=KZ "%6"!*%&N(6!C4

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6 G+ ",‘a6! "`˜>

4.4 Frauds in the computation

"! 'C„63(N!L%'N("Kc"N+.=:

(&236!J%!63!q8;-/'!w16!8>F–C5

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8;!/9.4 KN&/(: Š!L!"¦N9„(& *

8=7!6!(/(@9,/< =,>+MG!8=,Hr!J#6-' ,/(

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N6! 93BzB! 'C/(N!O"¡žšqJ6&F(6!B¤

-(K!" N6! 93BJ1)9±)!K5 "%!6&

(@q÷5¡+

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o ¤.4!Ca4/=a%B• "/(I)!9Wq>

pq(!(&-8=,H&8O,="C44(/(*3N="9a6!8A=

#6!8A=563Dž6!8‘ *=*3B+(& ((ƒ à!# 8=56!,

ƒ64 "%6 'w>

5. ENHANCEMENTS OF THE SCHEME 5.1 Time Stamp server

+2=&2 #q)%&+ "J>

m

!H

F6F.4+-/‘F()3B•GV.4-!F3B ]

,+6,Fƒ<!„ "%6 )B„ 8=,><•H

6.4!-/58=\/(Y3Bƒ -(Y.4!@

!/$3B5•6/(3B/6+G "qH- 96!2%&'#-'

8‘q„(r>„!/^,6,=,H,L "42 "%L2=2 Ÿ

6/(!6q-1‘-/#3B@%Bƒ "/(53B•J>

5.2 Anonymity

?A¬6-$%-%BDE3-/(…7%&,H$72a9

<]N!/.4wH/23,=9‘6&!8o=,><‹„6F8F3!/8=

(6#)$!(),w!43BN(&=5‘A.4>

m

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2‘r!q1%-'%-,‘-(* "2„6!58=!

zN6! 93BO! _‘/^,6(r>Œa/H-3B6!+O!

( O• *'/H "J9/"-=&$/73B)%Bƒ "/(H

'1 *,=*!)%&,>

pqw!/^,6!!H'8z */=13B<,‘%&CG!,=& "8A=

¡!B¤oDEH.4!#3B)3!49/((&6!3DE!&><pq

6!6!H3!-(@-6&—êF')"N!,1,Gƒ+%&N(&D

!6! `:!N=& "8A=,H!.4!%,a(&63!DE6!)‘

8;!/>

ü

C*6!*(:3!(V!6I ! "a-'%&,

‡A!DE "q•‰a—™›ž>„S#!1/58!C‘6!1!#S#] "D

ù& !Z—™¹,H9š™'9½.4! ½73-(Tß!Y(&8c"6!8o=

F "%!6&*(J8! "Hq)S#] "DAò#6!6&D

9#6!^,6-'—™¹NH<˜<;'<7.4! $3!/(I7)(&8c"6!8o=

"%!6AR/Dž@,$ "N(65* "%B89N6! 93B/>#S#!

*=9 *]L-F!4C‘(,„!(!("N=) "%6 '-

6 "%½K!Yn#ùp %3! 2!7-½!

m

!DEn1(6!#%3 @>

p1Qq-/N"%'a!q(&,o=,H-/CI6q/$!C.

%-8<B *'CqF!(J%6!3]=&>GwH.4L!„!%

B‘6&!o=r¡ ¤CH,!#8;-‘)/H6!!+q3!-(

!6"%&,>5Sq„6&Hr!"6!8A=7-/J@%&

J, 4'‘ "2‘!1.48aL6!/>F/'ƒ/H/ 5

%B‘A6!29!"8;-/(K%6!3!"]=,Hr.4! Y *=`3B"6/(

1)%6-(&,=& @>

ù&„.4$.‘,O$%&,G63Bo.q6!C

!(a!J6&!H.„9!'L"6)*(&236/(@3!(@D

-'6„.4CL/^,6!8<N!„6&!8O!CO#!%J!

6!C+aCJ(`%!6-(,= @>9•HO8 ""6&!D

8+.6(`])a%K56!LaL *,=I%6-(&,=& "q

@ "$-aH81.„6Q(3B$ "%B3!>

pq#¦ "%!O6CL4(&236/(#8;-/'L&%!6!(N=& "

4@a%!%B-(8¦7˜/œN>Ì>

5.3 Receipt-Free and Incoercible Properties

!*!%95‡o%Džƒ!‰.‘J;!29%N(K3N=

‹„!)!(Jw6!!2 L—™'ž><!G696!G1!43B,

-'a%,=/=R6! ,aA.„Dž.‘=…@ " )6!/'

3Bo.$F68G-($/C+/>z¥#¦N/H'!r%D

ƒa`%&,‘3-/(\\ "8¦NDE!J -!„-)3B

%%B/(@3,=@ù]"!(0+QZ—šN˜G.4L=aDž.‘=@CD

a " 96!…Q :68aZ!

6 "/(r>

pq#Q$F6&!#6!)Î,§CÊ!Àž©Èª§"§CÊB¿ ÃC¾-¨ÀÄʅ—ÌHw¹/

6! I-'$!JC$/+/.P!9%!!C¦N$$&D

=&%(a!(5‘ !^N6#64!,3!LC>

†K@;2@(&;-I3Bb5¿§ÂC§CÀ¾-¨žËÅ ¿§§Cʁ§CÇÇY-(½À•Ê!Ë

ÂÄ'§C¿ÂÀžÈÀ•ªÀ•¨Ã>Y!rHF.„@(&3Ba6%&%B‘3-/(ZY!

¦&2!"1aC) " )6!/': -!!F3BCA.K/C

6!C#-(@-(&%B!(!,‘%!o=,H/(:¿©Ê-ÎÄÏ9À“’'§C¿C>

5.3.1 Definitions

} e Ö e

‚ƒ€Gxvâwi eNe

u e k/k

!6&)-'!5"/!!"%'N(

,=%q%'N!)+,,1<1H-@8<!

.‘,‘>

ã

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e

iւQÝO‚••‚ƒx

f_!69-'I9/!)3B*,/(K

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C@ƒ5!5JI/!9>†K*!8=K•,6!

…%&DEQaR3B/6!78aK2!

^N68 ",)81.6 "5!)5)!7 " 96!

3BCA.a!#,#!(5+1(&6!J!+%&,ž>

5.3.2 Physical Assumptions

wJ%&'&(&4%Džƒ!.1/*6!1+%!,=&/-6! "%&D

:]"" "%BDž2 ,L(&N@¡6! `+ * ( ¤1

!1!+ !( ·( 6/(@(6&!!L!"%!!L/!!

3BL/(@3N=@,=&3BN(&=,>4ƒHB!!]N4"!L *,LD

68A=1WO<`)C=%&2=&2 .„*"6&-J,=3BN(N=

!‘/'a,=&!J3B6„qHN".4*!1!%*

)/>

m

/./Hw+6 "% *=I3B),a2!

"$/H!.„$.4w8!8=a6! "$)1 D

96!C!!r3BA.„@,=6q!(a9 -( "[/>

5.3.3 Previous Work

46 "%F " )6!/) -!&-F

3BY6!/(KK*,J%-%BC

m

8-(Zù]Y—8˜/›'E>`‹„6&

# " )6!/"(5r!8‘ "8¦NDE!F9%&'&(&1%D

Qq8<C=*!r>

?o-(&/(rH<Y!J !Q^,6HO!=K6 "H9.„a(&H<-'

I!C=%&… ! "•”½ " "%!a3!65V/.4

-(& àDE=&%rH&.4Ca.„L(+3,=Z‡V–

—

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( Š¡=aœ¤

˜t™ ìßÐZ” ™ a´

씡2˜$šœ¤CÑ#” ™ aë1ì”F¡E™5šœ¤CÑÒÒÒrÑw” ™ a²

ì”F¡8µXš2œ¤Ó,Ò

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6!J-LC9!L%&!,,L1

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%&'‘-'q @ "+-13BaDE=&%/(*a!+.

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5.3.4 Independent Randomizers

S#6&+%%B+ "9`$!`‡A8QDoC )3

!N=& "[‰—™$;ž>­#1]@¦N!,8A=*9 -( D

[I8H.4865 "N(&Q=&:!I,,>ߋ„6*

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%/H#!!/¬%&`,,a9`D

!+=&%r>º‹„6&@!@%&,9 )6!2@R3B[D

]&/.4("K (L@/]`@ƒ *Y3B6L!

!.b -( à6!$6/(5!$=&%/(">!CD

•Hr!=#6!F+, CF[DE]N!/.4/(%&,oH

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6!), a%&,q-"\!=&%&/(Y "@

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6q " -! @H+ -(& "[q-1 -(& "[+!

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)%&,/^,6!8A=7„%!,¦N+/:

3B$6!/(>„ù&)˜œ&>¹9•qC!/w(C >

6. PRACTICAL COMPLEXITY

ü

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C=

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)6!/'a "%!¦&8A=5z!$ ! "#1!/'qa$[L

WOa "N(&6!6!HVaN6! 93B)+/-((*-(Z\!

N6! 93B1>+M!6 "H!J "%6 '7 "%C¦&D

8A=a$•Ÿ)¡8µIíKøý:ø¤1 "N(6!'# 96!8%!/Iƒ$3B7

! '7!(8;!/r>

?A+%& /N%%!/rH'.F */=#C,qøžYøì;œNH!øS9øì;œ

!(RøýKørìU˜œ™Ì&>M!J`˜œa/-((LKC "Hr!

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/53B16!("1(6!1q "%!6& !-24!(" " "=

2/>½†Ka/b@!*!I278! "7R!

N6! 93B‘G>

7. CONCLUSION

L2=&2 ß.4C+6- ,z!‘ ! "G^N68 ",

-("/Q3!$L-'‘,="N6 )3BFw-("/!((!'

6!("3B16!/(>FŒa'C/H&42=2 Ÿ/*3B#(!%&/(+/

8•L-.48I(&ŽC,4q<6&!>

p1G=&% %!)%&,ƒ$-/J3B7%& "[/(@5%&'&(&

@c"N1!) ! ">

8. ACKNOWLEDGEMENTS

I6&!*I '•61!I!N=& "6!)ƒ"ƒ

81!%ƒ6!w!(a(C /(a " ",>

9.

—˜C4 >-‹„-w>Q¡!§C¿CÀ

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