OPTIMALLY EFFICIENT MULTI AUTHORITY SECRET BALLOT E-ELECTION SCHEME

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1 OPTIMALLY EFFICIENT MULTI AUTHORITY SECRET BALLOT E-ELECTION SCHEME G. Aja Babu, 2 D. M. Padmavathamma Lectue i Compute Sciece, S.V. Ats College fo Me, Tiupati, Idia 2 Head, Depatmet of Compute Applicatio. S.V.Uivesity, Tiupati, Idia [email protected] ABSTRACT A electoic votig scheme is a set of potocols that allow a collectio of votes to cost thei votes, while eablig a collectio of authoities to collect votes, compute the fial tally, ad commuicate the fial tally that is checked by tallies. This scheme is based o the RSA ad factoig assumptios. We apply the potocols of [CDS 88] to Guillo Quisate s idetificatio potocol [GQ 88] to costat poofs of validity fo ballots.. INTRODUCTION: A electoic scheme is a set of potocols that allow a collectio of votes to cost thei votes, while eablig a collectio of authoities to collect votes, compute the fial tally, ad commuicate the fial tally that is checked by tallies. I cyptogaphic liteatue o votig schemes, thee impotat euiemets ae idetified. Pivacy: A system maitais pivacy if:. Neithe electio authoities o ay oe ca lik ay ballot to the vote who cost it. 2. No votes ca pove that he o she voted i a paticula way. Veifiability: A system is veifiable if all votes ca idepedetly veify that thei votes have bee couted coectly without sacificig pivacy. I additio each vote must be able to veify the fial esults of the tally. Robustess: A system is obust if it esues that all the system ca ecove fom the faulty behavio of ay (easoably sized) locatio of paties. The mai cotibutio of this pape is pesetig a efficiet votig scheme that satisfies uivesal veifiability pivacy ad obustess. 2. OVERVIEW OF THE APPROACH The paties i a votig scheme ae modeled as pobabilistic polyomial time pocess. Two meas of commuicatio ae typically assumed to be available fo these paties. A bulleti boad: The commuicatio model euied fo ou electio scheme is viewed as a public boadcast chael with memoy, which is called a bulleti boad. All the commuicatios though the bulleti boad is public ad ca be ead by ay paty (icludig passive obseves). No paty ca ease ay ifomatio fom the bulleti boad. Pivate chaels:

2 To suppot pivate commuicatio betwee votes ad authoities. Fo this task ay secue public key ecyptio scheme is suitable. The paties of the votig scheme pefom the followig steps to eecute a electio. To cast a vote, each vote costucts a ballot as a ecyptio of the desied vote ad post the ballot to the bulleti boad. At his poit, a poof of validity is also euied that covices all paties that the posted ecyptio cotais a valid vote, without evealig it. The auhoities, how eve, ae able to decypt the ballots (because of the eta ifomatio eceived fom the vote though the pivate chael). I the ed, the fial tally is published togethe with some auiliay ifomatio to eable uivesal veifiability. Moe techically, uivesal veifiability is achieved by euiig the ecyptio fuctio to be suitably homomophic. Cetal to ou esults is the way we achieve a efficiet poof of validity fo ballots. The poof of validity shows t ay iteested paty that a ballot actually epeset a vote e.g., that it eithe epesets a yes o a o, ad othig else. To maitai pivacy fo the votes, the geeal idea is to use some sot of zeo kowledge poof. The poblem is howeve that ZK poofs usually euie a lage umbe of epetitios befoe the desied level of cofidece is achieved. The efficiecy of the whole scheme is iflueces by these poofs. Ou cotibutio ow is two fold. We use a paticula efficiet homomophic ecyptio scheme, based o -th esiduay assumptio. a umbe is a -th esidue modulo if thee eists a α such that α (mod ), moeove, by applyig the esults fom [CDS 94], the poof of validity is simple thee move potocol which is witess idistiguisible ( i fact, witess hidig as well). Istead of ZK poofs. 3. CRYPTOGRAPHIC PRIMITIVES We implemet ou electio based o -th esiduosity assumptio. A umbe is a -th esidue modulo if thee eists a α such that α (mod N). Homomophic Ecyptio with Efficiet Poof of Validity: Iitializatio: Iitialized the paametes of the scheme ae a modulus N, Which is a poduct of two lage pimes, a pime with gcd (, Ф (N)). Also a elemet h Є Z * N - ae available to all paties. The fied umbe h is ot a -th esidue modulo N. Ecyptio: A paticipat ecypts V by choosig α Є R Z N ad computes B α h v. Opeig: A paticipat ca late ope B by evealig v ad α. A veifyig paty the checks whethe B α h v ad accepts v as the ecypted value. Homomophic popety: Ecyptio is homophic i the sese that; if B ad B 2 ae ecyptios of v ad v 2 espectively, the B.B 2 is a ecyptio of (v +v 2 ) mod. Poof of kowledge fo -th esiduosity: Usig theotatios above, we peset poof of kowledge fo -th esiduosity, whee by a pope shows possessio of a α Є Z satisfyig α. 2

3 PROTOCOL [ α ] Pove Veifie W Є R Z * N A w a C Є R Z c w α c Figue-? a c Theoem: : The above potocol is a thee moe public coi poof of kowledge fo -th esiduosity. The poof satisfies special soud ess ad special hoest veifie zeo kowledge. Poof: Special soudess ow holds because fo ay two acceptig covesatios (a,,) ad (a, c, ), c > c, it follows that c c. Sice 0 < c c <, we have that thee eist iteges tems k, such that (c c ) k l. Hece k k k ( c c ). l l 3

4 k which is cotadictio to the -th esiduosity assumptio. Futhe moe, by the esult of [CDS 94], the potocol of fig(2), i a poof of kowledge that a vote kows -th esidue of B (Bh) o. Thus the veifie leas that the h VOTER Joual of Theoetical ad Applied Ifomatio Techology V V- PROTOCOL-2 vote kows α ad v {, -} such that B α h v without obtaiig ay ifomatio about actual value of v. Poof of validity: i ou votig scheme to follow, it will be the case that a vote posts a ecyptio of a value v Є {, -}. To demostate that the ecypted value is ideed is {, -} without evealig it, the vote ad the veifie eecute the followig efficiet poof of validity. VERIFIER α,, d, w 2 Є Z B α. h a (Bh) d α, 2, d 2, w Є R Z B α / h a w a 2 w 2 a 2 2 B d 2. h B, a, a 2 C C Є R Z d 2 c - d d c - d 2 2 w 2. α d 2 w. α d d, d 2,, 2 d + d 2 c a (Bh) d (Figue 2) 2 B d 2. h 4

5 Veifiable secet shaig: To achieve obustess efficietly, o iteactive veifiable secet shaig, efficiet solutio fo out of case is possible ca apply. Ude the -th esidusity assumptio, ou electio scheme satisfies uivesal veifiability, obustess ad pivacy. 4. SECRET BALLOT ELECTION SCHEME We ow peset ou mai esult, a secet ballot electio scheme. The paticipat i the electio scheme ae authoities. A, A 2,., A ad m votes V, V 2,.V m. The scheme woks as follows: Each vote V i pepaes a vote by adomly selectig b i {,-}. The vote fist b ecypts b i by computig B i α i i. h, whee α i Z is chose adomly, ad post B i to the bulleti boad. Subseuetly b i is cosideed as a secet which is to be shaed amog the authoities. The vote also posts poof (B i ). I the ed the aggegate value T b i educed i l module epeset the esult of the electio. Ballot Costuctio ad Vote costig : Each vote V i posts b i {,-}. I the followig.. The vote adomly chooses b i {,-} ad computes B i α i., the vote also computes poof of (B i ) also the vote computes. bi h B i (α i ) bi h, < l < 2. The vote posts B i, poof (B i ), B i, B i2,., B i to the bulleti boad. 3. All paticipats veify which the ballot B i is coectly fomed by checkig poof of (B i ). 4. The vote chooses the shaes (a ij, b ij ) Whee i l b ij j l a ij Seds (a ij, b ij ) to the authoity a j usig a pivate chael. 5. Each authoity checks the eceived shaes (a ij, b ij ) by usig that (a ij ) h b ij B ij. Tallyig : Each authoity A j posts S j, T j ad seds to the bulletei b m S j i l b i α a ij, T j ij i l Each tallie checks the shae (S j, T j ) posted by A j by veifyig that S j. m T h j i l ( B ij ) The fial stage is the tally itself. Let us deote as A {j T j is coect}. The tally is the itepolatio of the polyomial ad may be calculated as T j A T j l A{ j} l l We assume that i the successful electio, the shaes of evey vote have bee accepted by all authoities. i.e., all veificatio by j i 5

6 the authoities i the last step of the ballot costuctio is successful. I case a authoity eceives a false shae, the authoity may post the shae so that ay body ca veify that shae is ot coect ad that it coespods to the posted ecyptio of step (4) i the ballot costuctio. Theoem: 2 Ude the -th esiduosity assumptio, ou electio scheme saties uivesial veifiability, obustess ad pivacy. Poof: To pove uivesal veifiability, fist ote that oly ballots ae cotact o accout of theoem (). Futhe the fial tally is coect, if the step (2) of the tallyig holds fo all authoities. This deals with uivesal veifiability ad obustess. The pivacy popety ca easily pove fom the fact that the secet shaig scheme used ad the poof of validity (potocol) ae ifomatio theoetical scheme. 5. CONCLUSION We have show a vey efficiet scheme fo secue electio based o -th esiduosity assumptio. The scheme satisfies well-kow euiemets pivacy, uivesal veifiability ad obustess. [2]. [CDS-94] R.CRAMER, I. DAMGARD, AND B. SCHOENMAKERS. [3]. Poofs of patial kowledge ad simplified desig of witess hidig potocols. I Advaces I Cyptology CRYPTO 94. Volume 839 of Lectue Notes I Compute Sciece, pages 74 87, BERLIN, 994.Spige Velag. [4]. [CFSY 96] R.CRAMER, M. FRANKLIN, B. SCHOENMAKERS AND M.YUNG. [5]. Multi authoity secet ballot electios with liea wok. I Advaces I Cyptology EUROCRYPT 96, Volume 070 of Lectue Notes I Compute Sciece, pages 72 83, BERLIN, 996. Spige Velag. REFERENCES []. [BEN 87a] J. Bealoh. Cyptogaphic capsules: A disjuctive pimitive fo iteactive potocols. I advaces i cyptology CRYPTO 86, Volume 263 Of Lectue Notes I Compute Sciece, pages , BERLIN, 987. Spige Velag. 6