A PRACTICAL PROXY SIGNATURE SCHEME

Transcription

1 A PRACTICAL PROXY SIGNATURE SCHEME Sattar J Aboud and Sufan Yousef Telecommuncatons Engneerng Research Grou Angla Ruskn Unversty, UK sattar_aboud@yahoo.com ABSTRACT A roxy sgnature scheme s a varaton of the ordnary dgtal sgnature scheme whch enables a roxy sgner to generate sgnatures on behalf of an orgnal sgner. In ths aer, we resent two effcent tyes of roxy sgnature scheme. The frst one s the roxy sgnature for warrant artal delegaton combnes an advantage of two well known warrant artal delegaton schemes. Ths roosed roxy sgnature scheme s based on the dffculty of solvng the dscrete logarthm roblem. The second roosed scheme s based on threshold delegaton the roxy sgner ower to sgn the message s share. We clam that the roosed roxy sgnature schemes meet the securty requrements and more ractcal than the exstng roxy sgnature schemes. KEYWORDS Proxy sgnature scheme, warrant artal delegaton, roxy unrotected scheme, roxy rotected scheme, threshold delegaton. 1 INTRODUCTION The dea of a roxy sgnature scheme was frst resented by Mambo et al. [1] n Ther roxy sgnature scheme allows an orgnal sgner to delegate hs sgnng rght to a roxy sgner to sgn the message on behalf of an orgnal sgner. Later, the verfer, whch knows the ublc keys of orgnal sgner and a roxy sgner can check a valdty of a roxy sgnature ssued by a roxy sgner. Tll now we have the followng tyes of delegatons. In general, there are three dfferent tyes of delegatons: full delegaton, artal delegaton and delegaton by warrant. In a full delegaton roxy sgnature scheme, a roxy sgner uses the same rvate key as an orgnal sgner and creates the roxy sgnature as an orgnal sgner does. The drawback of a full delegaton comes from a dffculty of dstnctve between an orgnal sgner and a roxy sgner. In a artal delegaton roxy sgnature scheme, the orgnal sgner derves the roxy key from hs rvate key and asses t to the roxy sgner n a secure channel. In the roxy sgnature scheme wth delegaton by warrant, an orgnal sgner rovdes the roxy sgner a secal message namely warrant. The warrant certfes that a roxy sgner s legal and contans sgner dentty, delegaton erod and the tyes of a message on whch roxy sgner can sgn. Also, there are two tyes n the second one: rotected and unrotected roxy sgnature schemes. In unrotected roxy sgnature scheme, a roxy sgnature s generated by both the roxy sgner and an orgnal sgner. In ths case, the verfer cannot dstngush the dentty of a sgner. In the rotected roxy sgnature scheme, a roxy sgnature s generated by the roxy sgnature key of an orgnal sgner and also wth a rvate key of a roxy sgner. 296

2 Later, a verfer valdates a roxy sgnature wth the ublc keys of both an orgnal sgner and a roxy sgner. Proxy sgnature scheme s useful n many uses such as e-ayment systems and wreless networks [2, 3, 4, 5]. In ths aer, we ntroduce an effcent tye of delegaton that s warrant artal delegaton. In delegaton by warrant, the orgnal sgner sgns a warrant whch descrbes the relatve rghts and nformaton of the orgnal sgner and roxy sgner such that a sgnature verfer can use the warrant as a art of verfcaton nformaton. Usually, delegaton by warrant ncurs more comutatonal cost than the other two. In ths aer we roose effcent and sold roxy sgnature schemes One s a roxy unrotected tye and the other s a roxy rotected tye. The roosed schemes are arorate to schemes usng a dscrete logarthm assumton. Also, because the roosed roxy rotected scheme needs a lnear combnaton of two shared secrets, so t can be develo to a threshold roxy sgnature schemes wthout dffculty.. 2 PRELIMINARIE In ths secton we resent some of the relmnares requred for roxy based sgnatures and threshold. 2.1 Securty Requrements A secure roxy sgnature scheme must satsfy the followng requrements: 1. Identfablty: Any erson can determne an dentty of a corresondng roxy sgner from a roxy sgnature. 2. unforgeablty: Only the desgnated roxy sgner can create the vald roxy sgnature on behalf of an orgnal sgner. 3. undenablty: Once the roxy sgner generates the vald roxy sgnature on behalf of the orgnal sgner, cannot deny a sgnature creaton aganst anyone else. 4. Verfablty: From a roxy sgnature, the verfer s convnced of an orgnal sgner agreement on the sgned message. 5. Proxy sgner devaton: The roxy sgner cannot generate the vald sgnature not detected as the roxy sgnature. 2.2 Notatons Used Throughout ths aer, we wll use the followng notatons. A : orgnal sgner entty A. B : roxy sgner entty B. V : verfer enttyv. : large rme number. g : generator Z. * h (.) : secure one-way hush functon. Id A : dentty of orgnal sgner Id B : dentty of roxy sgner m w : a warrant. WI (b) : reresents a comutng cost to acheve b bt modular nverson. WH (b) : denotes a comutng cost to fnd hash functon wthb btlong nut 2.3 Defntons Used Ths aer resents two new tyes of delegaton, called delegaton wth warrant and threshold delegaton. Defnton 1. (Partal delegaton wth warrant). In artal delegaton wth warrant, the new secret b s calculated from the secret s of entty A and the warrant and b s rovded to entty B n a 297

3 secure manner. From securty ont vew s must not be calculated from b and the warrant. Partal delegaton wth warrant also combnes a beneft of a artal delegaton and the delegaton by warrant. Thus ths delegaton has rad rocessng and s sutable for lmtng messages to be sgned. Also, snce a roxy for artal delegaton wth warrant can dentfy ts vald tme, the roosed scheme s not requred an extra roxy revocaton rotocol. Now n a grou-orented assocaton t s referred that a roxy sgner strength to sgn documents s shared. The entty A, for examle, has teach the secretary entty B to rely nstead of her accordngly. However, assume a secretary entty B s not follow gven nformaton rovded by entty A. It means that an entty B s not sgn the message whch requres to be answered radly or he sgn what entty A has teach to hold back. Therefore, for securty reasons, t may be an organzaton rules that messages be sgned by roxy sgners rather than one entty. Defnton 2. (threshold delegato In threshold delegaton, n roxy sgners are rovded shares for examle t n are requred to create the roxy sgnature rather than the orgnal sgner but less than t. Ths s enttled ( t, threshold delegaton that s n a ( t, threshold roxy sgnature scheme. t out of n roxy sgners should hel to ssue a roxy sgnature any t 1 corrut roxy sgners cannot forge a sgnature. 3. RELATED WORK Mambo et al. n 1996 [1] develoed a systematc aroach to roxy or delegated sgnatures. Neuman n 1993 [6] ntroduced the scheme for delegaton by warrant, whch was further extended by Km et al. n 1997 [7] to artal delegaton by warrant. Okamoto et al. n1999 [8] rooses a roxy-unrotected sgnature scheme. They analyze the securty of ther scheme by usng the reducton among functons. Y et al, n 2000 [9], roosed roxy multsgnature scheme whch allows a grou of orgnal sgners to delegate ts sgnng ower to a sngle roxy sgner. Hwang et al, n 2001 [10], ntroduced a new roxy mult-sgnature scheme. In 2001, Lee et al. [11] roosed a roxyrotected sgnature scheme. Unfortunately, ts securty roof s ncorrect by Wang 2004 [5]. In 2002, Ln et al. [12] resent a mult roxy sgnature scheme for artal delegaton wth cheater dentfcaton, they clam that ther scheme requred less comutatonal overhead comare wth other schemes. Zhou et al. n 2005 [13] roose two effcent roxy rotected sgnature schemes. They clam that ther schemes are more effcent than other schemes. Unfortunately, ther schemes nsecure. In 2006, Qn Wang and Zhenfu Cao [14] resent an attack on the aggregatesgnature based roxy sgnature scheme, they gve arguments for artal delegaton wth warrant roxy sgnature schemes. They construct a new roxy sgnature scheme and rove that t s secure aganst exstentally forgery on adatvely chosen-message attacks and adatvely chosen-warrant attacks under the random oracle model. Moreover, Lu et al. n 2007 [15] ont out that Zhou et al. schemes vulnerable to the un-delegated roxy sgnature attack In 2005 Sunder Lal and Amt Awasth [16] ntroduce a new mult-roxy sgnature scheme for artal 298

4 delegaton wth warrant, whch requres less comutatonal overhead n comarson to Ln et al, and also fulfll the requrement of artal delegaton wth warrant smultaneously. In 2008, Suntha and Amberker [17] roosed a roxy sgnature schemes for controlled delegaton. They fnd that the scheme can be used to control delegaton of fnancal ower to a roxy sgner. They use the dgtal sgnature algorthm n ther scheme. Shao n 2009 [18] roose roxy-rotected sgnature scheme. In 2010,, Lu and Huang [19] roosed a varant of threshold roxy sgnature scheme n whch all roxy sgners reman anonymous. The authors clamed ther constructon satsfes unforgeablty, roxy sgner devaton, dentfablty, undenablty and varablty. However, ths scheme does not rovde the roxy sgner devaton and dentfablty requrements. In 2011, Constantn Poescu [20] ntroduced a secure roxy sgnature scheme wth delegaton by warrant, the scheme s based on the dffculty of solvng the dscrete logarthm roblem. The rest of ths aer s organzed as follows. After we descrbed the related work n secton 3. we wll descrbe n secton 4 the Mambo et al. scheme for artal delegaton. In secton 5 the roosed roxy sgnature scheme s descrbed. However, n subsectons 5.1 and 5.2 we wll exlan and suggest an effcent and sold roxy sgnature schemes for warrant artal delegaton. The frst one s the roxy unrotected scheme and the second one s roxy rotected scheme. In subsectons 3.3 and 5.4 we dscuss the erformance analyss and securty analyss of the roosed scheme whch s extensvely arorate to scheme under a dscrete logarthm assumton. Lastly, ths aer s concluded n secton 6. 4 MAMBO et al. SCHEME It s suosed that the orgnal sgner entty A nvtes the roxy sgner entty B to erform sgnng on behalf of hm, and the verfer enttyv verfes the valdty of generated sgnatures. Also, suose that s a large rme number * and g s a generator for Z. Select a random nteger value e as a ublc key where e= g u mod, and u Z Descrton of the Scheme The stes of the scheme are as follows: Generaton: The orgnal snger entty A should do the followng: 1. select an nteger Z 1 2. comute t g mod 1= 3. fnd = u + * t 1 mod 1 b A 4. ass b, t ) to a roxy sgner entty ( 1 B n a secure channel. Sgnng: Entty B should do the followng: b 1. verfes g e t mod. A * 1 2. sgns the message m on behalf of entty A 3. emloys b as a substtute to u A 4. mlements an ordnary sgnng rocess. 5. the generated roxy sgnature on m s m, s, ( m ) t ) ( b 1 Verfcaton: Entty V should do the followng: 1. fnd e = e t mod as the new ublc key A * 1 299

5 2. a verfcaton of roxy sgnature s mlemented by the same verfyng rocess as n an orgnal sgnature scheme. 5 PROPOSED PROXY SIGNATURE SCHEMES we wll exlan and suggest an effcent and sold two roxy sgnature schemes for warrant artal delegaton. The frst one wth two tyes the frst tye s the roxy unrotected scheme and the second tye s roxy rotected scheme. The second scheme s also wth two tyes. The frst tye s a roxy sharng unrotected and the second tye s a roxy rotected sharng. 5.1 Proosed Schemes Usng Dscrete Logarthm Assumton There are two tyes whch are as follows Proxy Unrotected Scheme The stes of the frst tye are as follows: Generaton: The orgnal sgner entty A should do the followng: 1. selects an arbtrary nteger value Z 1 2. fnds t g mod 1=. 3. concatenates t ) ( m w 1 4. comutes = h m, t ) such that an nformaton on a delegaton must be descrbed n the warrant m for examle ts vald erod. 5. fnds b= * u A + mod asses m w, b, t ) to the roxy ( 1 sgner n the secure channel. Sgnng: entty B must do the followng: w 1. checks = h m, t ) b 2. verfes g e t mod (1) A * 1 3. for sgnng message m uses b nstead of u A and mlements an ordnary sgnng rocess. Thus the roxy sgnature on m s m, s ( m ), t, m ) ( b 1 w such that s ) ndcates the b ( m sgnng message m by secret key b. Verfcaton: A verfcaton of a roxy sgnature s erformed by the same checkng rocess as n an orgnal sgnature scheme excet for an addtonal calculaton: 1. comute = h m, t ) ' 2. comute e e t mod =. A * 1 ' 3. A comuted comonent e handled as a new ublc key clearly showng the artcaton of entty A. 4. A scheme has the followng tye smlar to the congruence (1) b h( t) 5. g = e t mod (2) * Proxy Protected Scheme The stes of the second tye are as follows: 1. Generaton and modfcaton: After checkng a valdty of ( m w, b, t 1) where must be created from an orgnal sgner Id A, roxy sgnature IdB and other data on the delegaton, a roxy sgner entty B comutes a substtute roxy, t ) ( 1 b : b = b+ u B * h(, t1) mod 1 (3) 2. Sgnng: To sgnng the message m, entty B uses b as a substtute to u A and erforms an ordnary sgnng rocess. Thus, a roxy sgnature on m s ( m, sb ( m ) t1, ). 300

6 3. Verfcaton: A verfer entty V erforms the same verfyng rocess as n an orgnal sgnature scheme excet for an addtonal calculaton: = h m, t ) e ' = ( e A * eb ) * t1 mod The calculated key e ' s rocessed as the new ublc key clearly showng an artcaton of entty A. 5.2 Proosed Scheme Usng Threshold Proxy Sgnature Assume that the orgnal sgner entty A needs to delegate the grou to sgn a document n such a way that a roxy sgnature can be generated by the set of t roxy sngers from the desgnated grou G of n roxy sgners but a subset wth t 1 roxy sgners cannot. Here, we llustrate the effcent ( t, threshold roxy sgnature scheme by usng a scheme that we roosed n secton 5.1 and Ceredo Schnorr tye threshold dgtal sgnature scheme [21]. For exedency, we suose that G= ( P 1. The ublc arameters are the same as those of secton 4 excet that 1has large rme factor q and selects g z wth the order of q Generaton Random Number Assume that the dealer wth an arbtrary secret D selects the random olynomal where f ( x) = D+ a1 x a t 1x asses u = f () to P rvately for = 1,..., n and transmts d g D a1 at 1 = mod and g,..., g mod Ths rocess s generated by the followng rotocol: 1. Every roxy sgner P selects y arbtrarly and transmts Z q d g y = mod to each roxy sgners. 2. To delver y every P arbtrarly chooses a olynomal f of degree n Z q where f ( 0) = y that s: f ( x= y + a a a 2, 1x+ a,2 x... Z + a, x wth, 1,...,, q and asses f ( ) mod q top n a secure way ( ). P also transmts a value a g, 1 a, t 1,..., g mod. 3. From delvered f ( )( ), P verfes f for every ( ), by g f l t ( ) a a 1,1, d ( g )...( g ) mod 4. Assume L P P) s not notced to be ( dshonest n ste 3. Every P fnds share u = f ) L ( rvately and fnds a a a 1,1 at 1 = = =, t 1 d d, g g,..., g g L L L Proxy Unrotected Sharng Ths scheme contans four hases whch are as follows: 1. Proxy Generaton The orgnal sgner entty A should do the followng: 1. selects arbtrarly Z q. 2. obtans t1 = g mod. 3. fnds ( m w t 1), 4. comutes = h(, t1). 5. calculates b = u A + mod q. 2. Proxy Sharng To share the roxy b n the threshold scheme wth threshold t entty A should do the followng: 1. chooses nteger Zq, = 1,..., b 301

7 b 2. ssues the result B = g, = 1,...,. 3. fnds a roxy share b as follows: ' 2 f ( x) = b+ b1 x+ b2 x +... b t 1x b = f () 3. Proxy Share Dstrbuton The orgnal sgner entty A should do the followng: 1. asses to every roxy sgner ( for = 1,..., a roxy share b n a secure way 2. transmts, t ) ( m w 1 4. Proxy Share Verfcaton To check the roxy share b a roxy sgner should do the followng: 1. comute = h m, t ) t 1 b 2. check f g ( e t ) B mod A 1 = Proxy Sgnature Broadcastng wthout Exosng Shares Assume that m s the document and L G ublsh the roxy sgnature. For exedency, suose L= ( P 1 t). Thus, the rest of the stes of the rotocol are as follows: 1. L mlements the rotocol for generaton random number and obtan the ublc key y a1 a d = ( g mod ), g,..., g mod and the rvate key of P, u such that a 1 u = f ( ) = y+ a Every roxy sgner P do followng: 1. fnds = h d, m ) ( the 2. comutes d = u + b mod q 3. dscloses d 3. Each roxy sgner P checks the followng: t 1 = d a l l g ( d ( g ) = 1 (( e t ) t 1 A 1 = 1 ( g b ) l ) h( d, m ) mod 4. Every P L fnds t satsfyng: for l. t = y + b = f (0) + f (0) mod q by usng Langrage equaton to d. A roxy sgnature s m, t,, t, m ) ( 1 w 5. The valdty of a sgnature m, t,, t, m ) s checked by: ( 1 w d t h(, t1 ) = g ( e t1) mod A h d, m ) ( In addton, by relacng an above roxy sharng wth the followng roxy rotected sharng, we can clearly extend an unrotected threshold roxy sgnature scheme nto a roxy rotected threshold sgnature scheme Proxy Protected Sharng There are fve hases n ths scheme whch are as follows: 1. Grou Key Generaton The G mlement rotocol for generatng random number and get a ublc key of a grou G ug c c g t 1,..., 1 eg ( = g mod ), g mod and rvate key of P, u G, (such that = u, = f ( ) = u + c c 1. G G t 2. Proxy Generaton Entty A should do the followng: 1. selects Z q. 2. fnds t g 1 = mod. 3. concatenates ( m w t 1). 4. fnds = h(, t1) where m w contans orgnal snger d,roxy sgner d. 5. fnds b = u A + mod q. 302

8 3. Proxy Sharng To share the roxy b n the threshold scheme wth threshold t entty A should do the followng: 1. selects nteger b Z, = 1,..., b 2. ssues a value B = g, = 1,..., q. 3. fnds a roxy share b such that: b = f ( ) = b+ b b 4. Proxy share delvery An entty A should do the followng: 1. asses to every P the roxy sgnature b n the secure way. 2. transmts m w, t ). ( 1 5. Verfcaton of the Proxy The verfcaton of ( m w, b, t 1) are as follows: The P comutes another roxy, ) as follows: ( b, t1 b, = b + u, h m G, t ) mod q utlzes b, as an oton to b 5.3 Performance Analyss When we select the dgtal sgnature standard scheme [22] n generatng the roxy sgnature and on checkng, a comutng tme s lesser n a roxy sgnature for warrant artal delegaton than that by the warrant. Thus, a warrant delegaton needs WI(512) comutng tme, whereas a roosed warrant artal delegaton want WI (9512) + 2WH( ) ( WI (512) + 2WH( ) comutng tme wth and wthout addton the value for the roxy unrotected sgnature scheme and that for the roxy rotected sgnature scheme resectvely. Numbers ndcate a comutng cost to acheve modular multlcaton n 512 bts modulus, WI (b) reresents a comutng cost to acheve b bt modular nverson, and WH (b) denotes a comutng cost to calculate the hash functon wth b bt long nut. When we comare the artal delegaton wth the roosed warrant artal delegaton. The roosed scheme requres WH ( )(642+ WH ( m w )) of comutng tme n the roxy generaton ste, 642+ WI (512)(642+ WI(512)) n sgnature generaton 875+WH ( )(876+ WH( m )) n a sgnature verfcaton ste. But, artal delegaton needs 641(642) of comutng tme n a roxy generaton ste, 642+ WI (512)(642+ WI(512)) n a sgnature generaton ste, 875 (906) n a sgnature verfcaton ste, and an extra 1282 of commutng tme for roxy revocaton algorthm. Observe that as n [1], the subsequent congruence can be emloyed as a substtute to a congruence (3) b = b+ ub * eb mod 1and ths requres ( 906+ WH( )) of comutng tme n a sgnature verfcaton ste. However, from a ont of comutatonal beneft, a warrant artal delegaton decrease an amount of comutng tme comare wth a delegaton by warrant, and from the ont of busness, the roosed scheme needs no addtonal roxy revocaton algorthm n a artal delegaton, the roxy sgnature scheme [23] s calculated n the same manner. 5.4 Securty Analyss In ths aer, we wll dscuss the followng attacks: Framng attack: In such attack, a hacker forges the roxy secret key and then creates vald roxy sgnatures such 303

9 that a verfer consders that these roxy sgnatures were sgned by a roxy sgner entty B on behalf of an orgnal sgner entty A. If the roxy sgnature s resented, entty A cannot reudate that snce s an orgnal sgner of a roxy sgner entty B. The result s that entty A and entty B wll be framed. To acheve ths attack, hacker wants to forge entty B roxy key ar x, d ). As ( forward-secure sgnatures are used by roxy sgner t s comutatonally hard to forge a roxy rvate key. Knowng a roxy ublc key d hacker cannot create a roxy rvate key gven as t s hard to factorze. Thus the roosed scheme ressts the above attacks. By ths we can state that ust a desgnated roxy sgner can generate the vald roxy sgnature on behalf of an orgnal sgner. Thus a requrement of unforgeablty, of the secure roxy sgnature s satsfed. Forgery by orgnal sgner: The roxy rvate key s deendent on both a roxy nformaton sent by an orgnal sgner and a rvate key of a roxy sgner. So, an orgnal sgner cannot make a roxy rvate key. Also, cannot obtan a roxy rvate key from a roxy ublc key as t s hard to factorze. Thus an orgnal sgner s unable to sgn lke a roxy sgner. So, forgery by an orgnal sgner s comutatonally mossble. Imersonatng attack: Suose that entty B s not desgnated as the roxy sgner by an orgnal sgner entty A. Whle entty B can create the roxy key ar x, d ) and sgn the message on ( behalf of entty A, a verfer after recevng sgnatures, frst valdates usng the verfcaton formula f a sgnature s from the vald roxy sgner or from the revoked roxy sgner. Durng ths test a verfcaton fals and a verfer consders hm as the revoked sgner. Thus entty B cannot become a roxy sgner unless he s desgnated by an orgnal sgner entty A. 6 CONCLUSION We suggested two new knd of roxy sgnature. The frst one s a warrant artal delegaton, for whch has the comutatonal gan comared wth an orgnal roxy sgnature by warrant and has also the constructon beneft over a roxy sgnature for artal delegaton. The second one s the roxy for threshold delegaton In the ucomng hghly grou orented socety, t s desred to delegate the ower to sgn messages to a grou of n roxy sgner. We llustrated the requrements related to a roosed thoughts and demonstrated that the roxy sgnature scheme for warrant artal delegaton and a ( t, threshold roxy sgnature scheme wthout exosng roxy shares. Furthermore, a roosed scheme could be extended to a elltc curve roxy sgnature scheme easly. 7 REFERENCES 1. Mambo M., Usuda K., and Okamoto E.: Proxy Sgnatures: Delegaton of the ower to sgn Foundatonal, Volume E79-A, Number 9, PP (1996). 2. Chaum D., Fat A., Naor M.: Untraceable Electronc Cash, Proceedng of the Cryto'88, (1990). 3. Oros H., Poescu C.: A Secure and Effcent Off-lne Electronc Payment System for Wreless Networks, Internatonal Journal of 304

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