An Anonymous Multi-Receiver Encryption Based on RSA

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1 Iteratoal Joural of etwork Securty, Vol.15, o.4, PP , July A Aoymous Mult-Recever Ecrypto Based o RSA Le Har 1, Ch-Che Chag 2,3, ad Hsao-Lg Wu 2 (Correspodg author: Ch-Che Chag) epartmet of Computer Scece Electrcal Egeerg & Uversty of Mssour- Kasas Cty 1 o Rockhll Rd.,Kasas Cty, MO 64110, USA epartmet of Iformato Egeerg ad Computer Scece & Feg Cha Uversty 2 o. 100, Wehwa Rd., Seatwe, achug, 40724, awa epartmet of Computer Scece ad Iformato Egeerg & Asa Uversty 3 o. 500, Loufeg Rd., Wufeg, achug, 41354, awa (Emal: ala3c@gmal.com) (Receved Mar. 22, 2013; revsed ad accepted May 26, 2013) Abstract Wth the rapd developmet of formato ad etwork techologes, commucato securty has become a mportat ssue may applcatos. here are may publc-key based ecrypto schemes proposed the lterature. etwork applcatos are o loger ust oe-tooe type of commucato; but t volves multple recevers (>1) (e.g., multcast trasmsso). Oe-to-oe type ecrypto s o loger satsfyg ths type of applcatos. Ghodos et al. proposed a threshold RSA cryptosystem her scheme s also called a multrecever ecrypto. I Ghodos et al. s scheme, each recever s decrypto key depeds o all publc keys of recevers. Whe the umber of recevers s large, Ghodos et al. s scheme becomes mpractcal. We proposed a ovel aoymous mult-recever ecrypto. I our scheme, each recever s decrypto key s fxed. I addto, the scheme provdes aoymty of recevers. We clude performace aalyss ad comparsos wth other schemes. Keywords: RSA cryptosystems, Chese Remader heorem, Aoymty, Mult-recever ecrypto 1 Itroducto Publc-key ecryptos [4, 17, 18] allow a seder to sed a ecrypted message to a recever (.e., oe-to-oe type of ecrypto). However, etwork applcatos are o loger ust oe-to-oe type of commucato; but t volves multple recevers (>1), (e.g., multcast trasmsso). Oeto-oe type ecrypto s o loger satsfyg ths type of applcatos. I 1989, Chou ad Che have proposed a secure broadcastg [7] usg the CR for protectg the messages. I ther scheme, recevers eed to have ther publc ad prvate key pars tally. he seder selects a radom sesso key to ecrypt the message. he, the seder uses ay publc-key ecrypto ad the CR to securely trasmt the decrypto key correspodg to the radom sesso key to all legtmate recevers. I summary, the scheme uses the key-ecrypt-key techque to protect the messages. If the key-ecrypt-key algorthm s a publc-key algorthm, the parameters of the publc-key algorthm ca be used for oly oe tme trasmsso; otherwse, f the key-ecrypt-key algorthm s a prvate-key algorthm, ther scheme s a mxed scheme sce t uses both the publc-key ad prvatekey cryptosystems. he oto of broadcast ecrypto was frst troduced by Berkovts [3] ad was gve a formal defto by Fat ad aor [13]. he goal of the broadcast ecrypto s to securely trasmt a message to all legtmate recevers va secure broadcast chaels. I Fat ad aor scheme, each user og the system eeds to obta multple prearraged keys durg regstrato ad the use these keys real-tme operato. here are several papers [2, 10, 19] related to Fat ad aor scheme to dscuss the umber of prvate keys, the sze of cphertext ad computatoal cost. I 2005, u et al. [11] frst proposed a I-based broadcast ecrypto based o Boeh ad Frakl scheme [4] usg matrx. I that scheme, recevers do ot eed to regster wth a ceter tally. However, 2007, Che [6] poted out that ther scheme s secure. I 2005, Wag ad Wu [24] proposed a I-based multcast scheme usg blear parg. here are a key geerato ceter to regster all recevers tally ad a group ceter to sed messages to recevers. Oly the group ceter ca be the seder. Later, Saka ad Furukawa proposed aother I-based broadcast ecrypto [22] that the seder ca be ay user. Most exstg mult-recever ecrypto schemes the lterature [2, 3, 4, 6, 10, 11, 13, 19, 24] caot provde the aoymty of recevers. Fa et al. [12] proposed a aoymous mult-recever I-based ecrypto scheme Wag et al. [25] poted out that ther scheme caot provde the aoymty of recevers ad s secure hey also proposed a modfed scheme. Ufortuately, Zhag et al. [27] poted out that Wag et al. s scheme caot provde the aoymty ad s secure. here are may papers o the parg-based mult-

2 Iteratoal Joural of etwork Securty, Vol.15, o.4, PP , July recever ecrypto scheme [1, 5, 6, 11, 12, 24, 25, 27]; but oly a few papers are focused o the RSA-based multrecever ecryptos [14, 23]. RSA [21] s oe of the most commoly used publc-key ecryptos. However, there exsts a well-kow attack o the RSA-based broadcast ecrypto whe all recevers use the same publc key. he detal of ths attack ca be foud [16]. Whe each recever uses dfferet publc key, ths attack ca be avoded. he publc keys of recevers are all dstct our proposed scheme. I [14], Ghodos et al. proposed a dyamc threshold cryptosystems based o the RSA. I ther paper, a threshold RSA cryptosystem s the same as a mult-recever ecrypto scheme. I ther scheme, a seder ca arbtrarly select a set of legtmate recevers wthout eedg ay key dstrbuto. Each recever s decrypto key depeds o all publc keys of recevers. Ufortuately, whe the umber of recevers s large, Ghodos et al. s scheme becomes mpractcal. I addto, ther scheme caot provde the aoymty of recevers. I ths paper, we propose a RSA-based mult-recever ecrypto scheme based o the Chese Remader heorem (CR). I the decrypto, each recever ca use hs/her fxed prvet key whch does ot deped o other recevers to decrypt the cphertext. he rest of ths paper s orgazed as follows. I Secto 2, we revew CR, RSA ad Ghodos s threshold RSA scheme. I Secto 3, we propose a aoymous multrecever ecrypto scheme. he, Sectos 4 ad 5 are the dscusso ad aalyses. Cocluso s gve Secto 6. 2 Prelmares I ths secto, we brefly revew basc fudametals used our scheme cludg the CR [8], RSA ecrypto [21], ad Ghodos et al. s scheme [14] based o the RSA ecrypto. 2.1 Chese Remader heorem (CR) Assume that there are postve tegers whch are parwse coprmes, ad arbtrary tegers where he, there exsts a uque teger satsfyg followg system of smultaeous cogruece: X a mod ; 1 1 X a mod ; 2 2 (1) 2.2 RSA Ecrypto I 1978, Rvest, Shamr ad Adlema proposed a publckey cryptosystem [21]. Here, we oly revew the RSA ecrypto ad assume that Alce performs the followg steps to geerate her publc ad prvate keys. 1) Frst, Alce chooses two strog prmes, p ad q, such that p 2p 1 ad q 2q 1, where p ad q are two large prmes. he detal about strog prme ca be foud [24, 25]. he, she computes the product, p q. 2) Secod, she chooses the publc key e, such that e ad are relatvely prme, amely gcd( e, ) 1. 3) After choosg the publc key ad gettg the product, Alce ca compute ( ) ad prvate key d as ed 1(mod ), where ( ) ( p 1) ( q 1). I addto, the Euler s totet fucto ( ) meas the umber of postve tegers whch are less tha ad relatvely prme to. 4) Fally, Alce keeps the prvate key ( d, p, q ) secretly ad publshes the publc key ( e, ). We also assume that Bob wats to sed the message M to Alce secretly, where 0< M<. Bob must employ Alce s publc key e to compute the cphertext as C M e mod ad seds t to Alce. Oce Alce receves the cphertext C, she ca use her prvate key d to decrypt d the cphertext C as C mod M. 2.3 Ghodos et al. s Mult-recever Ecrypto Scheme I ths subsecto, we brefly revew Ghodos et al. s multrecever ecrypto scheme [14]. here s a group G cosstg of users ad oe trusted publc regstry. Each user U has hs/her RSA key par (.e., the publc key s ( e, ) ad the prvate key s d ). Furthermore, the trusted publc regstry stores ad maages all publc keys of users. her scheme cossts of two phases: 1) ecrypto phase ad 2) decrypto phase. A seder S ca ecrypt oe message M ad broadcast the cphertext to all recevers ecrypto phase. I decrypto phase, the recever U eeds to compute a decrypto key frst ad the uses the decrypto key to decrypt the cphertext to obta the message M. where X a mod, X ( ( ) y ) mod, 1 a 1 2, ( ) y 1(mod ), ( ) y 0(mod ) ad 0 X for. Ecrypto Phase Assume that the seder S wats to sed oe message M to the group G. S has to get all recevers publc-key dgtal certfcates ad valdate ther publc keys [26]. If the verfcato s passed, S computes the ecrypto key as E e ad the modulus as 1 2 of 1 the group. he, the seder S computes the cphertext:

3 Iteratoal Joural of etwork Securty, Vol.15, o.4, PP , July E C M mod. he seder S broadcasts E, ad C to all recevers. C c mod ; 1 1 C c mod ; 2 2 (3) ecrypto Phase ad e of other recevers, for 1, 2,, ad. If ad E are vald, each recever U ca check to determe whether he/she s oe of the teded recevers. If he/she s, each recever U computes the decrypto key as E 1mod ( ). he, each recever U uses to decrypt the cphertext C as E C mod M mod M mod. 3 he Proposed Scheme I ths secto, we propose a mult-recever ecrypto scheme based o the RSA ad CR. Our scheme cossts of two phases: 1) ecrypto phase ad 2) decrypto phase. A seder uses all recevers publc keys to ecrypt the secret message ad the broadcasts the cphertext to the recevers the ecrypto phase. Whe a legtmate recever receves the cphertext, he/she ca decrypt the ecrypted message by usg a fxed decrypto key the decrypto phase. Ecrypto Phase Frst, we assume that there are recevers ad oe seder. Each recever U has oe RSA key par, ( d,( e, )), for 1,2,,. he publc key ad prvate key of each recever are geerated accordg to the procedures descrbed Secto 2.1. Whe a seder wats to sed the message M to recevers, he/she has to valdate publc-key dgtal certfcates of all recevers at the begg. he message M must be smaller tha, where s the mmal modulus the set of {,,, }. he the seder S computes 1 2 ad 1 2 c M mod ; e1 1 1 c M mod ; e2 2 2 e c M mod. Accordg to the CR, whe 1, 2,, are parwse coprmes ad 0 c1 1, 0 c2 2,, 0 c, there exsts a uque soluto C (3) satsfyg followg system of smultaeous cogruece: (2) C c mod. Whe each recever U receves the broadcasted he seder broadcasts C to all recevers. message, ( E,, C ), he/she eeds to valdate publc keys ecrypto Phase Oce recever U obtag the cphertext C, he/she ca perform the followg steps to retract the secret message M. 1) Frst, he/she uses the publc key to execute the modulo reducto, that s, C mod c. 2) After gettg c, recever U ca use the prvate key d to decrypt c as c d ( e ) d M mod M. Sce 0 M, each recever ca retreve the same message M. I decrypto phase, each recever uses hs/her fxed key par to retract the message M. 4 scusso I ths secto, we dscuss the smlartes ad dffereces betwee Ghodos et al. s scheme ad our scheme. here are two smlartes. Frst, both Ghodos et al. ad our scheme use RSA ecrypto to provde mult-recever ecrypto. Secod, the seder ca select ay set of legtmate recevers. O the other had, there are three dffereces betwee Ghodos et al. s scheme ad our scheme. he detals are gve below. 4.1 Computatoal Complexty I Ghodos et al. s scheme, each recever U obtas the cphertext ( CE,, ). U eeds to valdate publc keys e of other recevers, for, 1, 2,,, ad, ad verfyg 1 tmes. he tme complexty s O ( ), where s the total umber of recevers. Whe there are a large umber of recevers, ther scheme s mpractcal. I our scheme, each recever does ot eed to valdate publc keys of other recevers. Furthermore, the ext secto, we wll show that Ghodos et al. s scheme, the seder speds more computatoal tme to ecrypt the message tha the tme eeded our scheme. 4.2 Aoymty of Recevers.e., I Ghodos et al. s scheme, whe recever U gets the, he/she eeds to determe whether he/she s oe of legtmate recevers by computg. Sce ad are publc formato, ther scheme caot provde the aoymty of recevers. I our scheme, seder broadcasts

4 Iteratoal Joural of etwork Securty, Vol.15, o.4, PP , July C oly. Hece, obody, except the seder, kows who legtmate recevers are. etal aalyss wll be cluded Secto ecrypto Key I Ghodos et al. s scheme, recever U eeds to compute a decrypto key as E 1(mod ). However, E s the product of publc keys of all recevers. Hece, each recever s decrypto key s dyamc for each sesso. But, our scheme, each recever U ca uses hs/her prvate key d to recover the orgal message M as d C mod M. So the decrypto key s fxed for all sessos. 5 Aalyss I ths secto, we dscuss the propertes of correctess, performace, securty ad aoymty. Correctess meas that each legtmate recever ca retreve the message ad the aoymty meas that the dettes of recevers ca be protected. We compare the computatoal cost ad the sze of cphertext amog our scheme ad other schemes. 5.1 Correctess Whe seder wats to sed the message the rage of message M s restrcted to be betwee zero ad whch s the mmal modulus the set of { for 1 }. Accordgly, each legtmate recever havg vald values, d ad, ca retreve the message. 5.2 Performace I ths subsecto, we compare the performace amog ucast RSA, Ghodos et al. scheme ad our scheme. I ucast RSA, the seder eeds to perform RSA ecrypto for tmes ad seds each cphertext to oe of recevers separately. We assume to use the 1024-bts RSA 1024 ecrypto,.e., modulus 2 for 1. Sce modulo expoetato takes more computatoal tme tha other operatos, we oly cosder the tme eeded for RSA ecrypto ad decrypto the followg dscusso. We let, E _1024 E_ 1024 ad deote the 1024-bt RSA ecrypto operato, the ( 1024) -bt RSA ecrypto operato ad the RSA decrypto operato, separately. able 1 shows the comparso results. he computatoal cost ad cphertext sze of ucast RSA s the same as our scheme; but t eeds tmes trasmsso. I addto, there has other traffc formato eeded ucast trasmsso. I the followg aalyss, we adopt results publshed [9] that clude complextes of stadard hardware mplemetato of modulo operatos. For ( 1024) -bt operato, the hardware complexty of able 1: Comparso results amog dfferet schemes Seder Recever Ucast RSA E _1024 Ghodos et al. scheme 1 4 E _1024 (.e., E_ 1024 ) Our scheme E _1024 Sze of cphertext btes btes btes Rouds of 1 1 trasmsso Aoymty of recevers o o Yes : the umber of recevers ca get 4. I summary, the 1 E _ 1024 E _1024 computatoal cost of the seder s 1 4 E _1024 Ghodos et al. s scheme ad s E _1024 our scheme. Furthermore, our scheme, the seder eeds oe CR computato. Although the seder does ot eed to compute CR Ghodos et al. scheme, the decrypto key of each recever s dyamc. O the recever sde, both Ghodos et al. scheme ad our scheme eed oe RSA decrypto. he dfferet s that ther scheme eeds to compute oe multplcatve verse; but ours eeds to compute oe modulo reducto. Obvously, the computatoal complexty of multplcatve verse s much larger tha the computatoal complexty of modulo reducto. herefore, our scheme s more computatoally effcet tha Ghodos et al. s scheme ad the cphertext sze of ours s Securty of the cphertext sze of ther scheme. Sce our scheme uses RSA ecrypto, the securty of our scheme s the same as RSA. amely, the securty of ours s based o the teger factorg problem. If a llegtmate recever wats to decrypt the ecrypted message, he/she eeds to break the RSA assumpto. 5.4 Aoymty 6 Coclusos 1 modular expoetato, s 4 tmes the hardware complexty of 1024-bt modular expoetato. Hece, we We assume that seder S wats to sed the cphertext C to t legtmate recevers a group G cosstg of users, where t. he attacker Eve has tercepted the cphertext C ad kows users publc keys, for 1. Eve Eve ca use C ad to compute e M mod, where s the publc key of oe of the legtmate recevers, Eve caot recover the message ad to detfy dettes of legtmate recevers. Hece, our scheme ca provde the aoymty of recevers. I ths paper, we propose a ovel mult-recever ecrypto scheme whch ca provde the aoymty of recevers. Our

5 Iteratoal Joural of etwork Securty, Vol.15, o.4, PP , July scheme s a smple tegrato of both the RSA ad CR. he proposed scheme s more effcet tha exstg schemes terms of computatoal cost ad cphertext sze. Each recever oly eeds to keep oe prvate key ad uses ths prvate key to decrypt the cphertext. Refereces [1] J. Baek, R. Safav-a, ad W. Suslo, Effcet mult-recever detty-based ecrypto ad ts applcato to broadcast ecrypto, Proceedg of Publc Key Cryptography PKC 05, LCS 3386, pp , Sprger-Verlag, Les ablerets, Swtzerlad, Ja [2] M. Bellare, A. Boldyreva, ad S. Mcal, Publc-key ecrypto a mult-user settg: securty proofs ad mprovemets, Proceedg of Advaces Cryptology Eurocrypt 2000, LCS 1807, pp , Sprger-Verlag, Bruges, Belgum, May [3] S. Berkovts, How to broadcast a secret, Proceedg of Advaces Cryptology Eurocrypt 91, LCS 547, pp , Sprger-Verlag, Brghto, UK, Apr [4]. Boeh ad M. Frakl, Idetty-based ecrypto from the Wel parg, Proceedg of Advaces Cryptology Crypto 01, LCS 2139, pp , Sprger-Verlag, Calfora, USA, Aug [5] Z. H., Che, S.., L, C. Z., Wag ad Y. P. She, wo costructos of multrecever ecrypto supportg costat keys, short cphertexts, ad detty prvacy, Iteratoal Joural of etwork Securty, vol. 14, o. 5, pp , Sep [6] H. Y. Che, Commets o a effcet I-based broadcast ecrypto scheme, IEEE rasatos Broadcast, vol. 53, o. 4, pp , ec [7] G. H. Chou ad W.. Che, Secure broadcastg usg the secure lock IEEE rasatos o Software Egeerg, vol. 15, o. 8, pp , Aug [8] H. Cohe, A Course Computatoal Algebrac umber heory, Sprger, 4th ed., pp , ew York, [9] J. P., avd, K. Kalach, ad. ttley, Hardware complexty of modular multplcato ad expoetato, IEEE rasatos o Computers, vol. 56, o. 10, pp , Oct [10] Y. ods ad. Fazo, Publc key broadcast ecrypto for stateless recevers, Proceedg of gtal Rghts Maagemet 2002, LCS 2696, pp , Sprger-Verlag, Washgto, C, USA, [11] X. u, Y. Wag, J. Ge, ad Y. M. Wag, A Ibased broadcast ecrypto scheme for key dstrbuto, IEEE rasatos Broadcast, vol. 51, o. 2, pp , Jue [12] C. I. Fa, L. Huag, ad P. Ho, Aoymous multrecever detty-based ecrypto, IEEE rasatos o Computers, vol. 59, o. 9, pp , Sep [13] A. Fat ad M. aor, Broadcast ecrypto, Proceedg of Advaces Cryptology Crypto, LCS 839, pp , Sprger-Verlag, Calfora, USA, Aug [14] H. Ghodos, J. Peprzyk, ad R. Safav-a, yamc threshold cryptosystems: a ew scheme group oreted cryptography, Proceedg of Pragocrypt 96: the 1 st Iteratoal Coferce o thw heory ad Applcatos of Crytology, pp , Prague, Czech Republc, Sep [15] J. Gordo, Strog RSA key, IE Electrocs Letters, vol. 20, o. 12, pp , Jue [16] J. Hastad, Solvg smultaeous modular equatos of low degree, SIAM Joural o Computg, vol. 17, o. 2, pp , Apr [17] IEEE Stadard Specfcatos for Publc-Key Cryptography, IEEE Stadard , [18] IEEE raft Stadard for Idetty-Based Cryptographc echques usg Pargs, IEEE P1363/6, [19] K. Kurosawa, Mult-recpet publc-key ecrypto wth shorteed cphertext, Proceedg of Publc Key Cryptography, LCS 2274, Sprger-Verlag, pp , Pars, Frace, February [20] J. H. Moore, Protocol falures cryptosystems, Proceedgs of the IEEE, vol. 76, o. 5, pp , May [21] R. Rvest, A. Shamr, ad L. Adlema, A method for obtag dgtal sgatures ad publc-key cryptosystems, Commucatos of the ACM, vol. 21, o. 2, pp , Feb [22] R. Saka ad J. Furukawa, Idetty-based broadcast ecrypto, Cryptology eprt Archve, Report 2007/217, [23] E. Sgurd, A mult-recever publc key cryptosystem, Proceedgs of orwega Symposum Iformato Securty (ISK), pp , rodhem, orway, ov [24] L. Wag ad C. K. Wu, Effcet detty-based multcast scheme from blear parg, IEE Proceedgs Commucatos, vol. 152, o. 6, pp , ec [25] H. Wag, Y. Zhag, H. Xog, ad B. Q, Cryptaalyss ad mprovemets of a aoymous mult-recever detty-based ecrypto scheme, IE Iformato Securty, vol. 6, o. 1, pp , Mar [26] R. W. Youglove, Publc key frastructure. How t works, Computg & Cotrol Egeerg Joural, vol. 12, o. 2, pp , Apr [27] J. Zhag ad Y. Cu, Commet a aoymous multrecever detty-based ecrypto scheme, Proceedg of 4th Iteratoal coferece o Itellget etworkg ad Collaboratve Systems ( ICoS), pp , Bucharest, Romaa, Sep

6 Iteratoal Joural of etwork Securty, Vol.15, o.4, PP , July Le Har receved hs BS degree Electrcal Egeerg from the atoal awa Uversty I 1980, he receved hs MS degree Electrcal Egeerg from the State Uversty of ew York-Stoy Brook ad 1984 he receved hs Ph.. degree Electrcal Egeerg from the Uversty of Mesota. Curretly, he s a Full Professor at the epartmet of Computer Scece Electrcal Egeerg, Uversty of Mssour- Kasas Cty, USA. Hs research terests are cryptography, etwork securty ad wreless commucato securty. He has publshed umber of papers o dgtal sgature desg ad applcatos, wreless ad etwork securty. Ch-Che Chag receved hs Ph. computer egeerg 1982 from the atoal Chao ug Uversty, awa. He was the head of, ad a professor, the Isttute of Computer Scece ad Iformato Egeerg at the atoal Chug Cheg Uversty, Chay, awa. From August 1992 to July 1995, he was the dea of the College of Egeerg at the same uversty. From August 1995 to October 1997, he was the provost at the atoal Chug Cheg Uversty. From September 1996 to October 1997, r. Chag was the Actg Presdet at the atoal Chug Cheg Uversty. From July 1998 to Jue 2000, he was the drector of Advsory Offce of the Mstry of Educato of the R.O.C. Sce February 2005, he has bee a Char Professor of Feg Cha Uversty. He s curretly a Fellow of IEEE ad a Fellow of IEE, UK. He also publshed several hudred papers Iformato Sceces. I addto, he has served as a cosultat to several research sttutes ad govermet departmets. Hs curret research terests clude database desg, computer cryptography, mage compresso ad data structures. Hsao-Lg Wu was bor Kaohsug, awa, She receved the BS degree Appled Mathematcs from Feg Cha Uversty, achug, awa She s curretly pursug her Ph.. degree formato egeerg ad computer scece from Feg Cha Uversty, achug, awa. Her curret research terests clude electroc commerce, formato securty, cryptography, ad moble commucatos.

Infinite Series Forms of Double Integrals

Infinite Series Forms of Double Integrals Iteratoal Joural of Data Evelopmet Aalyss ad *Operatos Research*, 4, Vol., No., 6- Avalable ole at http://pubs.scepub.com/jdeaor/// Scece ad Educato Publshg DOI:.69/jdeaor--- Ifte Seres Forms of Double