An ID-Based Public Key Cryptosystem based on Integer Factoring and Double Discrete Logarithm Problem

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1 Informton urnc n Scurt Lttr (00) n ID-B Puc K rptotm on Intgr Fctorng n Dou Dcrt Logrthm Prom hnrhkhr Mhrm Shm Sunr grw Dprtmnt of pp Mthmtc Shr Shnkrchr Engnrng og Junwn Bh (G) In Em: c_mhrm@rffmcom Dh Inttut of Mngmnt & Tchnoog Rpur Em: hmpkc@rffmcom trct In 984 Shmr [] ntrouc th concpt of n ntt- crptotm In th tm ch ur n to vt k uthntcton cntr (K) n ntf hm f for jonng communcton ntwork Onc ur ccpt th K w prov hm wth crt k In th w f ur wnt to communct wth othr h on n to know th ntt of h communcton prtnr n th puc k of th K Thr no puc f rqur n th tm Howvr Shmr not ucc n contructng n ntt crptotm ut on n contructng n ntt- gntur chm In th ppr w propo n crptotm on th ntgr fctorng n ou crt ogrthm prom n w conr th curt gnt conprc of om ntt n th propo tm n how th pot of thng mor cur tm K wor: Puc k rptotm Intt rptotm Dcrt Logrthm Prom Dou Dcrt Logrthm Prom n Intgr Fctorng Introucton In ntwork nvronmnt crt on k n to hr twn two ur to th crt communcton Wh th numr of ur n th ntwork ncrng k truton w com rou prom In 976 Dff n Hmn [5] ntrouc th concpt of th puc k truton tm (PKDS) In th PKDS ch ur n to ct crt k n comput corrponng puc k tor n th puc rctor Th common crt on k whch w hr twn two ur cn thn trmn thr ur on h own crt k n th prtnr puc k though th PKDS prov n gnt w to ov th k truton prom th mjor concrn th uthntcton of th puc k u n th crptogrphc gorthm Mn ttmpt hv n m to wth th puc k uthntcton u Kohnfr [6] u th RS gt gntur chm to prov puc k crtfcton H tm nvov two kn of puc k Rcv ugut 00 crptogrph: on n mour p whr p rg prm numr; th othr n mour whr pq n p n q r rg prm Bom [] propo mmtrc k gnrton tm (SKGS) on crt hrng chm Th prom of SKGS howvr r th ffcut of choong ut thrho vu n th rqurmnt of rg mmor pc for torng th crt how of ch ur In 984 Shmr [] ntrouc th concpt of n ntt- In th tm; ch ur n to vt crptotm K uthntcton cntr (K) n ntf hm f for jonng th ntwork Onc ur ccpt th K w prov hm wth crt k In th w ur n on to know th ntt of h communcton prtnr n th puc k of th K togthr wth h crt k to communct wth othr Thr no puc f rqur n th tm Howvr Shmr not ucc n contructng n ntt- crptotm ut on n contructng n ntt- gntur chm Snc thn much rrch h n vot pc n Jpn to vrou kn of ID- crptogrphc chm Okmoto t [] propo n ntt- k truton tm n 988 n tr Oht [3] xtn thr chm for ur ntfcton Th chm u th RS puc k crptotm [] for oprton n mour n whr n prouct of two rg prm n th curt of th chm on th computton ffcut of fctorng th rg compot numr Tuj n Itoh [] hv o propo n ID- crptotm on th crt ogrthm prom wth ng crt xponnt whch u th EGm puc k crptotm Hrn n Rn [3] propo ID- RS for mugntur Mhrm & grw [3] hv o propo n ID- crptotm on th ou crt ogrthm prom wth ou tnct crt xponnt whch u th Puc k crptotm on th ou crt ogrthm prom ow w gnrz th crptotm for ntgr fctorng n crt ogrthm prom wth tnct ou crt xponnt cu w fc th $ 0350 Dnmc Puhr Inc

2 030 Mhrm n grw prom of ovng ntgr fctorng n ou tnct crt ogrthm prom t th m tm n th mutpctv group of fnt f compr to th othr puc k crptotm whr w fc th ffcut of ovng mutnou th ntgr fctorng n crt ogrthm prom n th common group In th ppr w prnt n ID crptotm on n ntgr fctorng n ou crt ogrthm prom wth tnct crt xponnt (th c of th propo tm com on th puc k crptotm on fctorng n ou crt ogrthm prom) hr w cr furthr conrton uch th curt of th tm th ntfcton for nr tc our chm o not rqur n ntrctv prmnr communcton n ch mg trnmon n n umpton xcpt th ntrctt of th crt ogrthm prom(th umpton m to qut ron)thu th propo chm concrt xmp of n ID crptotm whch tf Shmr orgn concpt [] n trct n Th Puc k ncrpton on Intgr Fctorng n DDLP In th cton w ntrouc om notton n prmtr whch w u throughout th ppr: Two rg prm numr p n q r f prm n t pq on m u mtho n [5] to gnrt trong rnom prm functon ϕ ( ) ( p )( q ) ph Eur functon n two ntgr g n g r prmtv mnt n Z wth orr tfng g n g n n (mo ) Th gorthm cont of thr u gorthm K gnrton Encrpton n Dcrpton K gnrton: Th k gnrton gorthm run foow (ntt hou o th foowng) Pck rnom n ntgr ϕ( ) from Z ϕ uch tht gc ( ( )) ( ) ϕ Sct two rnom ntgr n uch tht ϕ ( ) (wth no uppr oun) 3 omput g (mo ) n g (mo ) 4 U th xtn Eucn gorthm to comput th unqu ntgr ϕ( ) uch tht ( ϕ( )) mo Th puc k form ( ) corrponng prvt k gvn ( ) n th Encrpton: n ntt B to ncrpt mg M to ntt hou o th foowng: Otn puc k ( ) Rprnt th mg M [ ] 3 Sct two rnom ntgr n j uch tht j ϕ ( ) (wth no uppr oun) 4 omput α (mo ) n g α (mo ) g j j 5 omput M ( )( ) (mo ) β 5 omput α α n β Th cphr txt gvn ( ) Dcrpton: To rcovr th pntxt M from th cphr txt ntt hou o th foowng: ϕ ( ) omput ϕ ( ) Rcovr th pntxt M comput ( ) (mo ) 3 Vrfcton of th gorthm In Encrpton: - α (mo ) n n g g j j β M (mo ( g ) g j j α g g (mo j β ( M ) α ( ) ( ) ) α ( ) ( ) In Dcrpton: - ϕ ( ) ( ) ) ϕ ( ) j Thn ( ) j j ( M ) M M

3 n ID-B Puc k rptotm on Intgr Fctorng n Dou Dcrt Logrthm Prom 03 4 Impmntton of th ID B rptotm 5 Prprton for th cntr n ch ntt Stp Ech ntt gnrt k-mnon nr vctor for h ID W not ntt ' ID ID foow: ( x x x ) x {0 } ID k j ( j k) () Ech ntt rgtr h ID wth th cntr n th cntr tor t n puc f Stp : Th cntr gnrt two rnom prm numr p n q n comput pq () Thn th cntr choo n rtrr rnom numr ϕ( ) gc ϕ whr uch tht ( ( )) ϕ ( ) ( p )( q ) th Eur functon of thn cntr puh ( ) th puc k n ntt cn comput th ntt foowng: ' xtn ID EID th EID ( ID) x {0 ( j t) (3) ( ) } t j whr t th numr of t of Stp 3 ntr crt nformton: - Th cntr choo n rtrr rg prm numr p n q n comput pq n o gnrt n-mnon vctor n m-mnon vctor ovr Z ϕ ( ) whch tf ( ) ( ) n m (4) ϕ ( ) ( n) ( m) ( m n) I J (mo( p )) I J (5) Whr I n J r n-mnon nr vctor n tor t th cntr crt nformton Th conton of quton (5) ncr to vo th ccnt concnc of om ntt crt k mp w to gnrt th vctor n to u Mrk n Hmn chm [4] Th cntr choo upr ncrng qunc corrponng to n ( n) n ( m) tf n < ϕ ( )( m n) (6) Stp 4: Th cntr o choo w whch tf gc ( w ϕ ( )) o comput n-mnon vctor n m-mnon vctor foow w(moϕ ( ))( n) w(moϕ ( ))( m) (7) Whr ( ) ( ) n m (8) Rmrk : t cr tht th vctor n fn (8) tf (4)-(5) th ov chm on mtho of gnrtng n n m mnon vctor n tf (4)-(5) In th ppr w opt th ov chm Howvr nothr mtho mght po Stp 5: Th cntr o choo n unqu ntgr ϕ( ) uch tht (moϕ( )) (9) Stp 6: ntr puc nformton: Th cntr choo two rtrr gnrtor α n β of Z ϕ ( ) n comput n- mnon vctor h ung gnrtor α & m-mnon vctor g ung gnrtor β corrponng to th vctor n ( h h ) g ( g g ) h h g h n g m (0) α ( n) β ( m) () Th cntr nform ch ntt ( β h g) α puc nformton Stp 7: Ech ntt crt k: Entt ' crt k n r gvn nnr prouct of n (th cntr crt nformton) n qn) EID EID EID (ntt (moϕ( )) j n j j j n j (moϕ( )) j ' xtn ID (moϕ( )) (moϕ( )) () (3)

4 03 Mhrm n grw 5 Stm Intzton Prmtr 5 ntr Scrt nformton : n -mnon vctor m-mnon vctor n - n ntgr { (8)-(9)} 5 ntr puc nformton h : n -mnon vctor & g m-mnon vctor { qn(0-)} p n q :rg prm numr : rnom ntgr two gnrtor α n β of 53 Entt ' crt k ( ) { qn ( 3)} Z ϕ ( ) 54 Entt ' puc nformton ID k mnon vctor { qn ()} 6 Protoco of th propo crptotm Wthout o of gnrt uppo tht ntt B wh to n mg M to ntt 6 Encrpton Entt B gnrt EID (Entt ' xtn ID qn) from ID It thn comput n from corrponng puc nformton h n g n EID : ( α ) h n n mo( ϕ ( )) n α α g ( β ) m n mo( ϕ ( )) m β β Entt B u n n Puc k crptotm on ntgr fctorng n ou crt ogrthm prom M M Lt ( ) ntt B' mg to trnmtt Entt B ct two rnom ntgr u n uv ϕ ( ) n comput v uch tht ( ) u Y α v Y β u v δ M ( ) ( ) M Y n comput ( Y )(mo ) ( Y ) (mo ) ( Y ) (mo ) ( δ ) E Th cphr txt gvn ( E) 6 Dcrpton To rcovr th pntxt M from th cphr txt Entt hou o th foowng ϕ ( ) omput ϕ ( ) n Rcovr th pntxt M ( E) (mo ) 7 Scurt n In th cton w h how thr po ttck whch n vrr m tr to tk own th nw ncrpton chm For ch ttck w fn th ttck n gv ron wh th ttck cou f 7 Drct ttck vrr wh to otn crt k ung nformton v from th tm In th c vrr n to ov fctorng n crt ogrthm prom wth ou tnct crt xponnt Th t w to fctorz pq ung th numr f v mtho (FS) [8]ut th mtho jut pnnt on th z of mouu It computton nf to fctor 04-t ntgr n to ncr th curt of our chm; w hou ct trong prm [9] to v ttck ung pc purpo fctorzton gorthm To mntn th m curt v for crt ogrthm prom wth ou tnct crt xponnt on mut u pq p q wth n rpctv prouct of two 5-t prm 7 Fctorng ttck um tht th vrr uccfu ov th fctorng prom o tht h know crt Thu h m otn ( ) M (mo )

5 n ID-B Puc k rptotm on Intgr Fctorng n Dou Dcrt Logrthm Prom 033 Unfortunt t th tg h t o not know crt n n cnnot xtrct th pntxt M from th ov xpron 73 Dcrt og ttck n ttckr hou ov crt ogrthm prom twc to otn th prvt k gvn th puc foowng: In th ncrpton th puc k gvn α β h g n th corrponng crt k ( ) gvn ( ) To otn th prvt k ( ) h hou ov th DLP og To otn th prvt k ( ) DLP og α β ( α ) h hou ov th ( β ) Th nformton quvnt to computng th crt ogrthm prom ovr mutpctv ccc group Z ϕ ( ) n corrponng crt k n w nvr rv to th puc S tht ttckr to otn th crt ntgr u n v from ov th DLP u ogα Y n v og β Y H cou rv th pntxt M f n on f h mng to gt ( ) (mo ) 3 n ttckr mght tr to mpront ur vopng om rton twn w n w nc Y w w n Y Smr Y w (mo ) n Y w (mo ) knowng w w th ntrur cn rv w w n n w w wthout knowng n howvr trng to otn w from α n β quvnt to comput th crt ogrthm prom 8 oncuon In th prnt ppr n ID- crptotm on fctorng n ou crt ogrthm prom wth tnct crt xponnt n th mutpctv group of fnt f Th propo chm tf Shmr orgn concpt n trct n t o not rqur n ntrctv prmnr communcton n ch t trnmon n h no umpton tht tmpr fr mou r v Th kn of chm fnt prov nw chm wth ongr n hghr v of curt thn tht on fctorng n ou crt ogrthm prom wth tnct crt xponnt Th propo chm o rqur mnm oprton n ncrpton n crpton gorthm n thu mk t vr ffcnt Th prnt ppr prov th pc rut from th curt pont of vw cu w fc th prom of ovng fctorng wth ou n trp tnct crt ogrthm prom t th m tm n th mutpctv group of fnt f compr to th othr puc k crptotm In othr wor on mut rk th fctorng n crt ogrthm prom wth ou tnct xponnt tm mutnou to rk our propo tm cu w fc th ffcut of ovng th trton crt ogrthm prom n th common group Rfrnc [] Shmr Intt- crptotm n gntur chm vnc n rptoog: Procng of rpto (Lctur ot n omputr Scnc 96) Brn Wt Grmn: Sprngr-Vrg vo 84 pp [] S Tuj n T Itoh n ID-B rptotm on th Dcrt Logrthm Prom IEEE Jounr on ct r n communcton vo 7 pp [3] SMhrm n SSgrw n ID-B rptotm on th Dou Dcrt Logrthm Prom Intrnton Journ of omputr Scnc n twork Scurt vo0 no7 pp [4] T EGm Puc K rptotm n Sgntur Schm B on Dcrt Logrthm IEEE Trn Inform Thor vo 3 pp [5] W Dff n ME Hmn w rcton n rptogrph IEEE TrnInformThor vo pp [6] L M Kohnfr mtho for crtfcton L omput Sc M Int Tchno mrg M M 978 [7] Y Dmt n J J Ququtr Puc-k tm on th (I thr ffrnc twn DES n ffcut of tmprng vnc n rptoog: Procng of rpto 86 (Lc- RS?) n ctur ot n omputr Scnc 63) Brn Wt Grmn: Sprngr-Vrg pp [8] H Tnk rzton chm for th ntt crptotm vnc n rptoog: Procng of rpto 87 (Lctur Sprngr- ot n omputr Scnc 93) Brn Wt-Grmn Sprngr Vrg pp [9] S Tuj T Itoh n K Kurow ID- crptotm ung crt ogrthm prom Ectron Ltt vo 3 o 4 pp

6 034 Mhrm n grw [0] S Pohg n M E Hmn n mprov gorthm for computng ogrthm ovr GF (p) n t crptogrphc gnfcnc IEEE Trn Inform Thor vo IT-4 pp [] E Okrnoto n K Tnk K truton tm on ntfcton nformton IEEE J SIcr r ommun 989 vo 7 pp48485 M 989 [] R Born n optm c of mmtrc k gnrton tm In Proc Eurocrp 84 Pn Frnc 984 pr 9- pp [3] K Oht Effcnt ntfcton n gntur chm Ectron Ltt vo 4 no pp [4] W-Bn L n Kun-hh Lo ontructng ntt- crptotm for crt ogrthm crptotm Journ of twork n omputr ppcton vo 7 pp [5] Mn-Shng Hwng Jung-Wn Lo n Shu-hn Ln n ffcnt ur ntfcton chm on ID crptotm omputr Stnr & Intrfc vo 6 pp [6] Eun-Kung Ru n K-Young Yoo On th curt of ffcnt ur ntfcton chm pp Mthmtc n omputton vo7 pp [7] Mhr Br hnthp mprmpr n Grgor vn Scurt Proof for Intt-B Intfcton n Sgntur Schm J rpto vo pp [8] K Kom n K Oht Intt- confrnc k truton tm n vnc n rptoog: Procng of rpto 87 (Lctur ot n omputr Scnc 93) Brn Wt Grmn: Sprngr- Vrg pp [9] T EIGm puc-k crptotm n gntur chm on crt ogrthm IEEE Trn Inform Thor vo IT- 3 pp [0] S Pohg n M E Hmn n mprov gorthm for computng ogrthm ovr GF (p) n t crptogrphc gnfcnc IEEE Trn Inform Thor vo IT-4 pp [] K kmur E Okmoto K Tnk n S Mur prvt communcton ug 987 [] D opprmth prvt communcton ug 987 [3] R L Rvt Shmr n L mn mtho for otnng gt gntur n puc-k crptotm omrnun M vo no pp [4] R Mrk n M E Hmn Hng nformton n gntur n trpoor knpck IEEE Trn Inform Thor vo IT- 4 pp [5] S Brntt Mtrx mtho for ngnr n cntt McGrw-H Book ompn 979 [6] STujJho n Krk Smp ID-B Schm for K Shrng IEEE Jounr on Sct r n ommunctonvono5 pp [7] LHrn Puc k crptotm gn on fctorng n crt ogrthm IEE Pro omput Dgt Tch vo4 no3 pp [8] K Lntr HW Lntr MS Mn n JMPor Th numr f v Proc n M Smp On Thor of omputng Btmor Mrn US pp [9] J Goron Strong RS k Ectron Ltt vo0 no pp [30] Srn n UPrmp Effcnt ntt gntur n th tnr mo IET Inf Scur vo no4 pp [3] K n HSZhou Hn ntt gntur IET Inf Scur vo3 no3 pp [3] L Hrn n J Rn Effcnt ntt RS mutgntur computr n curt vo7 pp [33] Jnun Sun h Zhng Yncho Zhng n Yugung Fng n Intt-B Scurt Stm for Ur Prvc n Vhcur Hoc twork IEEE Trn on Pr n Dtrut Stm vo7 no9pp uthor Bogrph hnrhkhr Mhrm tchng n tnt Profor n Dprtmnt of pp Mthmtc Shr Shnkrchr Engnrng og Junwn Bh (G) In H rcv th MSc (Mth) n MPh (rptogrph) gr from Pnt Rvhnkr Shuk Unvrt Rpur hhttgrh In n 007 n 008 rpctv H ong h rrch n th f of rptogrph n t ppcton H mmr of Intrnton octon of Engnr Hong Kong omputr Scnc Tchr octon (ST M) US n Intrnton octon of omputr Scnc n Informton Tchnoog (ISIT) Sngpor n Lf -tm mmr of Inn Mthmtc Soct n rptoog Rrch Soct of In Shm Sunrgrw workng n oct Profor n th Dprtmnt of pp Mthmtc n Dh Inttut of Mngmnt & Tchnoog Rpur In H rcv th MSc (Mth) n PhD Dgr from Smpur Unvrt Or In n 997 n 008 rpctv H ong h rrch whr h ntrt ncu Dcon Mkng unr Fuzz Logc omntorc n rptogrph H mmr of IMS ISTE In n Intrnton octon of Engnr omputr Scnc Tchr octon (ST M) US n Intrnton octon of omputr Scnc n Informton Tchnoog (ISIT) Sngpor