Security of Reconciliation in Continuous Variable Quantum Key Distribution *

Transcription

1 Scurty of Rconclaton n Contnuous Varabl Quantu Ky Dstrbuton * Y-bo Zhao, You-zhn Gu, Jn-an Chn, Zhng-fu Han **, Guang-can Guo Ky Lab of Quantu Inforaton of CS Unvrsty of Scnc & Tchnology of Chna, Chns cady of Scnc Hf, nhu 3006, P. R. Chna bstract W show that n contnuous varabl quantu y dstrbuton, whch s norally blvd to b scur undr any loss of th quantu channl, thr s stll a scurty loophol vn whn rvrs rconclaton s usd. Quanttatv analyss shows that th scurty of contnuous varabl quantu y dstrbuton s anly dtrnd by rconclaton, spcally by th sz of ach corrcton bloc. W also drv th rlatonshp aong th corrcton bloc sz, bt rror rat and scrt y rat. PCS nubr(s: Dd, p, c For hgh y rats, contnuous varabl quantu y dstrbuton (CVQKD has rcntly attractd or attnton copard to sngl photon countng schs. Th CVQKD schs typcally us th quadratur apltud of lght bas as nforaton carrr, and hoodyn dtcton rathr than photon countng. So of ths schs us non-classcal stats, such as squzd stats [, ] or ntangld stats [3,4], whl othrs us cohrnt stats [5-0]. ong th lattr, th sch n rfrnc [0] can provd hgh y rats whn ang sultanous quadratur asurnts. caus squzd stats and ntangld stats ar snstv to losss n th quantu channl, cohrnt stats ar uch or attractv for long dstanc transsson. For nstanc, an xprntal tabl-top stup at 780n that ncods nforaton n th phas and apltud of cohrnt stats has bn donstratd [], and a rcnt xprnt has also shown th fasblty of CVQKD n optcal fbrs up to a dstanc of 55 [,3]. lthough t has bn thortcally provd that only rvrs rconclaton can guarant th scurty undr any losss of th quantu channl [9], how to ralz propr rconclaton s a bg probl for practcal CVQKD. In th followng w wll dscuss th strct rqurnts of th rconclaton thod. Rconclaton convrts lc and ob s corrlatd raw y lnts nto * Supportd by:scnc Foundaton of Chna undr Grant No and No ; Th Knowldg Innovaton Proct of Chns cady of Scncs; ** To who corrspondnc should b addrssd, Eal: zfhan@ustc.du.cn

2 bnary bt strngs and prfors rror corrcton wth hgh probablty, to nsur that thy shar dntcal bt strngs. Unl th bnary bts gnratd fro th sngl photon schs, th bnary bts convrtd fro th contnuous varabls crtanly has hgh bt rror rats (ER. lso, th bnary bts Ev taps hav so rlatonshp wth lc s and ob s, unl that of th sngl photon schs, n whch Ev s bts ay hav no rlatonshp wth lc s and ob s. For th CVQKD, t s ust th cas that th ER of ach bt btwn Ev and ob s only slghtly hghr than that btwn lc and ob. Thn usng vry prcs rror corrctng thod, lc and ob can corrct th rrors btwn th, but Ev has so rrors that cannot b corrctd. ftr that, usng prvacy aplfcaton thy can gt th scrt ys. Th y pont s that lc and ob should guarant that thy can corrct thr rrors and Ev ust hav so rrors that cannot b corrctd. Othrws, thy cannot gnrat th scrt ys. Howvr, for th probablstc fluctuatons, t s vry dffcult to achv. Hr w wll gv a brf dscusson of rvrs rconclaton as dscrbd n rfrncs [4, 5] wth a quanttatv analyss of th dpndnc of th scurty on th rconclaton paratrs. ftr copltng th quantu councatons, lc and ob shar a st of contnuously E dstrbutd y lnts, dscrbd as {,, }, {,, }, {,, } ( N. To convrt C th nto bnary strngs, thy can ploy, for xapl, rvrs rconclaton. ob frst convrts C C ach C nto GF( l usng th slcd bnary functon S( x = ( S( x,, Sl ( x and consquntly obtans a st of l -bt strngs s = S ( C ( l. Gnrally, thr s a hgh probablty that lc can corrspondngly ap hr lnts nto GF( l bt by bt through usng prvously appd and corrctd bts. For hgh ffcncy, larg corrcton blocs should b usd for corrcton nforaton xchang. In th scond stp, ob sts bnary T strngs S = ( S ( C,, S ( C ( l as a corrcton bloc to corrct th rrors of th th bt, whr dnots th sz of th th corrcton bloc. Thrdly, ob snds th rror-corrcton nforaton T R of th bnary strng S = ( S ( C,, S ( C to lc va a publc authntc channl. Thn accordng to R, lc aps ach of hr lnts nto bts usng th optzd stators s ( C = S ( C, S ' ( C,, S ' ( C ( > and

3 s ( C = S ( C ( = whr dnots th th bt, ( l and S ( 0 < < dnots ' lc s corrctd bts fro S = ( s,, s T va R by prvous stps. Thn lc corrcts S usng R. Howvr, Ev also attpts to us C E and R to gt s, and hr fnal stats ar E E S. For th purpos of scurty, w rqur S' = S S. For convnnc, lt M = ( M, and dnot th corrcton,, M T M = ( M,, M T M = ( M,, M T unts S, S and Ev s uncorrctd strng S, rspctvly. E E E E In th followng t s assud that th channl s bnary sytrc. W dfn Pr(, and a b= M M = b Pr( M E M Pr( a = M M as th bt rror rats. E Suppos R s th rror corrcton nforaton that ob snt to lc va th publc channl and nrc n + dnots th axu rrors that can b corrctd va R. Thn, only rrors that ar lss than or or than rc rc th lnar party-chc codng [6]. n can b corrctd usng R, whch s th gnral cas for ost of Lt n dnot th nubr of rrors n M rlatv to M. Provdd ach rror n M s ndpndnt, thn n obys th bnoal dstrbuton probablty:! n n n n pn ( = ( = C ( n!( n! n ( n n W ntroduc th paratr and st n=. Usng th Strlng forula n! n π n. W can gt:! pn ( = C ( = ( n!( n! n n n n n π n πn n π n n n n ( n [( ( ] ( ( [ ( ] ( ( π = ( π π = ( ( π ( ( ( ( Thn th dstrbuton of can b wrttn as:

4 p( ( ( π( ( ( Lws, th dstrbuton p ( btwn M and E M s p ( ( ( π( ( (3 Th llustratons of p( and p ' ( ar shown n fgur. Fg : llustraton of p( and p'( ( = 00, dot ln -- p( ; sold ln - p'( = 0. = 0. a b b Lt rc n rc =, and dfn β as: rc rc ( β = p( d+ p( d ( ( d 0 rc 0 (4 π( Norally, p( d s uch sallr than rc rc p( d and can b nglctd. It s asy to 0 s that β s ust th probablty that lc obtans ob s M corrctly aftr rror corrcton. Slarly, th probablty of Ev obtanng M s rc rc ( α p ( d = ( ( d 0 0 π( (5 For scurty, β and α 0 ar ncssary. For convnnc, w us a Gaussan approxaton. Th varanc of p( n s ( and t s an s a b. Thn,

5 through th cntral-lt thor, Eqs. ( and (3 bco, rspctvly: ( p( xp[ ] π ( ( ( p '( xp[ ] π ( ( (6 (7 Fro Eqs. (4 and (6 w can gt: ( β xp[ ] d rc π ( ( ( = xp[ ] d ( rc π ( ( ( = xp[ ] d( ( rc π ( ( y xp( = ( rc π ( dy ( Thus = Q ( β (8 ( rc whr Q(x s th soluton of th quaton y dy x Q x =. ( π Slarly, fro Eqs. (5 and (7 w obtan ( = Q ( α ( rc (9 For th scurty of practcal CVQKD, < rc< ust b satsfd. Fro Eqs. (8 and (9 w hav > Q Q ( ( ax[ ( α, ( β ] (0 Eqs. (8, (9 and (0 show us th basc rqurnts of rconclaton. Ths ans that should b largr than th largst of ts valus n th abov thr quatons. In fact, th prforanc of actual rror corrctng algorth s only dtrnd by th nubr of rrors rathr than th ERs. For short, t ay b possbl for Ev to obtan a raw y wth fwr rrors than lc bcaus of th statstcal fluctuatons, although th ERs btwn hr and ob ar hghr than that btwn lc and ob. Consquntly, th sz of ach corrcton

6 bloc for scur CVQKD should b suffcntly long, and ths ay b a crucal ltaton for practcal systs. If undr th condton of asytry or whn th bts ar corrlatd wth ach othr, w can us th quvalnt avrag ERs to calculat th nu ltatons. W dfn th bt rror rats of th th appd bt for lc and Ev as and, rspctvly. In gnral, th condton n[ h ( h ( ] =Δ I < Δ I= ( I I E s satsfd, whr ΔI s ntroducd to dnot th scrt y rat that can b dstlld fro th th bt, as Δ I s th scrt bt rat dfnd n Rf. [] and ( h s th Shannon ntropy dfnd H ( = log ( log ( ( It s asy to s fro Eq. ( that dh ( = log d ( Thrfor, whn Δ s vry sall, t follows that ( log =Δ I (3 Usng Eq. (0, w obtan > Q α Q β ( (log ( ax[ (, ( ] ΔI (4 whr dnots th nu sz of th corrcton bloc for th th bt. It s asy to undrstand fro rfrnc [5] that whn th ER of on bt s about 0., th contrbuton fro ths bt Δ I ay b largr than that of othrs. Furthror, Q ( α s a slowly varyng functon of α, so α and β do not affct th staton rsults sgnfcantly. Eq.(4shows that s anly dtrnd by Δ I, so w only gv so quanttatv stats of th varaton of wth Δ I n th followng Tabl Ⅰ. Lngth of th Thortcal axu Maxu scrt y Mnu sz of quantu channl ( y rat ΔI [] (bt rat Δ (bt I corrcton bloc ^0 3

7 ^ ^0 5 Tabl Ⅰ: Mnu sz for dffrnt lngths of th quantu channl, whn 0 7 β =, Q ( β 7, th ER s 0., and th quantu channl attnuaton coffcnt s 0.d/. In concluson, w hav dscussd n dtal th crucal rqurnts of practcal rconclaton for scur CVQKD. W show that th sz of ach rror-corrcton bloc should b largr than th nu rqurnt to guarant th fasblty of CVQKD. It s also shown that th longr th lngth of th quantu channl, th largr ust b th sz of th rror-corrcton bloc. In addton, t should b notcd that n th rfrnc [] thy choos th bloc sz as for 3d loss, whch s uch largr than th gvn nus. cnowldgnt: Spcal thans ar gvn to Profssor Wu Lng-n for hr grat hlp n th syntax. W also gv sncr thans to P. Grangr who has pontd so probls n our prvous wors. Ths wor s supportd by Scnc Foundaton of Chna undr Grant No and No ; Th Knowldg Innovaton Proct of Chns cady of Scncs;. Rfrncs:. D. Gottsan and J. Prsll, Scur quantu y dstrbuton usng squzd stats. Phys. Rv. 63, 0309 (00. M. Hllry, Quantu cryptography wth squzd stats. Phys. Rv. 6, 0309 ( Ch. Slbrhorn, N. Korolova and G. Luchs, Quantu y dstrbuton wth brght ntangld bas. Phys. Rv. Ltt. 88, 6790 (00 4. M. D. Rd, Quantu cryptography wth prdtrnd y, usng contnuous-varabl Enstn-Podolsy-Rosn corrlatons, Phys. Rv., 6, ( F. Grosshans, and P. Grangr, Contnuous varabl quantu cryptography usng cohrnt stats. Phys. Rv. Ltt. 88, (00 6. S. Iblsdr, G. Van. ssch and N. J. Crf, Scurty of quantu y dstrbuton wth cohrnt stats and hoodyn dtcton, Phys. Rv. Ltt, 93, 7050 ( R. Na and T. Hrano, Practcal ltatons for contnuous-varabl quantu cryptography usng cohrnt stats, Phys. Rv. Ltt, 9, 790 (004

8 8. T. Hrano, H. Yaanaa, M. shaga, T. Konsh and R. Na, Quantu cryptography usng pulsd hoodyn dtcton, Phys. Rv., 68, 0433 ( Ch. Slbrhorn, T. C. Ralph, N. Lutnhaus and G. Luchs, Contnuous varabl quantu cryptography: batng th 3d loss lt, Phys. Rv. Ltt, 89, 6790 (00 0. C. Wdbroo,. M. Lanc, W. P. own, T. Syul, T. C. Ralph and P. K. La, Quantu cryptography wthout swtchng, Phys. Rv. Ltt, 93, (004. F. Grosshans, G. Van. ssch, J. Wngr,, R. rourl, N. J. Crf and P. Grangr, Quantu y dstrbuton usng Gaussan-odulatd cohrnt stats, Natur, 4, 38 (003. M. Lgr, H. Zbndn and N. Gsn, Iplntaton of contnuous varabl quantu cryptography n optcal fbrs usng a go-&-rturn confguraton, arxv:quant-ph/053, ( J. Lodwyc, T. Dbusschrt, R. T. rour and P. Grangr, Controllng xcss nos n fbr-optcs contnuous-varabl quantu y dstrbuton, Phys. Rv., 7, ( F. Grosshans and P. Grangr, Rvrs rconclaton protocols for quantu cryptography wth contnuous varabls, arxv: quant-ph/0047, ( G. Van. ssch, J. Cardnal and N. J. Crf, Rconclaton of a quantu-dstrbutd Gaussan y. IEEE Trans. Infor. Thory, 50, 394 ( S. S. Pradhan and K. Rachandran, Dstrbutd sourc codng usng syndros (DISCUS: dsgn and constructon, IEEE Transactons on Inforaton Thory, 49(3, 66 (003