物理層安全劍橋大學(xué)英文PPT課件

1、1. IntroductionThe main ways of key sharing:a) Transmission the keys over secure (encrypted) channels or a delivering them by special messengers;b) Using public key concept;c) Key sharing based on a presence of any noisy channel if adversary is passive, (wire-tap channel type I and II) 1,2,3d) Key s

2、haring based on a presence of active adversary if its channel is less noisy than channel of legal users. 4,5e) Key sharing using quantum channels.6f) Key sharing based on a concept of anonymous channel.g) Key sharing based on a concept of broadcasting channel.h) Key sharing based on ESPAR-like radia

3、tor over multipath channels. 7,81第1頁(yè)/共45頁(yè)Because method a) is trivial and b) is well known, we consider briefly methods c) g) and method h) in more details as a subject of our presentation. c) Source model with a passive eavesdropping .Aplication Key distribution via a satellite.Fact ( Maurer 3 ) Ri

4、fEEEKABE01 21 20/ ,/ ,2第2頁(yè)/共45頁(yè)4 Privacy amplification ( Bennett , Brassard , Crepeau , Maurer 9,10) The feature of keyless cryptography is :( i ) Share the secret key by legal parties using this concept( ii ) Use key - cryptography after receiving this key by legal parties (including perfect cipher

5、) KCSKSC,To share secret key , A and B perform the following steps1.A sends to B a truly random string x over public noisy channel .2.A sends to B the check symbols to x chosen in line with some error correcting code V3.A sends to B a truly random hash function h taken from universal class , which m

6、apsa string x of length n to string K of length k .4.B corrects errors in the string x using check symbols transmitted by A .5.Both A and B produce the key string as K = h ( x ) .Then the amount of information leaking over the wire - tap channel to eavesdropperE has the following upper bound 9,11 Ib

7、itntkr022 ()/ ln(),where n is the length of x , k - is the length of the key K , r - is the number of check symbols , t - is the amount of collision ( Renyi ) information leaking over the wire - tap channel to eavesdropper E .tPPnWW11222lo g()for BSC - wire - tap channel with BER=Pw第3頁(yè)/共45頁(yè)5 Wire -

8、tap channel type 2 . (Wyner 2)An eavesdropper can observe a subset of his ( her ) choice of size t n , where n is the block lengthMain applications - quantum cryptography (see in the sequel ) , optical fiber multiplexing , computer network containing eavesdroppers in some nodesRegular coding ( noise

9、less main channel )The key shared by A and B is the following :KxHTwhere H is the check matrix of some binary ( n , n-k ) code V , x is a binary string of length n radomly chosen by A and transmitted over the main public channel from A to B .Then the amount of information leaking over the wire - tap

10、 channel type 2 to easvesdropper is zero ( no easvesdropping at all ! ) providing the following inequality is true td 1where d is the minimum code distance of the code V which is dual of code V .第4頁(yè)/共45頁(yè)6 Example. V is ( 15 , 11 ) Hamming code . Then we have no easvesdropping about the key of length

11、 4 if t 7This concep can be exteded to noisy main channel ( Korjik , Kushnir 12) .Privacy amplification 9If A and B follow to the protocol described in the case type 1 in order to produce secret key, the amount of information leaking to eavesdropper has the following upper bound I0IntKP022()/ ln,whe

12、re n is the length of x , K is the length of the key , P is the number of check symbols , t is the maximum number of bits that cavesdropper can obseved of each block .第5頁(yè)/共45頁(yè)7d) A cryptographic scenario for source model (active illegal users )SatelliteAliceBobEveSY( )X( )Z( )B BAEe ee ee e1 .- Init

13、ialization phase ( S (X,Y,Z ) over BSC- s with BER-s : eeeA B E, ,respectively )第6頁(yè)/共45頁(yè)8e = e + e ( e ) = e + e ( e )2.-Authentication phase : ( M , a ) , where M - a string consisting of k information bits , a - authenticatora = f ( M , X ) , where f ( , ) is a public function . Intruders activity

14、 ( Upon receiving the pair ( M , a ) and knowing theauthentication algorithm , to form a pair ( M , a ) , where M = M - substitution attack )P - To be cheating by intruder ( the pair ( M , a ) is accepted by Bobas the original one )P - To be rejection the original message by Bob when an intruder has

15、not intervented into transmission at all .( The length of the string ,a as well as the length of the string X ( Y ) arevery important parameters . )BER - s between corresponding bits of X and Y , X and Z , Y and Z are ,respectively : ChRe = e + e ( e ) = e + e ( e )AB A B A B A B1 - 21 - 2AE A E A E

16、 A E1 - 21 - 2e = e + e ( e ) = e + e ( e )BE B E B E B E1 - 21 - 2 第7頁(yè)/共45頁(yè)9e e e e A E AB B E( It is easy to show that this inequality results in impossibility for Bob toauthenticate message sent by Alice ) b) ( It offers a positive solution for the authentication problem )a) A E AB B Ee e e e k n

17、MMM k122uuu122Code words of somebinary block code oflength n .The value 1 in the i - th position of some code word indicates that i - th bit of the string X should be taken as a bit of the autheticator corresponding tothe message compared with this code word .i -th positionk第8頁(yè)/共45頁(yè)10MMvvaaZBob acce

18、pts the message as original if and only if the fraction of bits in the received authenticator that agree with the corresponding bits of his string Yis not much smaller than 1 - ( In non - asymptotic case some fixed thresholdl should be chosen ) .The best substitution attackABeXX 11xxKeep the authent

19、icators bits as they were in a ,Put bits of Z - stringorThe positions of the authenticator can beremoved0 01xx 00 xx 10第9頁(yè)/共45頁(yè)11vvxxx= 011 0 0101v =11110111The probability of substituting the message Mfor M without detecting this fact by Bob is determind by 0 1 distance between the code words and .

20、 ( This distance property differsfrom the ordinary Hamming distance )vvDefinition 1 .vvdd,min0101vv Definition 2 .Constant weight authentication code : vl if=/,ilABlliliRPBee110VV),(V第10頁(yè)/共45頁(yè)12 idiBElidiChP01001BE01eeBiljjdldlj001010jAB1eeifdl010001ldif (, the upper limit in the first sum in ( )sho

21、uld be changed to 01dA simple construction of constant weigth codes ( due to Maurer-Wolf 4)Take some linear binary ( n , K , d ) code and replace every bit in its code wordsby pair of bits following the rule :001110 第11頁(yè)/共45頁(yè)13smsmdABABdBEBElChxxxPeeee)1()1(acababx22 , 1221BEABBEceeesmBEABdb)1 (ee)(

22、laBEABeeeeeeABABlABABllllP1)()1()()1(ReIt has been proved in 13第12頁(yè)/共45頁(yè)It gives the authentication code with parameters :ddlnXYnkk012=,/,Example 1 . BCH ( 1023 , 208 , 231 ) code . Let : eeABBE=,andthenOptimization procedure . ,BEkPPChRABeeGiven the parameters minimize the length l of the authentic

23、ator over all ( n , K , d ) linear codes .130,01770,2,101 , 14RP.1014ChP第13頁(yè)/共45頁(yè)1501000200030004000500060007000800090000.050.10.150.20.250.30.350.40.45kRRelative date rate (R=k/(w+k) as a function of information block length k for different BE and fixed parametrs AB=0.01 ,PRe10-4,PCh10-4 Rk1.BE = 0

24、.452.BE = 0.403.BE = 0.354.BE = 0.305.BE = 0.256.BE = 0.207.BE = 0.158.BE = 0.109.BE = 0.05123456789第14頁(yè)/共45頁(yè)1601000200030004000500060007000800090000.050.10.150.20.250.30.350.40.45kRRelative date rate (R=k/(w+k) as a function of information block length k for different BE and fixed parametrs AB=0.03

25、 ,PRe10-4,PCh10-4Rk234567891.BE = 0.452.BE = 0.403.BE = 0.354.BE = 0.305.BE = 0.256.BE = 0.207.BE = 0.158.BE = 0.109.BE = 0.051第15頁(yè)/共45頁(yè)17 Basic quantum key distribution protocol.1. A sends a random sequence of photons polarized horizontal ( ), vertical ( ), right-circular ( ), and left-circular ( )

26、.2. B measures the photons polarization in a random sequence of bases, rectlinear (+) and circular (o).3. Results of Bs measurments (some photons may not be recived at all).4. B tells A whicj bases be used for each photons he recived.5. A tells him which bases were correct.6. A and B keep only the d

27、ata from these correctly-measured photons, discarding all the rest.7. This data is interpreted as binary sequence according to the coding scheme:e) Quantum cryptography第16頁(yè)/共45頁(yè)18 f) Anonymous ChannelEavesdropper learns all bits transmitted between legitimate users A and B but does not know who ( A

28、or B ) is an “ author of any bit .Application .Key agreement protocol第17頁(yè)/共45頁(yè)SatelliteABEiaibiiicbaiiiFig. 1. The case g.0)/(;iEBAiiiBickIkkkabckakg) Key sharing based on a concept of broadcasting channel.18第18頁(yè)/共45頁(yè)h) Key sharing based on ESPAR-like radiators over multipath channels (general theor

29、y)2.1 Real word justification 7Legal user A transmits a series of packets each with a different beam pattern generated by electronically steerable parasitic array radiator (ESPAR)The packets are received by legal user B, which builds up a sequence of received signal strength indicator (RSSI).After t

30、hat B transmits packets back to A, where A builds up a sequence of RSSI data.Thanks to the reciprocity theorem of radio wave propagation between uplink and downlink, the sequence in A and B should be identical except for the random noise.Fig. 2. Key sharing procedure19第19頁(yè)/共45頁(yè)Security of such key s

31、haring is based on an assumption that the space locations of the eavesdropper and legal users are different. This results in a much greater disagreements key bits between legal users and eavesdropper. Raw disagreement bit distribution taken from 7 is shown in Fig.3. Sketch of experimental room is pr

32、esented in Fig.4. Fig.3. Raw disagreement bit distributionFig.4. Sketch of experimental room20第20頁(yè)/共45頁(yè)2.2. Our contribution.We present general theory based on some model in order to prove security of the key sharing system with the use of privacy amplification.We propose space diversity technique f

33、or increasing of security because our simulation of ESPAR-like system showed that the use of single omnidirectional antenna is not sufficiently for high security level.In order to present a disagreement in key bits of legal users we propose to use both “threshold-based” and “code-based” methods.It i

34、s interesting to note that there exist here two “seeming paradoxes”: - we do not need in a presence of noise at eavesdroppers point to provide security, - large eavesdroppers probability of bit error can be provided even so if mutual correlation between legal and illegal RSSI is rather significant.

35、21第21頁(yè)/共45頁(yè)2.3. Model of key sharing setting (without additive noise). , 10,0jjk, 10,0jjkLiijijx1Liijijy1;)1 ,0(,Nijij,RRR;),(1RyxLiii).(,diiijij). .()(1di ikjnjjHere are the key j-th bits of legal users and eavesdropper, respectively, are quadrature components of j-th RSSI of legal users and eavesd

36、ropper, respectively the attenuations on the i-th beam of legal user and eavesdropper, respectively, the number of beams (pathes of wave propagations) the radiation coefficients of the ESPAR-like system on the i-th beam in the j-th packet for legal user and eavesdropper, respectively.jjkk,jj,iiyx ,L

37、,ijij)1()2(otherwiseotherwisewherecorrelation matrices which are givenon index22第22頁(yè)/共45頁(yè)Assumption: and model (1),(2) are public.Particular case: (if an eavesdropper is located near the legal user)Correlation coefficient (general case):Particular case:where If , then we get by (4) that (nothing sec

38、urity)If , then in general. In a particular case when , then if N.B. (“Paradox” 1) LiiiyxL1,ijijTTTTTjjYYRXXRYXR),(TTTTTTjjYYRXXRYXR),(),.,(),.,(2121LLyyyxxxYXYX 1),(jj)3()4(YX 1),(jjLIR ,0),(),(yxyxjj0),(yx23第23頁(yè)/共45頁(yè)More strong model (for KDP designer) Eavesdropper is able to separate beams ; e.q.

39、 he (or she) has :Then this means that for a particular case ( ) an eavesdropper is able to find and hence to calculate the legal key bits exactly. This is not the case generally if Let us prove the key bit error probability for eavesdropper given the correlation coefficient and varianceThen we have

40、 after simple transforms (see Appendix 1) : NjyLiiji,.,1,1ijijijjkijijep),(jj 2jjVarVar)1(1)1 (22exp1212200222222arctgdxdyyxyxpe)5(24第24頁(yè)/共45頁(yè)It follows from (5) : (i) does not depend on but only on (ii) If , then ; if , then (in line with our intuition)The graph of versus is plotted in Fig.5. 210We

41、 can conclude that it is sufficiently to provide . (This is seeming “Paradox” 2).See Section 3 for detail. 95,0Fig.5. Dependence versus ep0ep21epep25epep第25頁(yè)/共45頁(yè)2.4. Two beam model.ESPARAEB(pathes 1)1122Fig.6. Two beam model of KDPGeneral model:(we drop index “j” for notation simplicity )Particular

42、 case: E is located very close to B.22112211yyxx22112211yyxx)21)(21 (1),(2222112112rrrr 1221212121,),(, 1yyxxrVarVarNew setting with a separation of beams by eavesdropper. given , ),(222111212211yyxx111222yy2211xx)6(2121,yyxxpathes 226第26頁(yè)/共45頁(yè)If E (as in Fig. 6 ) is between A and B, then , that is

43、reasonable. Particular cases: If r=1, then that is reasonable; If r=0, then ; If , then . 22112211xxxx) 21)( 21 () () 2)( 2() () ,(212221212221212221rrrrrrxxxxrxxxxrrxxxrx)7(,2122xx1) ,(lim)1 ()() ,( rrr1) ,()1 () ,(15 , 0) ,(27where),(22r),(12r).,(12 r第27頁(yè)/共45頁(yè)2.5. Simulation results of two beam mo

44、del with ESPAR-like system: 1. Using a random exciting of ESPAR-like system* elements results in a random beam-forming antenna diagram.(The number of radiation patterns can be provided as untractable by appropriated choice of the number ESPAR-like system elements “m” and the number of the bias volta

45、ge bits “ ”: ) 2. Radiation pattern amplitude can be approximated by Gaussion distribution with variable expectation and variance. 3. Radiation pattern amplitudes of ESRR with 6 radiators are uncorrelated for angle interval more than 1-4 degree.The last point gives a chance to justify a general mode

46、l in contrast to particular model (see slide 6). 1)2(m28* In our experiment we do not use ESPAR but electronically steerable ring radiator (ESRR) with 6 radiators equaly located on the circle of the radius 6 cm. We believe that ESRR gives more narrow beams than ESPAR第28頁(yè)/共45頁(yè)Let us consider two beam

47、 model (see slide (27)If ESRR system generates signal , then using two beam wave propagation scheme we get:where - is the attenuation of the signal s(t) over the path 1 from A to B (see Fig.6) - is the attenuation of the signal s(t) over the path 2 from A to B, - is the attenuation of the signal s(t

48、) over the path 1 from A to E, - is the attenuation of the signal s(t) over the path 2 from A to E.We let for simplicity that 22112211yyxxtwts0sin)( )8()(cos(),(cos()(cos(),(cos(1022001110220011twVytwVytwVxtwVx)9(1V2V1V2V2222112222111,1,1,1lVlVlVlV29第29頁(yè)/共45頁(yè) Substituting (9) into (8) and using the

49、relation (3), where the matrices are determined by ESRR system simulation results(depending on the users location), we can calculate the correlation coefficients as a function of interval between locations of legal user B and eavesdropper E. (The results are presented on Appendix 2 )From these resul

50、ts we can do the following important conclusions: 1. Correlation coefficients are changing by periodical manner depending on in the full interval (0, ) with the frequency propertional to (the radiated wave length). 2. It is can not be taken for granted that there exists some interval between legal u

51、ser B and eavesdropper E outside of which correlation is less than some threshold, that could provide in turn a large probability of bit key error for E. (See slide 26). We can say only about a probability of such event.These results somewhat contradict to a very optimistic conclusion presented in 7

52、. ,RRR),(30第30頁(yè)/共45頁(yè) In order to find a way out from this situation we propose to use antenna diversity. Then legal user B has m omnidirectional antennas which are randomly located in some area around of his presence. (The radius can be chosen of order , where is the length of radio wave used for co

53、mmunication) The protocol of key sharing has to be slight changed: The user B selects randomly one of m antennas and use it for a receiving and transmiting a series of packets.We can claim that if the probability of a random event is that the key bit error probability for E is at least for each ante

54、nna , then the probability that after “m” consequtive chosen antennas we get in all cases the probability less than , is less than . (See Table 1.) 0PriskP310PriskP第31頁(yè)/共45頁(yè)The probability (in percentages ) of the occurrence thatfor all points of eavesdropper presence at line between A and B dNumber

55、 of receiving antennas Number of receiving antennas123123h1=3m h2=3mh1=4m h2=2m/27.8 / 3.46 / 2.54.2 / 1.79 / 4.78.9 / 4.77.8 / 4.14.9 / 22.4 / 18.5 / 4.58.5 / 4.52 3 / 0.91.5 / 0.58 / 4.48 / 4.44 1.4 / 00.5 / 06.3 / 2.46.3 / 2.4Apathes 1, 2l1 =25 meters0 90 95( ,). /( ,). E(Path 1)(Path 2)1122B1 An

56、t 3 Ant2 Antddh2h1Table 1.第32頁(yè)/共45頁(yè)2.6. Privacy Amplification Theorem for local binomical channel.mnNn0ep0ep0ep0Ppe12m2ln210tlNIwhere is the total number of bits,n is the length of single substring,m is the number of substrings equal to the number of antennas,If legal channel is noisy with the error

57、 bit probability , then in order to correct errors we have to send over noiseless channel check bits, where . Then the inequality (10) has to be transformed to the following:mnN)1 (log20202PPnNtmP)10()(mPNhr )1 (log)1 (log()(22xxxxxh2ln210rtlNI)11(33第33頁(yè)/共45頁(yè)We can optimize the parameters n and N gi

58、ven and . The results of such optimization procedure are presented in Tables 2. 0,Pm 0IParametersResultsI0PmP0mnNRk25610-900,053198959670,043599450,02610198900,01312810-90,053110133030,039555050,02310110100,01225610-60,053192057600,044596000,02710192000,01325610-90085550050,05110100100,

59、02625610-90,2351515450,166525250,1011051500,05012810-90,1355416620,077527700,0461055400,023Table 2. Results of parameter optimization 34第34頁(yè)/共45頁(yè)For noisy legal channel with bit error probability the results of parameter optimization are presented in Table 3. 210mPParametersResults I0PmP0mnNRk 25610

60、-910-20,0533978119340,021511930596500,00410 12810-90,053220166030,01956599329950,0041025610-60,0533840115200,022511514575700,0041025610-900645172286100,030106186618600,00425610-90,2359217760,144565732850,0781090690600,02812810-90,1374022200,058595347650,027103422342200,004Table 3. Resul