Cryptanalyzing an image encryption scheme based on logistic map

0.38 MB·English

by  Chengqing Li

Nonlinear Dynamics manuscript No.
(will be inserted by the editor)
Cryptanalyzing an image encryption scheme based on
logistic map
Chengqing Li · Tao Xie · Qi Liu
3
1
0
Received: Jan 17 2013
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n
a1Abstract Recently,animageencryptionschemebased 27hotresearchtopic inmultimedia processingareainthe
J2onlogisticmapwasproposed.Ithasbeenreportedbya 28past two decades.
73researchgroupthat its equivalent secretkey can be re- 29 Because there are some subtle similarities between
1
4constructed with only one pair of known-plaintext and 30chaos and cryptography,chaos were used to design po-
] 5thecorrespondingcipher-text.Utilizingstabledistribu- 31tential secure and efficient schemes in all kinds of ap-
R6tion of the chaotic states generated by iterating the lo- 32plications in cryptography. Due to easy presentation
C7gistic map, this paper further demonstrates that much 33of image datum, most cryptographic schemes consider
s.8more information about the secret key can be derived 34image data as operation object [6,16,21]. According
c9under the same condition. 35to the record of Web of Science, about two thousands
[
v1 1110Kimeaygweoenrdcrsypctriyopnta·nloaglyisstisic·mchaposen-plaintext attack · 3367oinf tahreticplaesstotnwochdaeocsa-dbeass.eUdncfroyrptutonlaotgeyly,wmeraenypuobflitshheemd
9 38have been found to have security problems of differ-
8 39ent degree from the viewpoint of modern cryptology
0 40[?,2,3,7,9–12]. In general, cryptanalysis (breaking or
1 Introduction
412 41reporting some security defects) of a given encryption
.
1 42scheme requires rigorous theoretical support, so only
013With the rapid development of information transmis- 43aboutonetenthoftheproposedchaos-basedencryption
314sion technology and popularization of multimedia cap-
44schemes were reported being cryptanalyzed till now.
115ture device, multimedia data are transmitted over all
v:16kinds of wired/wireless networks more and more fre- 45At the same time, the conventionalcryptanalysis skills
46cannot be directly used to break the chaos-based en-
i17quently. Consequently, the security of multimedia data
X 47cryption schemes due to some reasons, such as differ-
18becomes a serious concern of many people. However,
r 48ence of essential structure, and some special security
a19the traditional text encryption schemes cannot protect
49defects caused by the properties of the adopted chaos
20multimedia data efficiently, mainly due to the big dif-
50system[5,13,23].Therefore,shortofsecurityscrutinyof
21ferences between textual and multimedia data, such as
51the proposed chaos-based encryption schemes has be-
22strongredundancybetweenneighboringelementsofun-
52come bottleneck of progress of chaos-basedcryptology.
23compressed multimedia data, bulky size of multimedia
53Some general rules about evaluating security of chaos-
24data,andsomespecialrequirementsofthewholemulti-
54based encryption schemes are concluded in [1,14].
25mediaprocessingsystem.Thischallengestirsthedesign
26of special multimedia encryption schemes to become a 55 Logisticmapisadiscretequadraticrecurrenceform
56of the logistic equation, a model of population growth
ChengqingLi,Tao Xie, Qi Liu 57first published by P. Verhulst in 1845. The application
MOEKeyLaboratoryofIntelligentComputingandInforma- 58oflogisticmapincryptologycanbetracedbacktoJohn
tion Processing, College of Information Engineering, Xiang-
tan University,Xiangtan 411105, Hunan, China 59von Neumann’s suggestion on utilizing it as a random
Tel.: [phone]numbergeneratorin 1947[22].Due to simple formand
Fax:[phone]relativelycomplexdynamicalpropertiesoflogisticmap,
E-mail: [email protected] 62itwasextensivelyusedtodesignencryptionschemesor
2 C. Li etal.
63pseudo-randomnumbergenerator[4,8,19].EveninWeb 109 RXkey(i,j)=Xkey(3k mod4),
64of Science, one can find more than one hundred and 110 GXkey(i,j)=Xkey((3k+1)mod4),
65eightyrecordsonapplicationoflogisticmapincryptog- 111 BXkey(i,j) = Xkey((3k+2)mod4), where k =
66raphy published between 1998 and 2012. A few papers 112 iN +j.
67reported some security deficiencies caused by logistic b) Iterate the standard map
68map [15,18,18].
x =(x+Ksin(y))mod(2π),
69 In [17], an image encryption scheme based on the
(cid:26)y =(y+x+Ksin(y))mod(2π),
70logistic and standard maps was proposed, where the
71twomapsareiteratedtogeneratesomepseudo-random 113 from the initial conditions (x0,y0) L times to
72number sequences (PRNS) controlling twice exclusive 114 obtainanewchaoticstate(x′0,y0′).Then,further
73OR (XOR) operations. In [20], it is reported that an 115 iterate it MN more times to get MN chaotic
74equivalentkeyoftheschemecanbeobtainedfromonly 116 states {(xi,yi)}Mi=N1.
75one known/chosen plain-image and the corresponding c) Iterate the logistic map
76cipher-image. However, the equivalent key can only be
77used to decrypt other cipher-images of smaller or the z =4z(1−z) (1)
78samesizeoftheknown/chosenplain-image.Basedona
′ ′
79dynamical property of logistic map on stable distribu- 117 fromtheinitialconditionz0 =((x0+y0)mod1)
′
80tionofitschaoticstates,thepresentpaperre-evaluates 118 for L times to get a new initial condition z0.
81the security of the scheme, and finds that the scope of 119 Then,furtheriterateitforMN timestogetMN
MN
82all the sub-keys can be narrowed much by an efficient 120 chaotic states {zi}i=1.
d) Generateachaotickeystream(CKS)imageI =
83brute-force search. CKS
{(CKSR(i,j),CKSG(i,j),CKSB(i,j))}
84 The rest of this paper is organized as follows. Sec- 00≤≤ij≤≤MN−−11
85tion 2 introduces the image encryption scheme under asfollows:CKSR(i,j)=⌊256xk/(2π)⌋,CKSG(i,j)
86study briefly. Our cryptanalytic results are presented =⌊256yk/(2π)⌋ and
87in Sec. 3 in detail. The last section concludes the pa-
88per. CKSB(i,j)=z˜k =⌊256zk⌋, (2)
121 where k =iN +j+1.
122 – Encryption procedure:asimpleconcatenationofthe
2 The image encryption scheme under study
89 123 following four encryption operations.
90Theplaintextencryptedbytheimageencryptionscheme 124 – Confusion I: masking the plain pixel values by
91under study is a RGB true-color image of size M ×N 125 the four XORing keys {Xkey(i)}3i=0. For k =
92(height×width), which can be denoted by an M ×N 126 0,⋆...,MN −1, set
R (i,j)=R(i,j)⊕R (i,j),
93matrix of 3-tuple pixel values I = {(R(i,j),G(i,j), Xkey
94B(i,j))}0≤i≤M−1. Denoting the cipher image by I′ = G⋆(i,j)=G(i,j)⊕GXkey(i,j),
95{(R′(i,j)0,≤Gj≤′(Ni,−j1),B′(i,j))}0≤i≤M−1,theimageencryp- B⋆(i,j)=B(i,j)⊕BXkey(i,j),
96tion scheme can be described0≤aj≤sNf−o1llows1. 127 where i=⌊k/N⌋, j =(k modN).
128 – Diffusion I: scanning all pixel values from the
97 – Secret key: three floating-point numbers x0, y0, K, 129 first one row by row (from top to bottom), and
98 and one integer L, where x0, y0 ∈ (0,2π), K > 18, 130 masking each pixel (except for the first scanned
99 100<L<1100. 131 pixel) by its predecessor in the scan.
∗ ⋆ ∗ ⋆
100 – Initialization:preparedataforencryption/decryption132 Set R (0,0) = R (0,0), G (0,0) = G (0,0),
∗ ⋆
101 by performing the following steps. 133 B (0,0) = B (0,0). For k = 1,...,MN − 1,
102 a) GeneratefourXORingkeysasfollows:Xkey(0)= 134 set
∗ ⋆ ∗ ′ ′
103 ⌊256x0/(2π)⌋, Xkey(1)=⌊256y0/(2π)⌋, Xkey(2) R (i,j)=R (i,j)⊕R (i,j ),
104 =⌊K mod256⌋, Xkey(3)=(Lmod256). Then, G∗(i,j)=G⋆(i,j)⊕G∗(i′,j′),
105 generate a pseudo-image IXkey = {(RXkey(i,j), B∗(i,j)=B⋆(i,j)⊕B∗(i′,j′),
106 GXkey(i,j),BXkey(i,j))}00≤≤ji≤≤WH−−11 byassigningan 135 where i = ⌊k/N⌋, j = (k modN), i′ = ⌊(k −
107 M×N matrixwiththefourXORingkeysrepeat- 136 1)/N⌋ and j′ =((k−1)modN).
108 edly: 137 – Diffusion II: scanning all pixel values from the
1 To make the presentation more concise and complete, 138 last one column by column (from right to left),
some notations in the original paper are modified provided 139 andmaskingeachpixel(exceptforthefirstscanned
thatits essential form keptunchanged. 140 pixel) by its predecessor in the scan.
Cryptanalyzingan image encryption schemebased on logistic map 3
∗∗ ∗
141 Set R (M − 1,N − 1) = R (M − 1,N − 1), 170is smaller than or equal to M ×N, encrypted by the
∗∗ ∗ ∗∗
142 G (M−1,N−1)=G (M−1,N−1),B (M− 171same secret key (x0,y0,K,L). In the following, we will
∗
143 1,N −1) = B (M −1,N −1). For k = MN − 172show that the scope of x0, y0, K, and L can be further
144 2,...,0, set 173narrowedunder the same condition.
∗∗ ∗ ∗∗ ′ ′ ∗∗ ′ ′
R (i,j)=R (i,j)⊕G (i,j )⊕B (i,j ), 174 The key idea of the enhanced attack is to search
G∗∗(i,j)=G∗(i,j)⊕B∗∗(i′,j′)⊕R∗∗(i′,j′), 175thevaluesof{Xkey(i)}3i=0 bybrute-forceandverifythe
∗∗ ∗ ∗∗ ′ ′ ∗∗ ′ ′ 176search with stable self-correlation of the blue channel
B (i,j)=B (i,j)⊕R (i,j )⊕G (i,j ),
145 where i = (k modM), j = ⌊k/M⌋, i′ = ((k + 177of ICKS, {CKSB(i,j)}00≤≤ij≤≤MN−−11, which is obtained from
146 1)modM), j′ =⌊(k+1)/M⌋. 178states of logistic map. Two main points supporting the
147 – Confusion II: masking the pixel values with the 179attack are described as follows.
148 CKSimagepixelby pixel.Fork =0,...,MN− 180 – Deterministicrelationshipbetween{Xkey(i)}3i=0and
149 1, set I∗
R′(i,j)= R∗∗(i,j)⊕CKSR(i,j), 118812 OXbkveyiously, every element of I∗Xkey comes from set
′ ∗∗
G(i,j)= G (i,j)⊕CKSG(i,j), 183 {Xkey(0),Xkey(1),Xkey(2),Xkey(3),Xkey(0)⊕Xkey(1),
B′(i,j)= B∗∗(i,j)⊕CKSB(i,j). 184 Xkey(0)⊕Xkey(2),Xkey(0)⊕Xkey(3),Xkey(1)⊕Xkey(2),
150 where i=⌊k/N⌋, j =(k modN). 185 Xkey(1) ⊕ Xkey(3),Xkey(2) ⊕ Xkey(3), Xkey(0) ⊕
151 – Decryption procedure is the simple reversion of the 186 Xkey(1) ⊕ Xkey(2), Xkey(0) ⊕ Xkey(1) ⊕ Xkey(3),
152 above encryption procedure. 187 Xkey(1)⊕Xkey(2)⊕Xkey(3), Xkey(0)⊕Xkey(2)⊕
188 Xkey(3), Xkey(0)⊕Xkey(1)⊕Xkey(2)⊕Xkey(3)}.
189 Given M, N and {Xkey(i)}3i=0, I∗Xkey is fixed. As-
1533 Cryptanalysis 190 signing {Xkey(i)}3i=0 with four different numbers,
∗
191 e.g., {1,2,4,9}, the construction of IXkey can be
∗
154In[20,Sec.4.2],ithasbeenreportedthatanequivalent 192 known.ScanIXkeyintherasterorder,andconvertit
155secretkey ofthe image encryptionscheme under study 193 into a one-dimensional sequence. Note that not ev-
156canbereconstructedwithonlyonepairofknownplain- 194 eryelementof{Xkey(i)}3i=0 haveindependentinflu-
∗
157imageandthecorrespondingcipher-image.Morerigor- 195 enceonIXkeywhenT =4,whereT denoteperiodof
∗
158ous presentation of the attack was represented in [12, 196 the one-dimensional version of IXkey. For example,
∗
159Sec. 3.1]. The equivalent form of the secret key can 197 the non-periodic component of IXkey is {Xkey(2),
160only decrypt other cipher-imageof smaller or the same 198 Xkey(2) ⊕ Xkey(3), Xkey(0) ⊕ Xkey(1) ⊕ Xkey(3),
161size of the known plain-image. In this section, we first 199 Xkey(0)⊕Xkey(1)} when M = N = 9, where dif-
162briefly introduce how the equivalent secret key is ob- 200 ferent combinations of Xkey(0) and Xkey(1) may
∗
163tained, then discuss how to derive further information 201 generate the same version of IXkey. Under condi-
164about secret key utilizing special dynamical properties 202 tion M = N, the relationship between T and N is
165of the logistic map. 203 shown in Proposition 1. Fortunately, from Proposi-
Denoting the horizontal and vertical diffusion pro- 204 tion 2, one can see that only a very minor portion
cesses by HD and VD, respectively, the image encryp- 205 of combinations of M and N making not every ele-
tion scheme under study can be represented as 206 mentof{Xkey(i)}3i=0 haveindependentinfluenceon
∗
′ 207 IXkey.
I =VD(HD(I⊕IXkey))⊕ICKS 208 – Self-correlation of {CKSB(i,j)}0≤i≤M−1
0≤j≤N−1
=VD(HD(I))⊕VD(HD(IXkey))⊕ICKS RecallEq.(2),onecancalculate∆k =µk−4,where
=VD(HD(I))⊕I ,
key
µk =(z˜k+1/256)/((z˜k/256)·(1−(z˜k/256))).
where
209 As well known, distribution of chaotic trajectories
∗
Ikey =IXkey⊕ICKS, (3) 210 generated by iterating the map Eq. (1) is stable.
211 To verify this, distributions of a great number of
and
212 chaotic trajectories generated by iterating Logistic
I∗ =VD(HD(I ). 213 mapEq.(1)for105 timesunderrandominitialcon-
Xkey Xkey
214 ditions were studied. All the distributions are quite
166As bothHDandVD areindependent onthe secretkey 215 similar to each other, so only one typical example
167andIkey isdependentonneithertheplaintextInorthe 216 is shown in Fig. 1. Furthermore, distribution of ∆k
′
168ciphertext I, one can see that Ikey can be used as an 217 is stable also under different values of z0. To verify
169equivalent key to decrypt any ciphertext of size which 218 this point, a greatnumber of initial values of z0 are
4 C. Li etal.
219 chosen randomly. For each value of z0, the corre- 101
220 sponding sequence {z˜k} is produced by transform-
221 ing the correspondinglogistical states with Eq. (2). Positive
222 As distributions of ∆k are similar to each other, 100 Negative
223 only the distribution of ∆k under the initial value
224 used in Fig.1 is shownin Fig. 2. To show the point 10−1
225 moreclearly,thedistributionof∆k oversomegiven y
226 intervalsareshowninTable1.Inaddition,distribu- 10−2
227 tion of ∆k is very sensitive to change of z˜k or z˜k+1
228 (See Figs. 2, 3, 4, 5). So, this stable self-correlation 10−3
229 of {CKSB(i,j)}0≤i≤M−1 can be used to verify the
230 search of {Xkey(0i≤)}j≤3i=N0−.1 10−40 0.2 0.4 0.6 0.8 1
x
Proposition 1 When M =N, T and N satisfies that
Fig.2 Thevaluesof∆k underdifferentvaluesofzk obtained
from oneinitial condition.
4 if (N mod2)=1;
T =2N if (N mod4)=0; (4)
 101
4N if (N mod4)=2.
231  100 Postive
Negative
232Proof This proposition has been verified by computer
233for N = 3,··· ,1204. We leave this proposition under 10−1
234the larger scope of N as a conjecture. y
Proposition 2 Thenecessaryconditionmakingnotev- 10−2
eryelementof{Xkey(i)}3 have independentinfluence
i=0
on I∗ is that 10−3
Xkey
|M −N|≤1. (5)
10−4
0 0.2 0.4 0.6 0.8 1
x
235
236Proof This proposition has been verified by computer Fig. 3 The values of ∆k under different values of zk when
237for 3 ≤ M,N ≤ 1204. We leave this proposition under z˜k+1 and z˜k are notright: z˜k isreplaced byz˜k+1
238the larger scope of M and N as a conjecture.
101
100 Positive
0.12 Negative
0.1 10−1
y
0.08 10−2
0.06
10−3
0.04
10−4
0 0.2 0.4 0.6 0.8 1
0.02 x
Fig. 4 The values of ∆k under different values of zk when
0
0 0.2 0.4 0.6 0.8 1 z˜k+1 and z˜k are notright: z˜k+1 is replaced byz˜k+1+1.
Fig. 1 Distribution of trajectory of the map Eq. (1) under
initial statez0=0.226. 1. Obtain Ikey via
′
I =VD(HD(I))⊕I;
key
239 Basedontheabovetwopoints,someinformationof
240the secret key, {Xkey(i)}3i=0, can be obtained with the 242 2. Searchthe valueof{Xkey(i)}3i=0,andgetthe corre-
∗
241following steps. 243 sponding version of IXkey;
Cryptanalyzingan image encryption schemebased on logistic map 5
Table 1 Thedistributionof ∆k corresponding totheinitial state z0=0.226.
z0 (−∞,−10) [−10i+1,−10i)−i=40 [−10−4,10−4) [10i,10i+1)0i=−4 [10,+∞)
0.326 0.0195 0.0474 0.1435 0.3216 0.0897 0 0.0267 0.0150 0.0672 0.1588 0.0709 0.0196 0.0202
0.761 0.0204 0.0481 0.1431 0.3177 0.0899 0 0.0274 0.0143 0.0678 0.1610 0.0709 0.0197 0.0198
0.539 0.0203 0.0469 0.1437 0.3222 0.0905 0 0.0270 0.0141 0.0672 0.1581 0.0705 0.0192 0.0203
0.487 0.0200 0.0480 0.1412 0.3230 0.0905 0 0.0265 0.0146 0.0660 0.1593 0.0716 0.0193 0.0201
0.194 0.0206 0.0491 0.1419 0.3199 0.0879 0 0.0276 0.0150 0.0666 0.1599 0.0719 0.0192 0.0204
0.875 0.0210 0.0484 0.1430 0.3201 0.0893 0 0.0271 0.0152 0.0662 0.1603 0.0698 0.0198 0.0199
0.942 0.0211 0.0483 0.1417 0.3217 0.0877 0 0.0270 0.0154 0.0663 0.1597 0.0712 0.0194 0.0205
0.293 0.0205 0.0479 0.1426 0.3201 0.0905 0 0.0267 0.0147 0.0662 0.1597 0.0725 0.0187 0.0199
101 225545cwohmerpele#xi(t·y)diserneodtuescecdartdoinOa(l2it3y6)o.fFaosreat.PInCtwhiitshcaCsPe,Uthoef
100 Positive 2562.83GHz and RAM of 2.98GB, the attack can be com-
Negative 257pletedwithinelevenminutes.Anumberofexperiments
258weremadetoverifythecorrectnessoftheaboveattack.
10−1 259Withthesecretkey(x0,y0,K,L)=(3.98235562892545,
y 2601.34536356538912, 108.54365761256745, 110), the out
10−2 261of the first stage of the attack is {160,54,108,108},
262{162,54,108,110}. Then, the 16 possible neighboring
10−3 263values of each set are verified further, the distributions
264of ∆k correspondigfor the second set are shown in Ta-
10−40 0.2 0.4 0.6 0.8 1 265ble 2. Obviously, the data shown in third row of Ta-
x 266ble 2 is closed to the data shown in Table 1 most. So,
Fig. 5 The values of ∆k under different values of zk when 267{Xkey(i)}3i=0 ={162,54,109,110}.Onecanfurtherob-
z˜k+1 and z˜k are notright atthesame time: z˜k and z˜k+1 are 268tainthat K =109mod256,L=110+n·256,n=1∼
replaced byz˜k+1and z˜k+1+1respectively. 2693,x0 ∈[81/64·π,163/128·π),y0 ∈[27/64·π,55/128·π).
244 3. Generat∗eestimatedversionof{CKSB(i,j)}00≤≤ij≤≤MN−∗−11, 2704 Conclusion
245 {CKSB (i,j)}0≤i≤M−1,fromthebluechannelofICKS =
∗ 0≤j≤N−1
246 Ikey⊕IXkey (See Eq. (3)). 271In this paper, the security of a new image encryption
247 4. Calculate the distribution of {∆k}Mk=N0−2, and out- 272scheme based on logistic map is re-analyzed in detail.
248 put the search value if the distribution match with 273Although it was reported that equivalent secret key of
249 the expected one (Refer to Table 1). 274the encryption scheme can be reconstructed with only
250 Once {Xkey(i)}3i=0 are determined, one can obtain 227765opnheerpteaxirt,otfhkisnopwapne-prlfauinrtthexert fiannddsththeactotrhreesspcoonpdeinogf acil-l
251K =Xkey(2)mod256,L=Xkey(3)+n·256,n=1∼4,
277the sub-keys can be narrowed further with relatively
252x0 ∈ [Xkey(0)/256 · (2π),(Xkey(0) + 1)/256 · (2π)) , 278lowcomputationunderthesamecondition.Theresults
253y0 ∈[Xkey(1)/256·(2π),(Xkey(1)+1)/256·(2π)). 279reported in this paper will help study security of other
Obviously, the complexity of the whole attack is
O(24·8·L), where L is the number of plain-bytes to be 280image encryption schemes based on logistic map.
calculated.ConsideringL=256is enoughfor counting
tahtteacdkistisribOu(t2io40n).oNf o{∆tekt}hMka=Nt0−th2,etchoemcpolmexpitlyexoitfythofetahte- 228812AtiocnkanloNwaletudrgaelmSecinetnsceTFhoisunredsaetairocnhowfaCshsuinpapo(Nrtoe.d6b1y1t0h0e21N6a)-,
283andScientificResearch Fundof HunanProvincialEducation
tack can be reduced much by dividing the attack into 284Department (No. 11B124).
the following two stages: 1) search for the possible val-
ues of {Xkey(i)}3 of regular interval with a weaker
i=0
matching condition; 2) verify the left possible values 285References
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Table 2 Thedistributionof ∆k corresponding to16 possibleneightouring valuesof {162,54,108,110}.
index [−10i+1,−10i)−i=40 [−10−4,10−4) [10i,10i+1)0i=−4
1 0.0629 0.1614 0.2716 0.0629 0.0000 0.0157 0.0078 0.1023 0.1811 0.1023 0.0275
2 0.0515 0.1666 0.2698 0.0238 0.0000 0.0357 0.0119 0.0753 0.2023 0.1230 0.0357
3 0.0476 0.1468 0.3373 0.0714 0.0000 0.0357 0.0119 0.0992 0.1626 0.0714 0.0119
4 0.0480 0.1600 0.3240 0.0480 0.0000 0.0240 0.0080 0.0720 0.2000 0.0800 0.0320
5 0.0557 0.1474 0.3187 0.0557 0.0000 0.0438 0.0119 0.0876 0.1434 0.1235 0.0079
6 0.0478 0.1673 0.2709 0.0557 0.0000 0.0239 0.0039 0.0876 0.2151 0.0956 0.0278
7 0.0632 0.1660 0.2885 0.0553 0.0000 0.0197 0.0079 0.0869 0.1699 0.1067 0.0316
8 0.0434 0.1778 0.2529 0.0395 0.0000 0.0316 0.0079 0.0869 0.2252 0.0909 0.0395
9 0.0634 0.1587 0.3055 0.0674 0.0000 0.0317 0.0079 0.0634 0.1587 0.1230 0.0158
10 0.0515 0.1666 0.2698 0.0714 0.0000 0.0198 0.0000 0.0753 0.2182 0.1031 0.0198
11 0.0634 0.1507 0.3492 0.0714 0.0000 0.0079 0.0079 0.0753 0.1468 0.0873 0.0357
12 0.0396 0.1507 0.3253 0.0595 0.0000 0.0277 0.0079 0.0873 0.1904 0.0793 0.0277
13 0.0632 0.1581 0.2964 0.0790 0.0000 0.0079 0.0039 0.0909 0.1620 0.1027 0.0316
14 0.0478 0.1513 0.3067 0.0438 0.0000 0.0358 0.0079 0.0756 0.1713 0.1314 0.0239
15 0.0553 0.1699 0.2885 0.0830 0.0000 0.0276 0.0039 0.0750 0.1818 0.0909 0.0197
16 0.0517 0.1713 0.2868 0.0637 0.0000 0.0239 0.0000 0.0597 0.2071 0.1075 0.0239
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