HAL Id: hal-00235464
https://hal.archives-ouvertes.fr/hal-00235464
Preprint submitted on 4 Feb 2008
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
SYNCHRONIZATION CHAOTIC COMMUNICATIONS WITH CLOSED
CRYPTOGRAPHIC SYSTEMS
Yahia Moussa, Skander Aris, Malek Benslama
To cite this version:
Yahia Moussa, Skander Aris, Malek Benslama. SYNCHRONIZATION CHAOTIC COMMUNICA- TIONS WITH CLOSED CRYPTOGRAPHIC SYSTEMS. 2008. �hal-00235464�
"#%$!& '( )'(*+, -$./%0
132547698;:<:<=?>@?AB2DC3EGFH:G>9>@94I2JLKNM;:POQ=9R=?>9>
SGTVUGWBXZY\[^]VUN_`[ba9c?dfegUNhi[^YPaj_9klh`mPn9oikpqUNe5rV_9ktsuUGYimPkD[^v9n5wfxy{z|n9}*et~BUGYPkHX
S9SGdeHUNh[^Yaj]3XG~{_BUi[^kmP]XN_ [UNegUhNa{]V]?_9kHhNX[kga9_{m-eHXGBa9YPX[^a5Y^v9ngnBfaj_Bm\[XG_[kg_9U?|999n9}*et~BUGYPkHX5
C3;:PEZ=9GP2
_[^9kmWBXGWBUGY[^ UIXNBXGW5[^ktsuUmv _BhN5YPa9_9kgX[kgaj_eQXi VkD[^XmPkg_9~{eHUsjXGYPkQXGBeHUIha BWBegk_9~k^m
9UisBUNega9WBUN [a¡UG_{m^BYPUWBXGY\[^kgXNe¢mv _BhN5YPa9_9kgX[kgaj_+a9cL[Q3a#_9a9_ kgBU_[^kghXNe5v9_BXN]kQhXNe¢mvum\[UN]mZ
£
UL_9UNUNm¤a{]VULhGYv9W5[aj~9Y¤XGW99kghLhNa9_9hNUZW5[Pm[^a.mPUNhBY¤ULhGa5]]3u_9kHhNX[kga9_{mi
¥Ki¦9§V69EZ¨;:G©
ª9XNa?mi«x¬XZY\[^kgXNe;jv _Bh|YPa9_9kX[klaj_?«|}*9XGW5[^ktsuULfaj_[^Ya9e«9®¢elajBXNe;jv _BhN5YPa9_9kgX[kgaj_u
¯29°iM?PEZ69¨;8;GPFQ69M
ªBXNa?m-9X`m-BUUN_[^9a9Ya5B~99etv'm^[ujklUNa5sBUGY¬[^9ULWBX`m^[±[HVa.9UhNXN9U`m³²<´Zµ\B¶P5}·hG9XNa5[^kgh!mvum\[UN]¸k^m
X _9a9_9eQkg_BUNXZY9U[UZY]Vkg_9km\[kHh¹mvum\[UN]º[^BXi[5kmWBeHXv·hNa9]VW{eHUG»Bn*_9a{kmPUµ¤egkg¼BU½XN_9½B_5W9YPUjkHh[XBe¾U
jUN9XGsBkga9Yi-¿% UUG»|[^YPUG]VUmPUN_?mPkD[ktsBU9UGWBUN_99UN_ hNU'BWBaj_ kg_BkD[kHXNe³hNa9_99kD[kla{_{m3X_ [^ Umv?m^[UN]'À^m
WBXZYPXN]VU[UGY¢sjXGYPkHX[^kga9_?mVXGYPUW9YPaj]Vkg_BUN_`[hG XGYPXNhi[UGYPkQm\[kgh5mVa9c*hN9XNa5[^kgh'BUN9XGsBkga9Yi-¿¬9UGYPUNc^a9YPUNn%[^ U
kH9UNX[^BX[³[a hG9XNa`[^klhmv?m^[U].mÁm^[XGY\[^kt_|~#cYPaB]XGYBkg[^YPXY^v kg_9kD[kQXNe³hNa9_99kD[kgaj_?mÂa?mPhNkgeQeHX[Uk_ X
mv _BhN5YPa9_9kt|UN']XN_9_9UGY³9X`mfuUNUN_mPUUN_X`m³kQeHegaj~BkHhNXNe-XN_9' U_ hU!kg]VWBaÃmqmPklueHUNnÃu_`[kge±[^ UxÄUNhajYPX
XZ_9.fXGYYPaBee^Àmfa5;mPUGY^suX[^kga9_²Å9¶ªkg_!9kHh.kÆ[¬VX`m³mP9a5_V[^9X[b[a'hZ9XNa`[^kghVmv?m\[UN].mfkg_![^ U!mPaBµ
hXegeHUN½]X5m\[UGYPµPmPegXGsBUhaj_9ckg~5BYPXi[^kgaj_¡hXN_#BU]XN[aÇmv9_BhN5YPa9_9kt|Ua9_9etvÇkQc[^ UhNa9_99kD[kga{_9Xe
È vBXZW5?_ a sÉUG»9WBa9_9UN_`[Pm¡a9c[^BU¸mPeHXsjUÊmv?m^[UN] XGYPU¸XNege_9UG~BXi[ktsBU|{kg_BhNUI[^9kmÇc\kgY<m^[3ajY¼Bn
XZW9W{egkHhNX[kga9_?m³ajchN9XNa?mLmv9_BhN5Y¤aj_9kgX[kga9_[^am¤UNhBYPUVhNa9]V]?_9kHhNX[kga9_{m³9XGsBUBUNUN_UG»5[UZ_?mPklsuUNeQv
m^[u5kgUN9vV]X_5v!X9[^9a9Y<mPnBXG]Va9_5~[^ U]²HËBµy9¶<
Ì _ Ua{c[^9Ukg]VWBajY^[XG_`[-kQmim^uU`mfkg_XN_9XNev Ua{cªmv _Bh|YPa9_9kX[klaj_Ía{c- v _BXN]!kHhXNe¬mv?m\[UN]m
km [^9UXN9XGW5[^ktsuU¸mv9_BhN5Y¤aj_9kt|X[^klaj_Îa{chN9XNa5[^kgh v _?XN]VkHhNXNemv?m^[UN]mÏ'Ba{]3UIWBXGW?UZY<m+9XGsuU
9UZYktsjUN'XN_XN9XZW5[ktsBUm¤hBUN]U!c^a9YLmv9_ h|YPa9_9kX[klaj_kg_[^9U!c^UUN9µBXhG¼Ðajc¬[^9U¢[^YPXG_BmP]kD[[UZYXZ_9
Y¤UNhNUNktsBUGY-hN9Xa [^kgh¢mvum\[UN].mVkD[^V]Vkm<]X[hZ9UNWBXGYPXN]U[UGYqmf²<´Z9µG´9´Z¶P
_[^9kmWBXGWBUGYVU½VkQeHe.W9YPU`mqUN_`[XZ_9a5[^9UGY XG_9ÊmPkH]WBegU¹3XGv[aÑmv _9hN5YPa9_9kgU¹[3a
kH9UN_`[kghNXNefhN XNa [^klhmvum\[UN]m¢sBkHX.XN_9a5[^9UZYmPk~B_9XNea{c_Ba9_9kg9UN_`[kghNXNemvum\[UN]VnmPkg_|~ XN_XNBXGW5[^ktsuU
cUNUN5BXhG¼ hNa9_`[^Ya9eeHUGYϱÒbktYim\[f[^aXNhG9kHUGsBUWBXGY\[kgXNemv _Bh|YPa9_9kX[klaj_BUi[VUNUN_'[3a_ aj_9kl9U_[klh`Xe
mvum\[UN]m¤n3kÆ[kmU5m<mqUN_`[^kHXNe¢[aIc^k_B7X½]Ui[^9aB9ajega9~9v½[^9X[XNeelajm.[^BU+hNaBajYP5kg_ X[U¹m^{umPU[a{c
ha BW9egUÍmvum\[UN].
¿ 9km-W?XGWBUZYªkmfa9Y~BXG_9kgUNX mc^a{eQela mi _BUh[^kga9_!Bn{3U¢kg_`[^YPaB`uhNU[^9U¢_ a [^klaj_a{cbW?XGY^[^kgX`e
mv _BhN5YPa9_9kt|X[^kga9_.c^ajY_9a9_9µkgBU_[^kHhNXNe%mvum\[UN].mi5w X`mPU'a9_![^|km XGW9W9YPaBXNhG nu[^ UhNX`m¤U 9UGYPUaj_9eQv
XmPkg_9~{eHUhNaBajYP5kg_ X[U'kQmL[aBUmv9_BhN|YPaj_9kgUkm9U[XNkgeQUk_¡BUNh[kgaj_Å9n%[^ U'hNBXNa`[^kHh È ajY¤UG_9
mvum\[UN]XN_9 `uX*hNktYqhukD[ÄXGYPULXNemPa3[^ UN_3;m¤UNXmªXN_VUG»jX]VWBeHU [akegetbm\[^YPX[Uª[^9kQmª]U[^ a{9a9ega9~jvu
Òbkg_BXNeQegv'3U.9UN]Va{_{m^[^YPX[U[^9X[-[^9UV~{ega9BXNe mv _BhN5YPa9_9kgX[^kga9_BU[3UNUG_'[3akg9UN_`[kghNXNefmv?m\[UN]m
sBkHX#[^9UÇhNa9_9hUGW5[Ð]VUN_`[kga9_9UNk_Ó[^9UImqUNhaj_ m¤U`hi[^kQa9_kQmWBaÃmqmPkt?eHU#;mPkg_9~[^ Uhaj_ jkD[kga{_ X`e
È
vBXZW5?_ a sUi»9WBa9_BU_[Pm¬a9c?[^9U¢mPeHXGsBU¢mvum\[UN]ÓW9YPa9W9a?mPUN 9v*xÄUhNajY¤X fXGYYa9ee;²¾Å9¶<
B2-=9EiPFQ=9O;Aj¦uM;;EZ69M;F=5PFH69M 69E69M;FQ¨;KNM?PFHN=5O±A9¦u:PPKNR:<2
fa9_{mPkg9UGY[^ U [a!c^a9egega 3k_9~V_ akg9UN_`[kghNXNe?5v9_BXN]kQhXe;mv?m^[UN]m
=
= ÁG´iÂ
£
9UGYPU
∈ n ∈
_~BUN_9UGY¤Xeln[^ U5v9_BXN]kHh5mVa9c[^9U[3a½m\v?m^[UN]mVXGY¤U_ aj_9eHkg_BUNXGYVXN_9[^ X[[^ UGYPUkmV_9a+hGYaÃmqm
ha BW9egkg_9~ÊjU[VUU_[^9U[3a mPUi[Pm+ajc m^[X[UsBXGYPkHXGBeU`mÏkg_ÎajYP9UZY[^aÎXNhG9kHUGsBUI[^9UIWBXGY^[kgXe
mv _BhN5YPa9_9kt|X[^kga9_9n¢3U9UNc^kg_BUXÇhNaBa5YP5kg_9Xi[Um\B;mPU[Ða9c
!
"
"
∈
#
ngXG_9¸mPkg]kHeQXZYegv $
% &
' ∈
9kHhNY¤UGW9YPU`m¤U_[f[^9UhNa9ajY¤5k_ X[U`mL9kQhNYPU(`uktYPUmvB_9hN|YPaj_9kgXi[kga9_?{a'[^ UÍd)(?ÁZ´iÂ*hNXG_uU
Ui»9W9YPU`mim¤UNVX`m
**
+,+
+.-
*
-/+
-,-
* 0
1
1
2
0
2 0
1
1
2
0
2 0
1
1
2
0
2 0
1
1
2
0
2 3
4
4
5
3
4 3
4
4
5
3
4 3
6
6
5
3
6 3
6
6
5
3
6 787 788 787 78
8
=
=
=
=
Á\9Â
¿¬9UN_ nÃkgc 9 : 9
; → X`m ∞→< [^9Umv?m^[U]km-WBXGY^[^kgXNeQevmv _Bh|YPa9_9kUN?`[ U¢5v9_BXN]kQh*a9c=/>? hNXZ_
jU'mv9_BhN5Y¤aj_9kmPkg_9~VkD[^'[^ Ua9_9U.a9c B @.A kgcª[^9U.haj_[^Ya9egeQUNYLkQm XN99UN'[^a'[^9UhNa5BW9eQUmv?m^[U]n
m^uhN¢[^9X[udC(?Á\|Â-hNXG_¢jULUG»9W9YPU`mqmqUN¢9v[^9ULc^a{egega5Vkg_{~U(`uXi[^kgaj_BmED
FF
GHG G.I
F
I/G I,I
F J
K
K
L J
K
K
L J
K
L
J
K
K
L
J
L J
K
K
L
J
L M
N
N
O
N M
N
N
O
N M
P
Q
M
R
R
O
M
R M
R
R
O
M
R STS STT STS ST
T
=
=
+
=
=
ÁÅ5Â
¿-aV9U`mPkt~B_VXLhNa9_`[¾Ya9egeQUZY c^a9Y±[^ U¢mv?m\[UN]Vn5d)(?ÁÅ5ÂkQmÄYPUN5uhNUN¢[^a [^ UcXNegegaj3k_|~c^a9YP] D
!
"
#
#
$ "
#
$
"
#
#
$
"
$ %
&
&
'
& %
(
)
%
*
*
'
%
* +,, +,+
=
+
= Á\Ë5Â
¿¬9UZYPUNc^ajYPUn`[^9ULUGYYa9Y¬`v9_ XN]kHh5m c^a9Y±[^ U¢mv?m^[U]hXG_uUc^ajYP]¢ueHX[UNVX`m
-
.
/
-
.
.
/
-
.
.
/
-
/ 01120
3 4
5
6
4
7
7
8
4
9
9
8
4
: ;
<
;
< −−
= ÁPÂ
9UZYPUNn5[^9ULUGYYa9Y¬n = > =
?
@ −= |¿%9kQm%hNXN_¢[^9UN_¢jULU» W9YPU`mim¤UNVX`mED
A
B
C
A
C
A
C
D E
F
G
E
H
E
I −∆= ÁKJ`Â
9UZYPUNn Q P R MOL R LNM
S
R −=
∆
TUVXWZY\[]^`_bacWad\]^fe gh\i]WjY[
_3[^9kmhNX`m¤U¢a9_9evÍXmPkg_9~{eHU*hNa{a9YP5kt_BXi[ULajclkm/XG_9 okm¬[a!BUmv9_BhN|YPaj_9kn UN9n?XN_9[^9UGY¤Uc^a9YPU p
kmªm¤hXeHXGYÏ5¿¬9UG_*dC(VÁKJ5ÂÄhNXG_¢[XG¼BUª[^9Uc^a9YP] D
q
r
s
q
s
q
s
t u
v
w
x
u
y
u
y −+−= λ n ÁKz`Â
V9UGY¤U |~} {
λ+∆= nXG_ >λ [ÍkmhNeHUNXGYÐcYPa{] dC(?LÁ~z5Â[^ X[ K
→− nXZ_9
>
λ 3keHe±m^[XBkQegkg|U [^9UYPU(`?ktY¤UNVU(`?kgeHkl YPklu]
=
n
f\
È
Ui[ ;m³hNa9_{mPkH9UGY³XN_ÐU»jXN]!WBeHUVa{cªha BWBegk_9~ÐXf`uXmv?m^[UN]ÊVkD[^ÐX
È
a9YPUG_|.mv?m\[UN] ²<´iB¶nÃ[^ U
ª;X¢mvum\[UN]km¬~BktsBUN_9v¢[^ ULX9[a9_9a9]Va ;m%jkgc^cUGY¤U_[kHXNeBU(`uX[kga9_{m D
δ
α α α
−
=
+
−
=
−
− +
− +
−
=
ÁzÂ
XZ_9VX
È
ajY¤UG_5mvum\[UN]
¡
¢
£
¡
¤ ¡£¢£
¢
¤ ¢£
£
¤ ¥
¦
§©¨
¨
¨
¨ ¨¨¨
ª¨
¨ ¨¨
¨
−
=
−
−
=
−
−
= σ
ÁyÂ
3Uc^kl»V[^ UWBXGYPXN]VU[UGYsjXNeluUm-X5mED
«
¬
®
¯
°®
¯
±³²
´
±³²²
¯
µ
«
²
¯
´¶±
«
²
¯
· ¸
¹
º
=
=
=
=
=
=
= »
¼½¾
¿σ
δ α
α α
w a`[^mvBm\[UN].m-XGY¤ULhG9XNa5[^kgh5
c?3U VkmP¢[^ahGa5BW9egU À XG_Bà5¿¬9UG_
( )
+
−
−
− +
− +
= −
ααα
ÁG´Z`Â
XZ_9
−
−
= −
ÁG´9´iÂ
_¢[^9kQm-UG»BXN]VWBeQU [^ UYPUNc^UGYPUN_9hU ]VaB9UNe{km
!
"
δ
−
= ÁG´i9Â
XZ_9¢[^9UGY¤Uc^a9YPUª[^9ULhNa9_`[^Yaje?mPkg~B_9XNeB]3;m^[;BULXGW9WBeQkHUN¢[^a
#
" m\uhG¢[^ X[
$
%
&
$
& '
(
' ( )
*
+
,
,
- +−−= σ ÁG´ZÅ5Â
¿-aV9U`mPkt~B_m\uhNVXLhaj_[^Ya9egeQUZY-aj_ U_9UNUN{mb[a3¼9_ a .
∆ 9k¾hGVkg_¢[^9km-UG»BXN]VW{eHULkm
/
0
1
/
1 2
3
2
2 3
32 4
5
6
7
7
8
9
9
9 −−+−=−=
∆ δ σ
[^9ULUGYY¤a5YsBXGYkHXGBeHU :
;
< =
>
=
>
? @@
−
=
−
= [^`;mbdC(?ÁG´iË9ÂÄhNXG_¢jU YPkD[[UZ_VX`mED
A
B
C
D
D
E F
G
H
I
J
J
K
L −−−+−=
∆ λ σ δ σ ÁZ´NÂ
3kD[^ M
>
=σ
λ nkg_¢[^9kQm%hNX5mPUNnBXG_9VcYPa{]U(`uXi[kga9_VÁKz5Â3U¢mPU<[
N
O
P
O
Q
P RR
S T
U
T
V
W
X
Y δσ −−== ÁG´³J`Â
}mPkH]?eHX[kga9_m\[u5vVkQmªmP a _Vkg_3Òbkg~;;´
_ [^ U sBXNeQkH9XG[^kga9_W99X5mPU [^ ULhNa9_`[^YajegeHUGY-kQm
kg_BkD[kQXNeHevmVkD[hN9UNVa9c\c [ Z
\
] =^_
`
XG_9¢[^ U [aÍmPk~{_ Xemba
c
XZ_9 d
e
XGY¤U B_{mv9_BhN|YPaj_9kg|UN9n
9kQeHULX[u[gf¶z{m;[^9ULhaj_[YajegeHUGY-kmb[BY_9UNVa9_VXG_ [^ U [amPUNegUh[UNVhNa9a9YP5kg_9X[U`m Dha
c
XZ_9 d
e
(`ukghG¼BevÍmv9_BhN5YPa9_9kgU5
ÁG´iË9Â
"!#$%'&'(*)+)-,./0#1+32 45.-1674
c98
1:41;7" 4
e <
7="7
17%# >
?
@
A
B
?
C
D =
E F
G'H
F
G
I
G
J
K
G'L
F
G
I
G
J M
N == O
O P
:%Q%41R T U S
=
V'WYX[Z\Z^]._.` acb_d'efbdhgi].d'jkac]\l'mne\` o\pd'e\_Qqracb_d'esdct*bvu;dhwyxkze^b{_` ac]\l'mn|bz }s|
~vRi' f%%'c[k%'ii^ cRc..%- {r. =¡ -
$¢ c ''£%'h' c.r[5 $'{[¤$¢- ¥R¦ ''¢i'§5 v§' k¨h.¦ c ©
' ª'¢« ¬ ''' ª'. % $%'' .Q¢'¢0® $-i7 «%' ¯5~ª'°{%''5±
' {^{ª¬'²% $¢ c ''%£%0±' ³{ ´'9'¬- v% f$¢' c ''%£%'
± %'' f%°{ %'fµ\¯¶'5·'h '¸ v$r.Q¢ %%''i c ©ir =
{$'{$¢- ¦¹$¯"c¯.º-% ' % »$¢- ¼^[7s ' '''. ' % $¢-r·¹{[½§\¯-µ
%;¼¯c¾[r ¿s hhv% #Q±¿h v. À$¢ c ''%£%Á ' f' 'hY%
$¢ c ''%'Â[ f' c ©nÂ$¢- Ãr ' {$'{Â$¢-v Äi.f
''i% %¢Å '¸%' 7% *Æ»¢rcnc©ÅÇ È»'f.©nf$¢-rÃ
%©c¯
É;p_Ênz¦|{m^|b{z } Ëhz |Z\d'ek|z¦|{mn|bz }
V'W̦ÍÎ'a'}sZk].z
Ïi '. ' 'cÐ;% ' $%rR$¢-rfÑ'h% #v''hh c ©nh ' == '' f¸
{$'{ * ÁY' s=\{ Áf%%'ÒÂc¡$¢-f¯Ï³i '^{¿'iÓn'Ô¦$¢-v
hQ%'Òh'¨;hÆR' £'Õ$¢-v fR R' c%¦ 'i='. %'-¯
• Ö o\×\almn|bz }غ
Ù
Ú
Û ÚÙÜ
Ù
Û ÜÜÚÜÙÜÙÜ
Ü
Û Ý
Þ
Þ
ß
Ý
ßà
à ààà
à ààààà
à
δ
α α α
−
=
+
−
=
−
− +
− +
−
=
¹áâc¼
• ã dpz e^q;lmn|bä }sz¦å$x\p_Ênzæ
ç
è é
ê ë
ìí
î
ïð
ñ
òRó
ô
õ»ö
÷
øù
ú
ïRû
ü
ý
þÿ ÿ ÿ ! "#$ % &'ÿ&(#*) * ÿ+,ÿ- .0/ 1
2
3 4
5,6 798
/ :
;
< =
>,? 7@
AB
(C&D &(#*)EF ÿ+ÿC GD HIJ A0vÿK L AG* *) !
−
=
−
−
=
−
−
= σ
• ! #"#$%&(') +*,-. +/
0
0
0
0 0000
0 00
0
12
2
2
2 222
32
2 22
2 456
4
7 4656
5
7 566
7 8
9
−
=
−
−
=
−
−
= σ
:;
<)=?>A@-BDCFEGHIKJMLNEPO-=-LRQSO-=-EUTHVBNHWLNEYXDVZOMI[EH\@^]_OV`E=baMcDCKC)QdCA>`EeI[BDC\OgfJ-LNEh?iKBj>A@?>A@E
>A@-BkV#hK\O-IlJO=gE=m>nOXpoq@mf!H-cDCr\BsV#\tf!Bu>+v
wxO&h-EIKO-=dCy>AV#Hz>`E{>A@-B|Cqi}EUC#ELNE\t> ~
.
H=gh
HmCq>A@gElXyBsVzCA>qJHBkVO-XT!HV#BsHWLNE;C# pH=-h
.
H=-h
HC>A@gE?C#E\O-=-h?JHBsVOMXqTHV#B|HWLNEmCFXAO-V\O;fJMLsBs=;pv.w@-EV#EXAO-V#E}i}EU=-O;i@HTE}>`O;i
EV)V`OVTHV#BsHWL|EC#
−
= H=-h = − >A@bBDCIEH=C>A@-Ht>(>A@gE\OgfJ-LsBs=-
Xf!=-\t>`BsO-=MCrHV#E-
¡
k¢ £
¤ δ−= A¥¦;
H=-h
§
¨
©
§
© §
¨
©
§
© ª«
ªA«
ª
«
ªA« ¬
®
¯
¯
° ¬
®
¯
¯
° ±±±± ²²²²
+
−
−
=
+
−
−
= σ
σ y¥³
w@gEV#EXAOVZE
´
µ
¶
´
¶ ´
µ
¶
´
¶ ·¸
¹
·y¸
¸k·
·
¸
¹
·y¸
¸k· º
»
¼
½
½
¾
¿
¿
¿ º
»
¼
½
½
¾
¿
¿
¿ ÀÀÀÀÀ Á
Á
Á
Á
Á
−
− +
−
=
−
=
∆
−
− +
−
=
−
=
∆
σ δ
σ δ
 f!\@Ã>A@-Ht>.i}E\H=ÃiV)Bu>`E
Ä
Å
Æ
Ä
Æ
Ä
Æ
Ä
Æ Ä
Å
Æ
Ä
Æ
Ä
Æ
Ä
Æ ÇÇÈ
É ÇÇÈ
É Ê
Ë
Ì
Í
Î
Í
Ê
Ï
Ê
Ï Ê
Ë
Ì
Í
Î
Í
Ê
Ï
Ê
Ï ÐÐÐ
Ð ÑÑÑ
Ñ
−
−
− +
−
=
−
−
− +
−
=
δ σ
σ
δ σ
σ
<y>.B|Cr\LNEHVrXDV#OMI A¥Ò;
H=-hÃ>A@-EV`BkM@>`Óy@-H=-hUC#BNh-EOMXdÔÖÕdv )×;
>A@-Ht>>A@-E\ØO=>AV)O-LsL|EVzCrHV#E
Ù
Ù
Ú ÙÙÚ Û
Ü
Û
Ý
Ü Û
Ü
Û
Ý
Ü ÞßÞ
à
á
â
ã ÞßÞ
à
á
â
ã äää ååå
δ σ
δ σ
−
−
=
=
−
−
=
= y¥-æ;
w@gEe=XAOV>A@-BDCÖEGHIKJL|EØ >A@-ElCy>`HWBsL|BDC+BR=-ç\O-=m>AV#O-LsL|EVq\H=KW!EÃHJ-JL|BNEhl>`OPE=MCAfV#E{>A@gE{JHVA>)BsHL
C)Q-=\@;V#O-=-BkabHt>ABsO-=PWEt>Di}EE=K>A@-EÃ>A@-BkV#hP\OIlJO=gE=m> è
é
O-Xroê@f.HbcyCÖ\BsV#\tf!Bu>_H=-hë>A@gE{>DiKOPXABRV+Cy>
\OIJO=gEe=>#C ì
í.î
H=-h ï
ð.ñ
O-X>A@gEh;V#BòTEH=-hlV#EC)JO-=dC#EÃC)QdCA>`EIYCnV#EmC)JEØ\t>ABkT!EeLkQv
wxOóE=CyfV#E&MLsO;WHLC)Q-=\@bV#O=-BsaH³>)BsO-=óO-X>A@-Eôh;VBkTE^H=hõV#EmC)JO-=MCZEöC)QdCA>`EIUC# xi}E÷IFf.Cy>
:-:-¦-ÿõ
A¥-¥-
A¥Ò;
]xQHJ;fd=gOgT EGgJO=gEe=>#COX>A@-EõC#LsHT!EõC)QdCA>`EI HV#E HeLsL=gEHt>`BòTE vr<)=õ>A@-B|C\H;C#E?]_O-V#E=baMcAC
C#L|HTEÃC)QdCy>`EIøBDCMBkTEe=lW-Q,
−
=
−
−
= y¥
H=-hUC#Ol>A@-E\OV`V#EmC)JgO=gh³H\O;WBNH=IKH³>AV)BkG\H=lWEiVBj>`E
−
= −
y¥;
H=-hKHt>d>A@gEEÕmf!BRLNBkW-V#BkfdIøJOBs=>
=
>A@gEIUH³>AV)BkG ëW!EØ\OMIKEmCt
!
"#
$%&
−
= − '
( ) )
*
y¥×;
XDV#OMIiK@gEV#E-
[ ] , +
-
./0
+
= +
−2 1
3
λ
λ λ4
4
5
y¥d
H=-hÃ>A@-E\@HV#H\t>`EV)BDCy>`Bs\JOLkQg=-OMIB|HLOMXd>A@gE{IKH³>AVBkG
[λ7 −6 ] B|CBkTE=lW-Q 8:9<;
=
> =λ
? [ ]67 −λ @ ABADCC ++ λλ E y¥:;
iKEÃCZEt>
F
G
H =λ
I
m>A@-E=FiKE@HTE
J
KL
= +
= + M
N
O M
N
P
O Q
λ
λ R TVUS
<
−
=
<
−
= W
W
X
Y
Z [
λ
λ `Ò-¦m
\.V#O-I >A@-BDCõLNHmCA>PÔÖÕmf!Ht>ABòO=(>A@-Eþ>Di}OEBkE=;THLkf!EgC÷HV+E =-EdHt>ABkT!EmCZ Ã>[email protected]?>A@gE h;V#BkT!E H=-h
V#EmC)JO-=MCZE]_OVZE=;alC)Q!Cy>`EIPCr\H=ÃWEÃC)Qg=\@;V#O-=-BsaEhdv
]:^`_badcfehgjilknmporqfilgjstcuqwvDs
\pOgfVD>A@-Ó`OV#h-E³Vyxf=bEÓ{z;f->N>`HþIKE³>A@-OhB|CPf CZEhú>`O C#O-LòTE>A@gEúC)Q!Cy>`EI O-XPh-B|XAX)EV#E=m>`BsHmL
EÕmf!Ht>ABsO-=dC³v-<)=K>A@-BCEGHeIKJLNE MiKEKHmC³CAfdIE{>A@gHt>>A@-Elh;V#BkTEKH=-hUVZEmC)JO-=dC#EUC)QdCA>`EIYCêHV#E{>Di}O
B|h-E=>)Bs\ØHeLC)Q!Cy>`EIPCKEG@-BkWdBu>#C}H^\@-HO;>ABs\YW!E@gHØTBòOgfVKH=hö>A@-EQHV#EöCA>`HV>`EØhöilBu>A@hBsX)X`EV#E=>
BR=Bu>`BNHeL\ØO=ghBu>`BsOM=C³v
<)=|\pBs.v Ò^
H W \ H=gh
h
iKEPC)@-O;iü>A@-EPC)Q-=-\@bV#O=-BsaH³>)BsO-=ôOMXx>A@-El>A@;V#EØEmCÃCA>`Ht>`EKOMX
]_OV#E=;alC)QdCA>`EIUC# }\pBR.v
æ
C)@gOgiC >A@-EE³TOMLuf->ABòO=dCrO-XCy>`Ht>EÃC)Q-=g\e@bV#O-=-BRaeHt>)BòO=KEV)V#O-VzC³v
k:^ !"#$&% '"()% *
+
,
-
.
+
. /
0
/
0 == 12 3
3 4
'65(7%
8
9
:
;
< =
>
=
> == ?@ A
A 4
'"&(B% C
D
E
F
G H
I
H
I == JK L
L 4
'"M( N
O P
QSR
P TT U
VXWY[Z"\^]`_badcfegh"aijak[glnmpogqgrmsoutvivwxlbyuainhzqgrh"aijmy)y)ay7o
{[Z}|j~[ W~[
7b)[d)"7[["vxB"S[ Sx¡¢£¤¥¦[¤£§d7[}
7¨uB©sp b[¥ ¤¢[[¤¨X^7[§S["vxB"[ª[[[¤d«d¬[^)¤¥¢7uB
)¡®§¤¢)[jd7[}§¯7¨)Bª°[7¯)[p¨X¡®±¥S¯¤[®[}))xB¤©¡²¡³ª"¤[dB"
¤[© 7uB® u« )[x )[ ¡³ "B¤ )B B
§ d«¨"}"
7[[[n¤ ¡¢6) ) §bB¤ [^[[d$ )¯[[[¤[dB"p7¨)Bb ¥µ¤[B¶¥S
Sv)7[" ¤ S¨§ d«dB£)[©"["[ S¤"xBv±¡®""±7bn[[b )[ª¥
jb¨¥«d j)[ µx)7 xB¥° [[myuk)ay qxibwm«
[S`§¤ª®x)[¤©¡®d$¤[®[}))xB¤©«7"n d«®[¤sSª7¨)Bbª ° ®[[6«¤
)[x)[±)«[duB¤©§©vX®BxB¥³«ª«[¯^«[xB"S°
"!$#%!'&!!
(v° pb*)®°,+.-®[0/ °213)v¤£ Sv)SX)¶ªSv7"v««54r)"© 7¨)B §
6760698:<;=>@?BADCB:EADFCG8H>IEJK>8L? \MON%POQ7SRPTO4URPP¨°K(PPT¨°
V°Wp[,)®°K Sv)7[ [¤ 7[[["vxB"[ sb¨ YX ¶"[[[ ¤[uB®d´
6sx°$6s°f[Z^[¥°\¢«))[¨°](PP^}°
T¨°`_ O$°$a °'+b 77[" °c$°dbn[[nx)"[©"®d)±7uBª°_feEJg>E?ihkjDl?nm*jc8o8u°
p[\[Kq'V0(4rq'Vs¨°](PPt}°
s°f_nd§uµ°v [dB"d««³Sv)7[s[ `j[ LjSv)[""[wBd ¤S$°xyeEJ]>z?nhkjl
m7j'8{8B?| (^t"sV0(@4sVKq°K(PPOV°
}° «}®uµ°[a °~+1 [[b"®°~ U{ "x«X"®}µbvBx)"[µ 7bn[[v"[
p¨ ¡³6)®[xB"[} B S®^«¨[7"[[GU yeJK>E?'hkjDl?m7j'8o8)°'K'N$(EQ7SR}'4%Rqn5(PPT
R [ [¤ d«%f ¯ ¡¢®B 7[§S[v±)¶¯7¨)Bbª¬¡®6Bµ[xB"[
r x«§¦[®±«[vx)¶S 7¨uBªG,<=S?,?CFO:@A$;g8BC{O=3;=e;SO>zSNUROQ§Y(R,(P4$(ROVS^
(PPT¨°
^¨°u³§x¥Y+_ f)vZ °"U{)±[©[[} )[ )¶ 7[§S["vxB"[ ¡®f)
["xB"Sj)©[¯^«[xB"SGUxyeEJ]>@?[hjlym7j'8o8L?{[\ONGVK}'Q}~tOVKqc4r}~tT,(,(PP}
q°a ""«])®°+a "]°%H "}fB 7"µ^¨ 7bn[[nx)"[ª¬vx¡®^µ
¡®f)®[^`§¤®¤¶¥"[ª"}"[U77 O:;=>E?g¡CL:EAFC¢8H>£IEJK>¤8B?SNU^OQB¥(R,(^~4$(RTOs,(PPq°
P¨°a ""« ¦)¯°S+§-¢)Sbf¨°¦_ }vBS[ u¥Sk)[ }¶SbX7[§S["vxB"[
7uB© ¡®6B«[)«s®d)«®[[["6BdS<=g8B?S?,G©CFO:EA$;g8LCoO=ª;=«*e;SO>
SNUTOQ§^tT4U^,(EV[VSttT}°
(t¨°"-®¥¤[¨°¬²°ªU_¬u)"£7[nS["v7"[ [[[¤[v7" b b)" 7¨)Bbª ¥
¤[)¶¥S^[d)[" ¡³6)ª[7"[¯B©¢}¤[""[d«["¤ª[[[" 7¨)Bbª@U=g8B??]o
CFO:EA$;g8LCoO=¬;=e;SO>®SNUT[Q©(R,(P4(ROVS^VSttOV¨°
((v°f\¢v"7®°yd°67}b¥\ °,+¯6sL4°-¢S¥a °fa °±oj¤[B¶¥S 7[["vxB"S²)S
³ ¨"X¤ d«X§x«¨f 7uBªEU'<=g8B?'¬G¬?´CFO:@A$;g8BC{O=¬;=*e;SO>µ®SNU^OQ§(}~^P4
(RP^[VSttOV¨°