1
2
3
4
5
6
7
Claude E. Shannon (1916∼2001)
A Mathematical Theory of Communication,
B.S.T.J, 1948
8
9
10
11
12
13
14
M
M
C
C
K
K
M
C
K
M
C
K
15
M
M
C
K
M
C
K KP
C
KS
M
K
M
C
K KP
KS
C
K
16
M
M
C
K
M
C
K KP
C
KS
M
K
M
C
K KP
KS
C
K
17
M
M
C
K
M
C
K KP
C
KS
M
K
M
C
K KP
KS
C
K
18
ap−1
p p≡aa1 (mod
1a≤p−1
ap)
≤≡
p −11 (mod
ap−1 ≡ p)
1 (mod p)
p a
M
KM
K
CP(mod
KSKK
≤ a ≤ p −M1 C
ap−1
=K
1C
p)P KKPS KS
K K
P 5 KS
p=
p=5
14 = 1 = 1 (mod 5)
24 = 16 =114 =(mod
5)
4=
34 = 81 = 41 (mod 5)
2 = 16 =
4
4 = 256 = 1 (mod 5)
p a
1≤a≤p−1
1 (mod
1 (mod
34p−1= 81 = 1 (mod
a
= 1 (mod p)
44 = 256 = 1 (mod
5)
5)
5)
5)
19
p a
ap−1
p p≡aa1 (mod
1a≤p−1
ap)
≤≡
p −11 (mod
ap−1 ≡ p)
1 (mod p)
M
KM
K
CP(mod
KSKK
≤ a ≤ p −M1 C
ap−1
=K
1C
p)P KKPS KS
K KP
KS
b1 ̸= b2 a1 = b1 (mod p) a2 = b2 (mod p)
mod p) a2 = b2 (mod p)
b1 p)
(mod p)
a×1 a×1
= b1 =
(mod
a×2
=
b2 p)p)
(mod p)
a×2
=
(mod
a1 ̸= aa21 ̸=b1a2̸= bb21 ̸=a1b2= ba11 (mod
p) a
p)
= b1 (mod
= bb22 (mod
2p)= ba22 (mod
...
...
4
1==5bp12 = (mod
5)
mod1p)= pb1=
5
1 = b(mod
(mod p)
a×p −a×p
1 =−
bp−1
p−1 p)
24 5)
= 16 = 1 (mod 5)
(mod
ap)1 −aa22 == bb21 −
b2 =p)0 (mod p) b1 = b2
b1 (mod
(mod
(1)
4
3 5)
= 81 = 1 (mod 45) 4
(mod
1 = 11=
=p−1
11 =
(mod
5) 5)
1 p)(mod
p−1
4
a
=
̸
a
b
=
̸
b
a
=
b
(mod
a2 =· b·×(p−1))
p)= 1×2×·
a
(1×2×·
· ·×(p−1)
(mod
1
2
1
2
1
1
2 (mod
= 1×2×·
· ·×(p−1)
(mod p)
4 5)
= 256 = 1 (mod4 5) 4 a (1×2×· · ·×(p−1))
(mod
2 = 216== 116 =
(mod
5) 5)
1 (mod
20
φ(n)
a
=(p1=−−
(mod
n)−−1)1)
pq
=1)(q
p, qn =
= pq
(p1)(q
−
p, q np,=q pq
φn =
(p φ−φ(n)
1)1)(q
p−1
pp, qa na=
p)
pq≡ φ1 =(mod
(pp, −
1)(q
−
1)
q n = pq φ(n) = (p − 1)(q − 1) n = 1
φ(n)
= bn)
1 (mod p)
a
=a×1
1 (mod
M
C
K a×1
KP a×1
K
b1 p)
(mod p)
=a×2
b1S =
(mod
= b2 (mod p)
p, q a×2
n=
φ(n)
a×1
=
bb... 12 pq=
(mod
p)= (pp)− 1)(q − 1)
b
(mod
2
a×2
=
(mod
p)
φ(15) = 2 · 4 = 8
− 1)(q − 1) n = 15 = 3 · 5
... b2 − 1(mod
a×2...
=a×p
= bp−1 p)(mod p)
18 =1 = 1.. (mod 15) a×1
= (mod
b1 (mod
p) (1)
.
a×p
−
1
=
b
p)
p−1 15)
1 =+ b1p−1
p)
28 =256a×p
= 15 −
× 17
= 1 (mod
(mod
a×1
=
b
(mod
p)
a×2
=
b
(mod
p)
p−1
1
2
=
bp−1
a a×p
(1×2×·
· ·×(p−1))
= (mod
1×2×·
·×(p−1)
p) (1)
(mod p)
48 =65536
=−
15 1×
4369
+ 1..×
1 = 1 ·p)
(mod
15) (mod
(1)
.
a×2
== bb22 (mod p)(1)
8
a
−
a
=
b
−
b
=
0
(mod
p)
b
1
2
1
2
1
7p−1
=5764801 = 15 × 384320 + 1 = 1 (mod 15)
(mod p)
a×p
−
1... = b·p−1
(mod p)
p−1
a
(1×2×·
·
·×(p−1))
=
1×2×·
·×(p−1)
(mod
p)
.
a
(1×2×·
·
·×(p−1))
=
1×2×·
·
·×(p−1)
(mod
p)
.
. a1 ̸= a2 b1 ̸= b2 a1 = b1 (mod p) a2 = b2 (mod p)
p−1
a (1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)
(mod
b2 (mod p)
a×p
bp−1
2 = 0 p)
1=
a −aaφ(n)a=1=−b1 a(mod
−2 =b bn)
=1 −0 b(mod
b p)=−b1b=
21
pq
=1)(q
(p=−−
p, qn =
= pq
(p1)(q
−
p, q np,=q pq
φn =
(p φ−φ(n)
1)1)(q−−1)1)
p−1
pp, qa na=
pq≡ φ1 =(mod
(p − p)
1)(q
n − 1)
M
C
φ(n)
= bn)
1 (mod p)
a
=a×1
1 (mod
K a×1
KP a×1
K
B
=
, b2, · · p)
· , bφ(n)}
b1{b1p)
(mod
= bS =
(mod
1 = b (mod p)
a×2
2
n
p, q a×2
n=
= (p ·−· ·1)(q
− }1)
a×1
=
bb... 12 pqA=
(mod
p)
(mod
=bφ(n)
, abφ(n)
2{ab
a×2
=
(mod
p)
n
1 , ab2 ,p)
n B = {b , b , · · · , b .}
1 a×2
2
... φ(n)
=a×p
p)(mod p)
.. b2 − 1(mod
=
b
p−1
C = b1 × b2 × · · · × bφ(n)
B = {b1, b2, · · · ,. bφ(n)}
B =A{b=1,{ab
b2, ·,·ab
· ,.. b, φ(n)
}, ab }a×1
= (mod
b1 (mod
p) (1)
·
·
·
1
2
a×p
−
1
=
b
p)
φ(n)
p−1 p)
n
n
a×p
−
1
=
b
φ(n)(mod
p−1
a C ==Cb2(mod
n) p)
A = {ab1, abp−1
2 , · · · , abφ(n) }
a×2
(mod
1φ(n)
=
a a×p
· ·×(p−1))
1×2×· ·p)
·×(p−1) (mod p) (1)
A =C {ab
· · ·,−
ab
} bp−1 = (mod
2b, ·(1×2×·
= b11, ab
×
×
·
·
×
b
2
φ(n)
... = {b , b , · · · , b (1)}
B
=
{b
,
b
,
·
·
·
,
b
}
1
2
φ(n)
B
φ(n)
C = b1 × b2 ×a1· −
· ·a×
b b1 − b2 = 0 (mod p)1 b12 = b2
(1)
2 =φ(n)
8
φ(n)
p−1
C
=
b
×
b
×
·
·
·
×
b
2 C (mod φ(n)
1= =1
==1 b·p−1
(mod(mod
15) p)
n)
a×p
−
1
p−1 a 1a C =
(1×2×·
·
·×(p−1))
1×2×·
·×(p−1)
(mod
p)
A
=
{ab
,
ab
,
·
·
·
,
ab
}
φ(n)
a
(1×2×·
·
·×(p−1))
=
1×2×·
·
·×(p−1)
(mod
p)
1
2
φ(n)
A
{ab
,
ab
,
·
·
·
,
ab
}
n)
8=
1
2
p−1aφ(n) C = C (mod
φ(n)
a
=
̸
a
b
=
̸
b
a
=
b
(mod
p)
a
=
b
(mod
p)
1
2
1
2
1
1
2
2
2 =256
= 15 × 17 +(mod
1 = 1 p)(mod 15
a a(1×2×·
·×(p−1))
= 1×2×·
· ·×(p−1)
C =aC· −
(mod
n)
(mod
p)
b1 = b2
2· ·=×0bp)
8
b12−
×
a1 − 8a2 =1Cb=1a−2b1=b×2 b=
0 b·(mod
b
=
b
φ(n)
4 =65536
=2 15 × 4369 + 1 × 1 = 1
1
22
pq
=1)(q
(p=−−
p, qn =
= pq
(p1)(q
−
p, q np,=q pq
φn =
(p φ−φ(n)
1)1)(q−−1)1)
nn nee e dd
e× d×
1 φ(n))
(mod
d ee×
d =d11 =
(mod
(mod
φ(n)) φ(n))
a×1 φ =
= (p
b1 −(mod
p)− 1)
p,
q
n
=
pq
1)(q
d
e
×
d
=
1
(mod
φ(n))
B
=
{b
b
,
b
}
a×1
=
b
(mod
p)
1b,n
2 ,· ··e···a×1
φ(n)
=
b
(mod
p)
1
B
=
{b
,
,
,
b
}
1
B = {b1 1 ,2b2 , · · ·φ(n)
, bφ(n)
}
a×2 = b2 (mod p)
n , ·a×2
e· , ab d
ee××
dbd
==
1 (mod
φ(n))
n
e
1
(mod
φ(n))
.
a×2
=
(mod
p)
.
2
=
b
(mod
p)
A = {abB
,
ab
·
}
1 =2 {b1 , b2 , φ(n)
· · · },. 2bφ(n)}
A
=
{ab
,
ab
,
·
·
·
,
ab
1 1 , ab
2 2 , · · · φ(n)
A = {ab
, a×p
ab
}
... φ(n)
...
−· ·1 =
bp−1
(mod
p) p)
=
b
(mod
1
{b
,, bba×1
,
·
,
b
}
C = b1 ×B
b2B
×=
·=
· ·{b
× b1φ(n)
2 , · · · ,φ(n)
b
}
1
2
A
=
{ab
,
ab
,
·
·
·
,
ab
}
φ(n)
1
2
φ(n)
(1) e
e− 1 = b
C
=
b
×
b
×
·
·
·
×
b
1
2
φ(n)
a×p
(mod
p)
M
C
C
=
M
(mod
n)
a×2
=
b
C a=φ(n)bC1 ×
b
×
·
·
·
×
b
p−1
M C C = M (mod
a×p
−
1
=
b
(mod
p)
2
2
φ(n)
p−1
= CA
(mod
n)e 1 , ab2 , · · · , ab
p−1= {ab
}
φ(n)
d
a
(1×2×·
·
·×(p−1))
=
1×2×·
·
·×(p−1)
(mod
p) n)
.
M
C
C
=
M
(mod
n)
C
=
b
×
b
×
·
·
·
×
b
φ(n)
M
=
C
(mod
M
(1)
.
1
2
φ(n)
A
=
{ab
,
ab
,
·
·
·
,
ab
}
aφ(n)C = C n(mod
(1)
e dn) e 1× d .= 2
1 (mod φ(n)) φ(n)
n
e
d
e
×
d
=
1
(m
a
C = C a(mod
n)
1 − a2 = b1 − b2 = 0 (mod p) b1 = b2
C
=
b
×
b
·
·
·
×
b
φ(n)
d
e
×
d
=
1
(mod
φ(n))
115)
2d ×
φ(n)
a×p
−
1
=
b
(mod
p)
d
e
ed
kφ(n)+1
18 n=1aep−1
=
1
(mod
a
C
=
C
(mod
n)
p−1
p−1
(1×2×·
·
·×(p−1))
=
1×2×·
·
·×(p−1)
(mod
p)
B
=
{b
,
b
,
·
·
·
,
b
}
C
=
(M
)
=
M
=
M
=
M
(mod
n)
B
=
{b
,
b
,
·
·
·
,
b
}(
1
2
φ(n)
1
2
φ(n)
a 8(1×2×·
·
·×(p−1))
=
1×2×·
·
·×(p−1)
(mod
p)
n
e
d
e
×
d
=
1
C
=
b
×
b
×
·
·
·
×
b
a1 ×
̸= 17
a12 +b11 ̸==b212 a(mod
(mod p)φ(n)
a2 = b2 (mod p)
1 = b1 15)
8 2 =256 = 15
φ(n)
(1)
B=
,−b(mod
,M
· ·C
·=,b15)
b=
}
1 8 =1
= {b
1A
dC
1a
2a
φ(n)
(mod
n)
a
=
−
b
=
0
(mod
p)
b
=
b
C
·2· , abn) }
1 = {ab
2 1 , ab
1 2 , ·(mod
1A = {ab
2
, ab , · · · , ab
8
23
aφ(n) = 1 (mod n)
e
e
M C C =p,M
(mod
n) φ(n) = M
C
C
=
M
(mod
q
n
=
pq
(p
−
1)(q
−
1)
n
=
15 =
= 177147 = 11809 × 15 + 5 (mod 15)
d
1 n)
M Cn Ce = dM eM
(mod
M
=
C
(mod
n)
M
=
C
1
=
177147
=
11809
×
15
+
5
(mod
15)
e
×
d
=
1
(mod
φ(n))
n e d e × d = 1 (m
= M 3 = 343 = 22 × 15 + 3 = 3 (mod 15)
n= 1e(mod
d= 11)
e ×, bd ,=· ·1· ,(mod
φ(n))
=
4
×
11
+
1
B
{b
b
}
=
2 ·M4 3==8343 = 22Bn×=15e{b+1,3db=2, 3·e·(mod
1)(q − 1) n = 15 = 3 · 51 2Mφ(15)
· ,dbφ(n)
= 7 φ(n)
C=
×
=15)1} (m
x (mod p) eB==
15d1 ,=
b32768
,{ab
· · · =,, ab
b2184
}×
e
×
d
=
45
=
4
×
11
+
1
=
1
(mod
11)
e ==3C{b
215
φ(n)
15
+
8
=
8
(mod
15)
1 (mod p) M
A==
,
·
·
·
,
ab
}
AB=={ab
, · · · , ab }
1
2
φ(n)
{b11,,bab
2 , 2· · · , bφ(n)φ(n
3, · ·3 · , ab
=
{ab
,
ab
}8 x×
(mod
ggxxrr == yyrr (mod
p)
MAM
=
8
C
=
M
=
512
=
34
×
15
++
2 5=p)
2 5(mod
15)
1
2
φ(n)
15
15
a×1
=
b
(mod
p)
M
=
5
C
=
M
=
125
=
5
=
(mod
15)
M
=
C
/C
(mod
e
=
3
d
=
15
C
=
b
×
b
×
·
·
·
×
b
1=
470184984576
31345665638×15+6
6
M ==C C ==470184984576
=2=31345665638×15+6
=
6
C
=
b
×
b
×2,· ·· ·· ·×, ab
bφ(n
1
2
φ(n)
1
1
2ab
A
=
{ab
,
1
φ(
mod p) (mod
15+ 1 = 1 (mod 11)
e ×e15)
d×15)
=d45
=
4
×
11
M
=
C
=
32768
=
2184
×
15
+
8
=
8
(mod
15)
(mod
=×33b =×3y·×=
8g×
+xx b1(mod
= 1 (mod
11)
=rr b(mod
·
·
a×2
=
b
(mod
p)
od p)
p) C
C22 =
=C
My
p)
od
My
p)
(mod
p)
1aφ(n)
2
φ(n)
2
φ(n)
r
x
rx
x
r
n)
aC =Cb1=×Cb2(mod
n)
x C = 3CC (mod
mod p)
=
g
=
g
=
y
(mod
p)
3
×
·
·
·
×
b
3
M
=
C
/C
(mod
p)
e
=
3
d
=
15
1
15
φ(
26 C C
x=M8==
M
C
=
M
=
512
=
34
×
15
+
2
=
2
(mod
15)
.
1
=
=
216
=
14
×
15
+
6
=
6
(mod
15)
MM
=
6
=
M
216
=
14
×
15
+
6
=
6
(mod
15)
M
=
C
=
.
M
=
C
/C
(mod
p)
e
=
3
d
=
11
φ(n)
2
.
1
15
a C =M
=n)
470184984576
= 6(mod 15)
eCx×r=(mod
dC=r45
=
4 × 11 + 1 ==131345665638×15+6
(mod 11) φ(n)
x
rx
15
ra
x
15 = =
C
=xry=C
(mod
C
=
C
(mod
n) p)
32768
=2184
×
15
8 =8 8C
(mod
15)
M1M=
=x=gC C
=
×rp)
15
++
8 p)
=
(mod
15)
=
g
(mod
=
My
(mod
p)
y
=
g
(mod
rx8g32768
r2184
(mod
15)
1
2
8
C1 = g 1 ==1
g ==1y x (mod
(mod p)15) a×p − 1 = bp−1
x
1 =1(mod
= 1 p)(mod 15)
24
pq
=1)(q
(p=−−
p, qn =
= pq
(p1)(q
−
p, q np,=q pq
φn =
(p φ−φ(n)
1)1)(q−−1)1)
nn nee e dd
e× d×
1 φ(n))
(mod
d ee×
d =d11 =
(mod
(mod
φ(n)) φ(n))
a×1 φ =
= q(p
b1p,−q
(mod
p)
n
=
pq
φ
=
(p
−
1)(q
p,
n
=
pq
φ
=
(p
−
1)(q
−
1)
p,
q
n
=
pq
1)(q
−
1)
n
e
d
e
×
d
=
1
(mod
φ(n))
B
=
{b
,
b
,
·
·
·
,
b
}
a×1
=
b
(mod
p)
1
2
φ(n)
=a×2
b}1 (mod
p) p)
= {b
BB =
{b1,1b,2b, 2· ,· ·· a×1
·, b·φ(n)
, b}φ(n)
= b12 (mod p,
q= (p
n=
pq
φ−=1)(p −
p,
q
n
=
pq
φ
−
1)(q
n , ·a×2
e· , ab d
ee××
dbd
==
1 (mod
φ(n))
n
e
1
(mod
φ(n))
.
a×2
=
(mod
p)
.
2
=
b
(mod
p)
A = {abB
,
ab
·
}
1 =2 {b1 , b2 , φ(n)
· · · },. 2bφ(n)}
A
=
{ab
,
ab
,
·
·
·
,
ab
1 1 , ab
2 2 , · · · φ(n)
A = {ab
, a×p
ab
}
... φ(n)
...
a×1= b1 =(mod
b1 (m
−
1
=
b
(mod
p)
a×1
p)
p−1
a×1
=
b
(mod
p)
1
{b
,, bb2 ,,··· ·· ·, b,φ(n)
}
C = b1 ×B
b2B
×=
·=
· ·{b
× b1φ(n)
b
}
1
2
A
=
{ab
,
ab
,
·
·
·
,
ab
}
φ(n)
1
2
φ(n)
(1)
a×1
=
b
a×1
=
b
(mod
p)
e− 1 = b
C
=
b
×
b
×
·
·
·
×
b
1
a×2
=
b
(m
1
e
1
2
φ(n)
2
a×p
(mod
p)
M
C
C
=
M
(mod
n)
a×2
=
b
(mod
p)
a×2
=
b
C a=φ(n)bC1 ×
b
×
·
·
·
×
b
p−1
M
C
C
=
M
(mod
2
a×pn)− 1 = bφ(n)
2 p)
2(mod
p−1 (mod
= CA
...=
p−1= {ab
,
ab
,
·
·
·
,
ab
}
e ·1·×(p−1))
2
φ(n)
p,
q
n
pq
φ
=
(p
−
n
e
d
e
×
d
=
1
(m
a×2
=
b
a×2
=
b
(mod
p)
.
φ(n)
a
(1×2×·
=
1×2×·
·
·×(p−1)
(mod
p)
.
2
2
n
e
d
C
C
=
M
(mod
n)
=
b
×
b
×
·
·
·
×
b
φ(n) M C
.
(1)
.
=
·φ(n)
· · φ(n))
, abnφ(n)
}d e(1)
.
.= 21 ,(mod
aφ(n)C = CA
n(mod
e1 {ab
dn)2e 1×,dab
e
×
d
=
1
(m
.
.
a
C = C a(mod
n)
n ed 1.. =db e de
1 − a2 = b1 − b2 = 0 (mod p) ..b1 = b2 a×p −
p−1
C
=
b
×
b
×
·
·
·
×
b
8 n ep−1
φ(n)
d
e
×
d
=
1
(mod
φ(n))
1
2
φ(n)
C
=
(M
) b2=
a×p
−
1
=
b
(mod
B
=
{b
,
b
,
·
·
·
,
b
}
a×p
−
1
=
b
(mod
p)
B
=
{b
,
,
p−1
1 =1a=
1(1×2×·
(mod
15)
1
2
φ(n)
a
C
=
C
(mod
n)
p−1
p−1
1
·
·×(p−1))
=
1×2×·
·
·×(p−1)
(mod
p)
B
=
{b
,
b
,
·
·
·
,
b
}
B
=
{b
,
b
,
·
·
·
,
b
}
1
2
φ(n)
1 2 p)
φ(n)
a 8(1×2×·
·=
·×(p−1))
=
1×2×·
·
·×(p−1)
(mod
d =
C
b
×
b
×
·
·
·
×
b
a×p
−
1
b
(mod
p
1
2
M
=
C
(mod
n)
M
C
C
φ(n)
a×p
−
1
=
b
p−1
a1 ×
̸= 17
a2 +b11 ̸==b12 a(mod
(mod p) a2 =B
b2 =
(mod
p)
1 = b1 15)
8 2 =256 = 15
{b
,
b
,
·
a×1
=
b
1 d 2 (1)
φ(n)
B=
b(mod
, · ·C
·= ,b15)
b=
}
1 8 =1
= {b
1aA1a,−=
2a
φ(n)
(mod
n)=
p) {ab
b =, bab M
, · ·=· C, ab(mo
{ab , ab−,C
·b· · =
, ab0 (mod
}A
8
1
2 1 12
2
2 , ab
A, =
A1 = {ab
· · ·{ab
, ab , a
25
pq
=1)(q
(p=−−
p, qn =
= pq
(p1)(q
−
p, q np,=q pq
φn =
(p φ−φ(n)
1)1)(q−−1)1)
nn nee e dd
e× d×
1 φ(n))
(mod
d ee×
d =d11 =
(mod
(mod
φ(n)) φ(n))
a×1 φ =
= q(p
b1p,−q
(mod
p)
n
=
pq
φ
=
(p
−
1)(q
p,
n
=
pq
φ
=
(p
−
1)(q
−
1)
p,
q
n
=
pq
1)(q
−
1)
n
e
d
e
×
d
=
1
(mod
φ(n))
B
=
{b
,
b
,
·
·
·
,
b
}
a×1
=
b
(mod
p)
1
2
φ(n)
=a×2
b}1 (mod
p) p)
= {b
BB =
{b1,1b,2b, 2· ,· ·· a×1
·, b·φ(n)
, b}φ(n)
= b12 (mod p,
q= (p
n=
pq
φ−=1)(p −
p,
q
n
=
pq
φ
−
1)(q
n , ·a×2
e· , ab d
ee××
dbd
==
1 (mod
φ(n))
n
e
1
(mod
φ(n))
.
a×2
=
(mod
p)
.
2
=
b
(mod
p)
A = {abB
,
ab
·
}
1 =2 {b1 , b2 , φ(n)
· · · },. 2bφ(n)}
A
=
{ab
,
ab
,
·
·
·
,
ab
1 1 , ab
2 2 , · · · φ(n)
A = {ab
, a×p
ab
}
... φ(n)
...
a×1= b1 =(mod
b1 (m
−
1
=
b
(mod
p)
a×1
p)
p−1
a×1
=
b
(mod
p)
1
{b
,, bb2 ,,··· ·· ·, b,φ(n)
}
C = b1 ×B
b2B
×=
·=
· ·{b
× b1φ(n)
b
}
e 1 , ab
d, e·2· · ,e ab
ed φ(n)
1
A
=
{ab
d}φ(n)
ekπ(n)+1
ed = bkπ(n)+1
2
(1)
a×1
=
b
a×1
(mod
p)
C
=
(M
)
=
M
=
M
=
M
(mod
n)
e
C
=
b
×
b
×
·
·
·
×
b
C
=
(M
)
=
M
=
M
=
M
(m
1
a×2
=
b
(m
1
e
1
2
φ(n)
2
a×p
−
1
=
b
(mod
p)
M
C
C
=
M
(mod
n)
a×2
=
b
(mod
p)
a×2
=
b
C a=φ(n)bC1 ×
b
×
·
·
·
×
b
p−1
M
C
C
=
M
(mod
2
a×pn)− 1 = bφ(n)
2 p)
2(mod
p−1 (mod
= CA
...2(mod
p−1= {ab
,
ab
,
·
·
·
,
ab
}
1
2
φ(n)
d
e
=
b
a×2
=
b
(mod
p)
.
d
e a×2
a
(1×2×·
·
·×(p−1))
=
1×2×·
·
·×(p−1)
p) (1)
.
2
C
=
b
×
b
×
·
·
·
×
b
φ(n)
C
=
M
(mod
n)
M
=
M
(mod
n)
.
.
C
=
M
(mod
n)
M
=
C
(mod
n)
1
2
φ(n)
A
=
{ab
,
ab
,
·
·
·
,
ab
}
.
.
aφ(n)C = C n(mod
n)
(1)
1
2
e d e × d = 1 (mod φ(n))
φ(n)
n
e
d
e
×
d
=
1
(m
.
.
a
C = C a(mod
n)
.
b1 = b2 a×p − 1.. = bp−1
.
1 − a2 = br1 − b2 = 0 (mod p)
N
C
=
b
×
b
×
·
·
·
×
φ(n)
r bφ(n)
N = bp−1
1215)
2(C
a×p
−
1
(mod
M
=
C
/y
,
C
)
Z
a×p
−
1
=
b
(mod
p)
18 =1ap−1
=
1
(mod
1
2
e
d
e
ed
kφ(n)+1
M
=
C
/y
(C
,
C
)
Z
a
C
=
C
(mod
n)
p−1
p−1
(1×2×·
·
·×(p−1))
=
1×2×·
·
·×(p−1)
(mod
p)
2
1
2
B
=
{b
,
b
,
·
·
·
,
b
}
B=M
{b
,p)
· · · , n)
bφ(n)}
1 (M
2
=
)b
=φ(n)
M ·=
M
1 , b2(mod
a 8(1×2×·
·C
·×(p−1))
=
1×2×·
·
·×(p−1)
(mod
C
=
b
×
×
·
·
×
b
a×p
−
1
=
b
(mod
a×p
p−1
a1 ×
̸= 17
a12 +b11 ̸==b212 a(mod
(mod p)φ(n)
a2 = b2 (mod
p)− 1 = pb
1 = b1 15)
8 2 =256 = 15
φ(n)
(1)
1 8 =1 = a1Aa−
(mod
e
k=
kC
−,db15)
lab
=−H(S)
C
(mod
n)
a{ab
=
b
=
0
(mod
p)
b
=
b
k − l } H(S)
, · ·k· , ab
8
1 C =2 M1 1(mod
2 n)
2
1A = n)
2
M = C (mod
{ab
, ab , · · · , ab
26
p = 13 g = 2
p a φ(n)
ap−1 ≡g 1 (mod p)
p)n) g g x =
1
x d
2
= 2 = 2 (mod 13)
e y
d 1 ≤edx, y ≤ kφ(n)+1
y
(mod
p)
x,
p
−
1
y
=
g
(mod
p)
C
=
(M
)
=
M
=
M
=
M
(mod n)
mod p) M C K K K
P
S
22 = 4 = 4 (mod 13)
e d p) edx
kφ(n)+1
x
=d =
y (M
(mod
y
≤
≤x,pn)
) =φ(n)
M M
=
= x,
M
(mod
g M
g
(mod
p)
y− 11≤ x,23y=≤8 p=−
8 1(mod 13)
= C=d 1y(mod
n)y
4
x≤ p − 1
odxp)
x
y
1
≤
x,
y
2
= 16 = 3 (mod 13)
y
=
g
(mod
p)
x
=
log
y
g
d
d p = 13 e gd = 2
ed
kφ(n)+1
ed
kφ(n)+1
e n)
=
(mod
C n)= (M
M
=
n)
M
M
(mod
M )=
C=M
C=
MM
(mod n)= M (mod
ed C =
kφ(n)+1
25 = 32 = 6 (mod 13)
x
kφ(n)+1
φ(n)
g
g
= y (mod p) 2x6 = 64
y = 112≤ x,
y ≤13)p − 1
p
=
13
g
=
2
=e M
=
M
(mod
n)
e
d
(mod
dC C =
edM
kφ(n)+1
M =(mod
n) d n)e × d = 1 (mod φ(n))
Cn (mod
e
d n)
27 = 128 = 11 (mod 13)
2d1 = 2 = 2e d (mod 13)
ed
kφ(n)+1
C 2 =e (M ) = M = M
= M (mod n)
8
e d e×
φ(n))
Md =C1B (mod
C {b
= 12M
n)
2 = 256 = 9 (mod 13)
=2 ,4·(mod
=
4, bφ(n)
(mod
13)
=
,
b
·
·
}
e
M (mod n)
3 21 =
d 28 =(mod
2 n)
(mod
= 8C=
13) 13) 29 = 512 = 5 (mod 13)
M2 =
(mod
= {bn)
×41d,2ab
= 21, ·4(mod
1 , b2 , · ·n· , beφ(n)
eod
A}d= e{ab
· ,4(mod
abφ(n))
} 13) 210 = 1024 = 10 (mod 13)
(mod
2 = 216=
= ·3=
13)
φ(n)
e
× d = 1 e(mod φ(n)) M25 =2C332==C
=
M
(mod
n) 211 = 2048 = 7 (mod 13)
6
(mod
13)
8
=
8
(mod
13)
= {ab1, ab2B
, · ·=· {b
, ab
b2,b}·1·×
· ,bb2φ(n)
}· · · × b
1 ,φ(n)
C
=
×
φ(n)
(mod φ(n))
212 = 4096 = 1 (mod 13)
26 = 464 = 12 (mod
13)
(mod p) x y 1 ≤ x, y ≤ p − 1
g = y (mod p) x y 1 ≤ x, y ≤ p − 1
M =M
= M (mod n)
) =M =M
= M (mod n)
n)
(mod n)
(mod n)
C = M (mod n)
13) φ(n))
n e2 =d16 e=×3 d (mod
= 1 (mod
27
x
y = g (mod p)
p a φ(n)
ap−1 ≡g 1 (mod p)
x
y
=
g
(mod
p)
p)
x y = g x (mod
x
d p) p)
e y
d φ(n)
edx,gy ≤gkφ(n)+1
(n)
g
g
=
y
(mod
x,
1
≤
p) n)x
C
=
(M
)
=
M
M (mod
mod p) M C K K K = M p −=1y=(mod
P
S
y 1≤
p−1
x
φ(n)
g
p
a
a
φ(n)
g
g
=
y
(mod
p)
x
y
1
=
(mod
≤yx,
y
≤
p
−
1
y =y(mod
x xM
x,
≤
p
−
1
gy =y1C ≤1(mod
p)
x,
y
1
≤
x,
y
≤
p
−
1
(M
)p)=p)
M
=
(mod
n)
n)
d
e d
ed
kφ(n)+1
xp) x y 1 ≤ x, y ≤
C
=
(M
)
=
M
=
M
=
(
od
p
−
1
d
e M
d
g
=
y
(mod
p)
x
y
1
≤
x,
y
≤
p
−
1
d
e
d
ed
kφ(n)+1
C
=
(M
)
=
M
d
d
e
d
ed
kφ(n)+1
ed (mod
kφ(n)+1
M =
C MK
e= n)
C
=
(M
)
=
M
=
M
=
M
(mod
n)
MM
=
C
n)
C
(M
)
=
M
=
M
=
=
M
=
M
(mod
M
C
C
=
M
(mod
n)
M ed =kφ(n)+1
M kφ(n)+1 = M (mod n)
d
d
=
M
=
M
d (mod n)
M
=
C
(mod n
e
e
d
ed
kφ(n)+1
=
C
(mod
n)
M ) C= M
C =M
M
(mod
n)
d
M
=
C
(mod
n)
= Mn e d =eM
(mod
n)
× d=
=C
1 (mod
φ(n))
(mod
n)
d n)
C = g r (mod p) C
d
e d
φ(n)ed
=d y (mod
Mg xkφ(n)+1
=M
e
M
C
C
=
M
(mod n)
=n)gex (mod
p)
x
=
log
y
r
e
gφ(n))
Md =C1 (mod
C=M
(mod n)
d e×
1
M C C = M2
B = {b1, b2, · · · , bφ(n)
}= g r (modeep) C = M y r (mod p) y
e
C
M
C
C2 =
=my
Mr (mod
(mod
n)
1 C
M
C
C
M
(mod
n)
x = 2log
y= grx (mod p)
(mod
p)
=
p)
y
(mod(mod
n)x n) n Ce 1 =dgr eM
g
×
d
= 1 φ(n))
(modCφ(n))
d ep)
×d
x
rx
x r n er
n
e
d
e
×
d
=
1
(mod
=
{b
,
b
,
·
·
·
,
b
}
=
g
=
g
=
y
(mod
1 2gr = yφ(n)
n)e C(mod
gn)
(mod
p)
x
1
≤
x,
y
1 ≤ p−1
A
=
{ab
,
ab
·
,
ab
}
mod
r2 , · ·xy
x
1
φ(n)
=
log
y
r
(mod p)x =
C2log
=M
ynr (mod
p)dgy =e g× (mod
p)
x
1 = g n)
x
φ(n)
g
g
= y (mo
y
e
d
=
1
(mod
φ(n))
× d = 1 e(mod φ(n))
g
M
=
C
/C
(mod
p)
2
B, =
{b
· · ·},ebφ(n)}d e C
B =φ(n))
{b1=, bM
1
r 1 (mod
1·, b
2b, n
2 , ·y·r · (,
×
d
=
B
=
{b
b
,
·
·
,
AC==
{ab
,
·
·
·
,
ab
}
=
g
(mod
p)
C
M1e, ab
(mod
n)
1
2
φ(n)
2
1
2
φ(n)
b2 × · · · × bφ(n)
x
d
edC = b1 ×
kφ(n)+1
y = g x (mod
x p)
28
φ(n) g g = y (mod p) x y 1 ≤
φ(n)px ga y ap−1
≡
φ(n) g g x = y (mod p)
1≤
C dd = (M ee)dd = M eded= M kφ(n)+1
=
(m
d
e M
d
Ckφ(n)+1
=
(M
)
=
M
r M
C K
e (M ) = M
C gC
(mod n)
g x (mod p) xM= log
y =CrM=
= g (mod =
p) MC2
=C1M
r
r d
d
C
=
g
(mod
p)
C
=
M
y
(mod
p)
y
M
=
C
(mod
n
1
2
r 13)
r
x
C2 = 8 = 32 =C61(mod
d
x
=
log
y
r
M
=
C
(mod
n)
=
g
(mod
p)
C
=
my
(mod
p)
y
=
g
(mod
p)
g
2
(mod
n e d eM
× d==C1 (mod
φ(n))
x n) rx
xr
r
x
C
=
g
=
g
=
y
(mod p) e
) C g= ggr (mod
= y p)
(mod
p)
x
1
≤
x,
y
1 ≤ p−1
r xy = log
x
Mg Cg x =
C y=(mo
M
C2log
=M
y r (mod p) gy y= gr (mod p)
1 x
5
9
x
φ(n)
x
=
y
C1 =M6 ==810077696
r=
C151,=
32, b=φ(n)
6 }(mod 13)
g b22, ·=
MM
=eeCr2(mod
/C1 (mod
p)
B 5= =
{b
·
·
r
M
C
C
=
n)
C
=
g
(mod
p)
C
=
M
y
(m
M
C
C
=
M
(mod
n)
1
2
x
e
d
ed
kφ(n)+1
g gxn)
= yrx(mod
n(Mexer)d d=y M
e1×
d x rp)
ed≤dx
x==(M
log
y
r
)
=
M
=
M
M
(mod
x= φ(n)
5
g
C
=
913)
φ(n)
g6aby(mod
g=
=
yab
(mod
p)
x g13)y= g1 ≤ =x,yy ≤(mod
p − 1p) =
C2 p=p=9=×
9=
3125
=
6=
(mod
13)
=
32==
=19,28125
C
=
13135 Cg=
282=
2g =
x{ab
2
512
=
5
(mod
A×
,
·
·
·
,
}
2
φ(n)x = 1log y r
x
9
n
ey6 C=1d d10077696
eye×
1 =(mod
φ(n))
d=dg5= ed
kφ(n)+1
x
C
=
=
(M
)
=
M
M
B
=
{b
, · ·r(m
· ,b
1
d
φ(n)
g
g
=
y
(mod
p)
x
≤
x,
≤
p
−
1
5
1 ,=b2M
d 2 = 32
e=d 6e(mod
ed
kφ(n)+1
d =9 r=
n
d
e
×
d
=
1
(mod
φ(n))
5
r
M
=
C
(mod
n)
M
5
C
=
13)
C
=
(M
)
=
M
=
M
=
M
(mod
n)
1 r = 5 C1 = 2 = 32 = 6 C
M =C8 =
(mod
= C (mod n)
p) C2 = M y (m
1 = g13)(mod
b1 × b2 × · · · × bφ(n) φ(n)
x
g 28125
g == y6 (mod
p) x
5
d
d
e d
ed
kφ(n)+1
C
=
9
×
5
=
9
×
3125
=
(mod
13)
1 = M dB
{b
,(mod
b39C
· (mod
·n)+13)
· 1, =
bn)
} 13)
M
=
A
=C{ab
, abe 2(mo
,··
92 =
1=
2×, 13
C p==(M
=2M2xφ(n)
=
φ(n)
1M
=
2
=
2
(mod
13)
5
×
8
=
40
1
(mod
13 )g =
=
9
y
=
2
=
512
=
5
(mod
M
C
=
1
(mod
n)= 2 =
pe= a132 M
gC2===2C2Cx (mod
=
512dlog
=g5y (mod
x
=
r}d 13)
x 9 13)
9yn)
=
(mod
e
ed
kφ(n)+1
C
6
=
10077696
=
5
(mod
13)
B
=
{b
,
b
,
·
·
·
,
b
2
C C d= M (mod
n)
5
C
=
(M
)
=
M
=
M
1
1
2
φ(n)
4 = 4 M(mod
= 9 13)
r =C
5 CC1 =
2M=e 32
= 6C
(mod
13)× b × · ·
M
=
(mod
n)
=
b
M = C (mod n)222 =
1e × d
2 =1
e
−1
A
=6×5
{ab5(mod
abφ(n)
,=· ·48· g=, ab
}
x
n
e
d
= 4 =C4 M
(mod
13)
1=, 6×8
2n)
φ(n)
C
=
M
=
13×3+9
=
9
(mod
13)
3 M
g
=
y
(mod
p)
x
d
2 = 8 = 8 C(mod
13)
=
9
×
5
=
9
×
3125
=
28125
=
6
(mod
13)
2
9
M
= Cy = (mod
n)
38
p
=
13
g
=
2
x
=
9
2
=
512
=φ(n))
5 (mod
13)
φ(n)
1
1
e
(5
×
8
=
40
=
3
×
13
+
1
=
1
(mod
13))
e M d C e C×=d2M
==212421(mod
(mod
φ(n))
=1
=
1
(mod
15)
n
e
d
e
×
d
=
1
(mod
=
8
=
8
(mod
13)
a
C
=
C
(mod
n)
2 e2=
(mod
13)
=3dA
2x(mod
=
2×9{ab
(mod
13)
=
,
ab
,
·
·
·
,
ab
}
1
2
φ(n)
=
13)
d
e
d
ed
kφ(n)+1
B
=
{b
,
b
,
·
·
·
,
b
n 16
e
d
=
1
(mod
φ(n))
2 M
φ(n) }
C =b ×b ×
C ·=· ·(M
) = M1 =
5× b
y = g x (mod
x p)
29
φ(n) g g = y (mod p) x y 1 ≤
φ(n)px ga y ap−1
≡
φ(n) g g x = y (mod p)
1≤
C dd = (M ee)dd = M eded= M kφ(n)+1
=
(m
d
e M
d
Ckφ(n)+1
=
(M
)
=
M
r M
C K
e (M ) = M
C gC
(mod n)
g x (mod p) xM= log
y =CrM=
= g (mod =
p) MC2
=C1M
r
r d
k
k
−
l
H
(S)
d
C
=
g
(mod
p)
C
=
M
y
p) yn
M
=
(mod
1
2
r M = C d (mod n)
r
xC(mod
x
=
log
y
r
C
(mod
p)
C(mod
myn)(mod p) yg = g (mod p)
1 =dg eM
2 = φ(n))
=
C
n
e
×
d
=
1
(mod
x
rx
xr
r
x
C
=
g
=
g
=
y
(mod p) e
) C g= ggr (mod
= y p)
(mod
p)
x
1
≤
x,
y
1 ≤ p−1
r xy = log
x
Mg=Cg1
=(mo
M
x C(mod
C
=×
M
y8r =
(mod
p) =
gr ×
(mod
p)
gy y= 3
1
2log
(5
40
13
+
1
x
φ(n)
=
y
x
=
y
MM
=eeCr2(mod
/C1 (mod
p)
B = {b1,g b2, M
· · · , bC
}C =
r
φ(n)
n)
C
=
g
(mod
p)
C
=
M
y
(m
M
C
C
=
M
(mod
n)
1
2
x
e
d
ed
kφ(n)+1
g gxn)
= yrx(mod
n(Mexer)d d=y M
e1×
d x rp)
ed≤dx
x==(M
logg )y =
r M =φ(n)
M
M
(mod
x= φ(n)
−1
C
=
g, ·×
=
yab(mod
p)6 =
x g 8y=
1≤
x,y=
y ≤(mod
p −×
1p) =
5
=
×
=
48
13
3
C
g
=
A =M
{ab1=
,gab6
·
·
,
}
2
φ(n)x = 1log y r
n
ey C1d d=
eye×
1 =(mod
φ(n))
d dg = ed
kφ(n)+1
x
(M
)
=
M
M
B
=
{b
, · ·r(m
· ,b
d
φ(n)
g
g
=
y
(mod
p)
x
≤
x,
≤
p
−
1
1 ,=b2M
d
e d e
ed
kφ(n)+1
d
n
d
e
×
d
=
1
(mod
φ(n))
r
M
=
C
(mod
n)
C
=
(M
)
=
M
=
M
=
M
(mod
n)
rb x× be × ·9· · × b
=C M
(mod
C1 = g (mod
C2 = M13)
y (m
x 5p) (mod
M=
/y=C
(C
C
)
1M
21 ,6
Cn)C2CC
=
(mod
n)
φ(n)
=
=
10077696
=
2
φ(n) g g = y (mod p) x
1 kφ(n)+1
d
d
e d
ed
{b
,(mod
bC2 , · (mod
·n)· , bn)
} A
M1=
=C{ab
, abe 2(mo
,··
C = (M ) = M φ(n)
=
M
=
φ(n)
1M
dB =
M
C
=
(mod n)
n) x = logg y r
e a MC ==CC B(mod
d ,b
e}d
ed
kφ(n)+1
5
=
{b
,
b
,
·
·
·
C Cnkd= eM
(mod
n)
C
=
(M
)
=
M
=
M
1
2
φ(n)
d
e
×
d
=
1
(mod
φ(n))
C
=
9
×
5
=
9
×
3125
=
28125
=
k
−
l
H(S)
e
2
Me 1C, abC2 ,=
M (mod
n)
C
=
b1e×
bd2 ×
·1·
M = C (mod n) M C A
=M
{ab
}
x
n
e
d
×
=
φ(n)
C=
(mod φ(n)
n) · · · g, ab
d g = y (mod p) x
M
=
C g x(mod
r e 8
r
5rn)
φ(n)
C
=
g
(mod
p)
C
=
my
(mod
p)
y
=
(mod
p)φ(n))
M
=
C
/y
B
=
{b
,
b
,
·
·
·
,
b
}
M
=
9
r
=
5
C
=
2
=
32
=
6
e M d C (5
e C×
=
(mod
φ(n))
1×
2 13
1(mod
=1
=
1×
(mod
15)
n1
e1=
d2 ,(mod
e
×
d
=
1
(mod
a
C
=
C
(mod
40
=
3
+
1
13))
2
1=12
1
φ(n)
=d8M
n)
A
=
{ab
,
ab
·
·
·
,
ab
}
d = {b1ed
· ·kφ(n)+1
, bφ(n)}
n e dC e=×bd =
φ(n))
2 , ·M
×1b(mod
×
· ·(M
×eb)Bφ(n)
C d ·=
= M , b=
30
1
1
k
n
k
n
1
k
n 1 k H(S
k |Z
n ) K I(S ; Z )
Z )
k H(S |Z ) K I(S ; K
R≡N
K
R≡N
NK
1
K
M = C2/y r (C
,
C
)
Z
1
2
P
≡
ED
(S
,
Ŝ
)
H
K
1
Ke
K
, Ŝ )
X N YPNe ≡ K EDH (SP (yz|x)
XN Y N
k
k
K − lK H(S)
K
P
(yz|x)
Pe ≡ Pr{Ŝ ̸= S }
P
≡
Pr{
Ŝ
K
e
S Ws
K Rk ≥ h
Rs ≥ H(S)
Rs ≥ H(S)
̸= S K
Rk ≥
S × 8 = 40 = X
Bob
N 1 =N1 (mod
K
Alice
(5
3 SN×
13
+
13))
Y
Z
Ŝ
K
(Z)
P
<
ε
P
>1−ε
log MN
Nlog Ms N
K
e K loge M
s
s
K
X
Y
Z
Ŝ
Rs = K S K Rk = −1
RŜs = ̸= S
Pe ≡ Pr{
} Rk =
K
K
M = 6 × 5 = 6 × 8 = 48 = 13 × 3 + 9 = 9 (mod
Rs ≥ H(S) Rk ≥ h
Ws ∈ {0, 1, · · · , Ms − 1} Wk ∈ {0, 1, · · · , Mk −W
1}s ∈ {0,
CZ 1, · · · , M
log
Ms
log Ms
C1x = 69 = 10077696 =
5 (mod
13)
R
=
R
=
N
N
s
k
1
1
K
K
K1
K 1
X
Y
N
N
K
H(S
|W
)
H(S
|W
)
≥
h
0
≤
h
≤
H(S)
Ys K
H(S
|Ws) K H
s
K X
K
K
K
CSZ ≡}max I(X; Z)
5K
r
N
P
≡
Pr{
Ŝ
=
̸
K
e
=(CŜ91×
9 × 3125 =W28125
=1, 6· · ·(mod
X, Ms −13)
M = 1C2/y
, ̸=C5S2) =
PeC≡2 Pr{
}Z
1} W
s ∈ {0,
1
; W2 ) = r
Bob
R
≥
H(S) Rn!
≥!
h1; W2) = r
1Alice
s
kI(W
Eve
n I(W
Rs−nr
≥ H(S) Rk ≥ h
1
K
K P(
−nr
5 1
P ≈2
1
k n 1
k n
|Z
)
I(S
1
k n k1H(S
k
n
1
k )Kn 1 ; Z )k
n
;
Z
1
H(S
|Z
)
I(S
;
Z
)
k
n 1k H(S k|Z )nK I(S
k K
K
H(S
|Z
)
I(S
;
Z
)
K
k
K
K
R≡
R
≡
N
R
≡
N
K
1
k n 1
k Nn
R≡N
H(S
|Z
)
I(S
1 ;KZ1 ) K KK
k
K
1
K
≡e,KŜ≡ED) HED
(S H,(S
Ŝ ) , Ŝ K )
Pe ≡KK EDHP(S
eP
K
R
≡
1
KN K
Pe ≡ EDH (S , Ŝ )
K
Alice
P (yz|x)
SK
XSNK
YN
31
P (y|x)
P (z|x)
P (yz|x)
P (yz|x)
P (yz|x)
K
K
PSe K≡ K1 EDH (S
,
Ŝ
)
KS K
S
Bob
P
(yz|x)
N
N
N N K max[I(X;
K
Y
)
−
I(X;
PX(yz|x)
NN
NK
K
Y
Z
Ŝ
N
N
K
S
XSX
Y Y Z ZŜ Ŝ
≥
K SK
S
∈
{0,
1,
2,
·
·
·
,
M
−
1}
K
S N
H
K N
Z
Ŝn
n
n
X
Y
Z
N
NS
N
K Ŝ
XS K Y
Z
Ŝ
K
min[I(X; Z
P (y|x) P (z|x)
≤
N
H
K
max[I(X;
Y )−−H(X|Y
I(X; Z),
I(X;
YY
)−
I(R
H(X)
)
=
I(X;
)
≥
N
H(S)
log M
R Z),
≥ H(X|Y
)
KR =min[I(X;
I(X; Y |Z)]
N
R≡N
R N≡k N n
K
1
1
k
n
R≡N
1 ;KZ1 ) K KK
k H(S 1|Z ) P
K I(S
K
≡e,KŜ≡ED) HED
(S H,(S
Ŝ ) , Ŝ K )
Pe ≡KK EDH (S
eP
C≡ZKN K
K
R
1
Pe ≡ K EDH P
(S(yz|x)
, Ŝ )P (yz|x)
P (yz|x)
1 P (y|x)K P (z|x)
K
P
≡
ED
(S
,
Ŝ
)
H K K
P (yz|x)
Se K K
S
32
CZ
C
S
Z
C
≡
max
I(X;
Z)
Z
Bob
P
(yz|x)
Alice
X(yz|x)
N
N
N N max[I(X;
K N
P
K
YNK) −ŜI(X;
Z), I(X; Y )
K
K
X
Y
Z
Ŝ
N
N
N
K
S
X
Y
Z
S
XS K ≥
Z Ŝ C ≡ max I(X
S KY
S ∈ {0,
1,(x,
2, y)
· · · , M − 1} Z H(S)
K
!
!
N
P
S
!
N
N
N
K
X
XYS K) = Y
Z
Ŝlog
I(X;
P (x, y)
n
n
n K
min[I(X;
Z),
I(X;
Y |Z)
X
Y
Z
I(X;
Y
)
=
N
NS P (x)P
N
K Ŝ
(y)
K
K
X(z|x)
Y
Z
Ŝ
x
yP
≤Pe ≡!
Pr{Ŝ!
̸= S }
P (y|x)
SK
N
H(S)
x
!
R
≥
H(S)
R
≥
h
s
kP (x, y)
I(X;
Y
)
=
H(S)
≡
−P
(s)
log
P
(s)
(Z)
H(X)
−
H(X|Y
)
=
I(X;
Y
)
R
≥
H(X)
Kε P
max[I(X;
Y
)
−
I(X;
Z),
I(X;
Y
)
−
I(Y
;
Z)]
Pe <
>
1
−
ε
≥s e
log
Ms y
log Ms
x
H(S)
Rs = K Rk = K ≡
RN= K/N
log M H(S)
R < CY R > CZ
R = N R ≥ H(X|Y
) C! P (y|x)
C
>
K min[I(X; Z), I(X; Y Y|Z)] Z
R = K/N CH(S)
Z
≤
s
Ws ∈ {0,
1, · · · ,−P
Ms −(s)
1}
H(S)
≡
N
H(S) CY
CZ
CY ≡ max I(X; Y )
P·se·max[I(X;
<
ε− 1}Pe
R
<
R
>
W
∈
{0,
1,
·
,
M
K
N
H(S)
H(S)
X
1
1
K
H(X) − H(X|Y
) = I(X; Y ) R ≥ H(X)
ZK |W≥
(Z)
H(S
)
H(S
|W
s
s
N
K
K
C
≡
max
I(X;
Z)
P
<
ε
P
>
1
−
ε
!
Z
e
CZ
e X
XCY
(Z)
N
"
#
W
Ŵ
S
(Z)
P
<
ε
P
>1−ε
R
<
R
>
1
1
1 P < ε log
e
e
1
−
ε
H(S)
H(S)
e R = PeM >
1
R ≥ H(X|Y )
I(W1; W2) = r
N
CCZ

K(.、Oヾ

RYO C. YANAGITA, PhD