1
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p
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a
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-1
, nx, x
n
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5
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p
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è 1 p,q1??￥
í
?￡
üí pq¨￥?ì .
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! R1 pq ¨￥?ì
5 <R,+>1 pq¨￥ Abel ? .
i p¨í a q¨í b.
?[ |a+b|=pq, <R,+>1?ì ?
7 c=a+b1
3?í .
R={ 0, c, 2c, … , (pq-1)c }
| x=pc, y=qc, 5
xy=(pc)(qc) = pqc
2
=0
x,y1
,y0 .
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6
?l F1× |F|μK
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p
, p1
í
?
Z
p
1?ì <Z
p
?{0},?>μK? ?
í
,í
a?h ?
p <Z
p
?{0},?>? Abel ?
2
? μK￥?ì?
^×
μK×￥ +?
F1μK× 1 <F,+>?￥¨1× F￥+? .
Z
p
￥+?1 p.
μK×
7
! F1μK×5i
í
? p
P¤ |F|=p
n
,
￡
ü
±X :
A=<1>={ 0, 1, … , p?1 }
Ax
1
= { 0, x
1
, 2x
1
, …, (p?1)x
1
} x
1
∈F
*
|Ax
1
| = p
? F=Ax
1
52
? ?5 ?x
2
∈F?Ax
1
, x
2
≠0,
Ax
1
+Ax
2
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1
x
1
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2
x
2
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1
,a
2
∈A}
V[￡
ü Ax
1
+Ax
2
?￥í
í
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1
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2
|=p
2
,
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1
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2
+Ax
3
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3
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a ≠ 0(modn) μ a
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8
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V[￡
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ê | 2—n-2 ?￥
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k?
k n 1?
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qà
1
1/2. ?
^??
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?￥ù5 .
10
í
??
k￥
6B?Hq
? ? 2 ?T n1
í
?5Z? x
2
≡ 1(mod n)￥?oμ
?
' x=1 x=?1
x=n?1
￡
ü x
2
(mod n)≡1 ? x
2
?1≡0(modn)
? (x+1)(x?1)≡0(modn)
? x+1≡0 x?1≡0
×?
àμ
,y0
? x=?1 x=1
? x±≠1￥?1d üO￥ .
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? . è ? :
x
2
(mod 5) ≡ 1 ? x=1 x=4
x
2
(mod 12) ≡1 ? x=1 x=5 x=7 x=11
5? 7
^d üO￥? .
11
Miller-Rabin
?E
! n1
í
?i q,m
P¤ n?1=2
q
m, (q≥1). ?
￥KaB[1 a
n
?
1
(mod n), O
?B[
^ -
?B[￥ üZ .
? ?i i
i=0,1,…q?1 , ?
^?1 1? n?1, O
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^?1 1.
?T a[1 1?'[??? 1? n?1, 5
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^
d üO￥?V7?? n?
^
í
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2
na
m
i
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242
nananana
mmmm
q
12
è ? n=561, n-1=560=2
4
? 35, L
! a=7, /￥?
1
1)561(mod7)561(mod7
,67)561(mod7)561(mod7
,166)561(mod7)561(mod7
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560352
280352
140352
70352
35
4
3
2
1
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V[|pà
q??à 2
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.
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13
18.20ì a ?X a
ì aì]
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? +,? ?ì .
L è nZ
^ <Z,+,?>￥ 0ì
0ìü
^0}
? üO0ìi
?Y 0F ? ?Y +? ? ?Y
0?ì a0"ì a0×
15
?l
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^ì <R,+,?>￥d b0" ?
(1) <D,+>? Abel ?
(2) ?r∈R, rD?D, Dr?D
5? <D,+,?>
^ì R￥ ?X .
a
ü
P ?X (o
?@ rD?D)D· ?X
D=
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a
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üO ?X {0}, R1
& .
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16
è 1 R1?Dì 1∈R, O 1≠0, 5 R1×? O?? Roc
μ üO ?X .
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! D1 ?X D≠{0} ?x∈D,
x≠0 ? x
?1
∈R ? 1= x
?1
x∈D ? ?r∈R, r= r?1∈D,
R=D
“?” ?x≠0, x∈R,
7 Rx = { rx | r∈R }.
?r
1
x, r
2
x ∈Rx,
r
1
x?r
2
x = (r
1
?r
2
)x∈Rx
yN <Rx,+>? Abel ? .
?r
1
x∈Rx, r
2
∈R
(r
1
x)r
2
= (r
1
r
2
)x ∈Rx , r
2
(r
1
x) = (r
2
r
1
)x∈Rx,
Rx
^ ?XyN Rx=R, i y
P¤ yx=1, xμ Ií .
è5
17
?l D1 R￥ ?X ?x∈R,
yxyx
yxyx
RxxDR
DdxdxDx
?=?
+=+
∈=
∈+=+=
}|{/
}|{
? <R/D,+,?>?ì1 R1? D￥
ì .
?
?l￡
yxxyxyd
xyydxddd
ydxdyx
ydyxdxyyxx
?==+=
+++=
++=?
+=+=?==
1221
21
21
))((''
','','
ì
18
L è <Z
6
,⊕,?>
?X {0}, {0,2,4}, {0,3}, Z
6
ì Z
6
/{0} = { {0}, {1}, {2}, {3}, {4}, {5} }
Z
6
/Z
6
= { Z
6
}
Z
6
/{0,3} = { {0,3}, {1,4}, {2,5} }
Z
6
/{0,2,4} = { {0,2,4}, {1,3,5} }
ì￥
L è
19
ì]
? f:R
1
→R
2
f(x+y) = f(x) + f(y)
f(xy) = f(x) f(y)
]
?? kerf ={ x | x∈R
1
, f(x)=0 }
L è f
c
:Z→Z, f
c
(x) = cx c1?
?
kerf
c
= cZ
ì]
?
20
ì]
?￥?é
1. f(0)=0, f(1)=1 f(?x)=?f(x), f(x
?1
) = f(x)
?1
2 (1) S
^ R
1
￥0ì5 f(S)
^ R
2
￥0ì
(2) T
^ R
2
￥0ì5 f
?1
(T)
^ R
1
￥0ì
(3) D
^ R
1
￥ ?X5 f(D)
^ f(R
1
)￥ ?X
(4) I
^ R
2
￥ ?X5 f
?1
(I)
^ R
1
￥ ?X
3 kerf = {x|x∈R
1
,f(x)=0}5 kerf
^ R
1
￥ ?X
4]
?'? ?
ì R￥ ??
ì R/D
^ R￥]
?^
? R～R’, 5 R’ ? R/kerf
21
￡ 2. (2) f
?1
(T)d b ?x,y∈f
?1
(T)
?a,b∈T
P¤ f(x)=a, f(y)=b,
f(x?y)=f(x)
?f(y) = a?b∈T, x?y∈f
?1
(T)
f(xy) = f(x)f(y) = ab ∈T, xy∈f
?1
(T)
(3) f(D)
^ f(R
1
)￥0F ? O1 Abel ? .
?x∈f(D), r∈f(R
1
),
?a∈D,
P¤ f(a)=x, ?b∈R
1
, f(b)=r,
xr = f(a)f(b) = f(ab)∈f(D)
] ? rx∈f(D)
?é￥￡
ü
22
3 kerf = { x | x∈R
1
, f(x)=0 }
￡ kerf
^ R
1
￥ ?X
kerf
^ <R
1
,+>￥??0 ? .
?x∈kerf, r∈R
1
,
f(xr) = f(x)f(r)= 0 f(r) =0
xr∈kerf] ? rx∈kerf
?é￡
ü
?
23
T<
?ˉ 1?
ì￥?l
+
yì￥ ?Y
μK×￥í
í
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I
1?é
ì￥?l
ì]
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?
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5
E? 4, 5, 6, 7, 33,
?1DμK×
￥ [
Tì
