dissertação - gilberto tenani
TRANSCRIPT
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2 × 2
n
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a
b
a
b
m
a ≡ b(mod m)
m|(a − b)
m
Zm = {0, 1, 2, . . . , m − 1}
m
a
m
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Z
N
a
b ∈ Z a b d
b = ad
a
b
a
b
a | b a b
13 | 65, −5 | 15, 13 | 169, 6 35 17 0
4
±1, ±2 ± 4
a,b,c ∈ Z a | b a | bc
a | b
d1 b = ad1 bc = (ad1)c = a(d1c)
a,b,c ∈ Z
a | b
b | c
a | c
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a | b b | c d1 d2 b = ad1 c = bd2
c = bd2 = (ad1)d2 = a(d1d2) a | c
a,b,c,m,n ∈ Z a | b a | c a | (mb + nc)
a | b a | c d1 d2 b = ad1 c = ad2
mb + nc = m(ad1) + n(ad2) = a(md1 + nd2) a | (mb + nc)
a,b,c ∈ N
a | b
a | c
a | (b ± c)
x ∈ R
x
[x]
x ∈ R x − 1 < [x] ≤ x
a, b ∈ Z b > 0
q, r
a = bq + r
0 ≤ r < b q ∈ Z
r ∈ Z
q = [a/b]
r = a − b[a/b] a = bq + r
r
0 ≤ r < b
a/b − 1 < [a/b] ≤ a/b.
b
a − b < b[a/b] ≤ a.
−1
−a ≤ −b[a/b] < b − a
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a
0 ≤ r = a − b[a/b] < b
q ∈ Z
r ∈ Z
a = bq 1 + r1 a = bq 2 + r2 0 ≤ r1 < b 0 ≤ r2 < b
0 = b(q 1 − q 2) + (r1 − r2).
r2 − r1 = b(q 1 − q 2)
b
r2 − r1 0 ≤ r1 < b 0 ≤ r2 < b
−b < r2 − r1 < b b r2 − r1 r2 − r1 = 0
r2 = r1 bq 1 = r1 = bq 2 + r2 r1 = r2
q 1 = q 2
a
b
r = 0
133 = 21.6 + 7
−50 = 8.(−7) + 6
p ∈ Z
n > 1
n1 n n1 = 1 n1 = n
n2
n = n1 × n2, 1 < n1 < n 1 < n1 < n
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a,b,p ∈ Z p p|ab p|a p|b
p|ab
p |a
p|b
p|ab
c ∈ Z
ab = pc
mdc( p, a) = 1
m, n ∈ Z
np + ma = 1.
b
b = npb + mab.
ab
pc
b = npb + mpc = p(nb + mc),
p|b
n > 1
n ∈ N
n = 2
n ∈ N
n ∈ N n
n
n1 n2 n = n1 × n2 1 < n1 < n 1 < n2 < n
p1, p
2, . . . , p
r q
1, q
2, . . . , q
s
n1 = p
1 × p
2 × · · · × p
r n2 = q
1 × q
2 × · · · × q
s p1, p2, . . . , pr
n = p1 × p2 × · · · × pr
q 1, q 2, . . . q s
n = p1× · · · × pr = q 1× · · · ×q s p1 | q 1× · · · ×q s p1 = q j
j
q 1, . . . , q s q 1
p2 × · · · × pr = q 2 × · · · × q s.
p2 × p3 × · · · × pr < n r = s pi q j
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m
a b
m
m | (a − b)
a
b
m
a ≡ b(mod m)
m (a − b)
a ≡ b(mod m)
a
b
m
m
21 ≡ 13(mod 2)
22 ≡ 4(mod 9)
(9|(22 − 4) = 18
a
b
a ≡ b(mod m)
k
a = b + km
a ≡ b(mod m)
m | (a − b)
k
km = a − b
a = b + km
k
a = b + km
km = a − b
m | (a − b)
a ≡ b(mod m)
m ∈ Z m
a
a ≡ a(mod m)
a
b
a ≡ b(mod m)
b ≡ a(mod m)
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a
b
c
a ≡ b(mod m) b ≡
c(mod m)
a ≡ c(mod m)
a ≡ a(mod m)
m | 0 = (a − a)
a ≡ b(mod m)
m | (a − b)
k
km = a −b
(−k)m = b −a
m | (b−a)
b ≡ a(mod m)
a ≡ b(mod m)
b ≡ c(mod m)
m | (a − b)
m | (b − c)
k
l
km = a − b
lm = b − c
a − c =
(a − b) + (b − c) = km + lm = (k + l)m
m | (a − c)
a ≡ c(mod m)
m ∈ Z m
Z
Z
m
m
m
m
Zm
· · · ≡ −10 ≡ −5 ≡ 0 ≡ 5 ≡ 10 . . . (mod 5)
· · · ≡ −9 ≡ −4 ≡ 1 ≡ 6 ≡ 11 . . . (mod 5)
· · · ≡ −8 ≡ −3 ≡ 2 ≡ 7 ≡ 12 . . . (mod 5)
· · · ≡ −7 ≡ −2 ≡ 3 ≡ 8 ≡ 13 . . . (mod 5)
· · · ≡ −6 ≡ −1 ≡ 4 ≡ 9 ≡ 14 . . . (mod 5)
a ∈ Z m > 1
a = bm + r
0 ≤ r ≤ m − 1 a = bm + r a ≡ r(mod m)
m
0, 1, . . . , m −
1
m
0, 1, . . . m − 1
m
m
0, 1, . . . , m − 1
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m
m
m
{a1, a2, . . . , am} ai = a j ai ≡ a j(mod m).
m m
{0, 1, 2, . . . m − 1}
m
a b c m m > 0 a ≡ b(mod m)
a + c ≡ b + c(mod m)
a − c ≡ b − c(mod m)
ac ≡ bc(mod m)
a ≡ b(mod m) m | (a − b) (a + c) − (b + 1) =
a − b
m | [(a + c) − (b + c)]
a + c ≡ b + c(mod m)
ac − bc = c(a − b)
m | (a − b)
m | c(a − b)
ac ≡ bc(mod m)
18 ≡ 2(mod 8)
25 = 18 + 7 ≡ 2 + 7 = 9(mod 8)
17 = 18 − 1 ≡ 2 − 1 = 1(mod 8)
36 = 18 × 2 ≡ 2 × 2 = 4(mod 8)
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2 × 7 ≡ 2 × 4(mod 6) 7 ≡ 4(mod 6).
a,b,c,m ∈ Z
c = 0
m > 1
mdc(c, m) = d
ac ≡ bc(mod m)
a ≡ b(mod
m
d )
m
d
c
d
ac ≡ bc(mod m) ⇐⇒ m | (b−a) ⇐⇒ m
d
| (b−a)c
d
⇐⇒ m
d
| (b−a) ⇐⇒ a ≡ b(mod m/d),
50 = 50 × 10 ≡ 20 = 2 × 10(mod 15)
mdc(10, 15) = 5
50 × 10
10 ≡
2 × 10
10 (mod
15
5 ),
5 ≡ 2(mod 3).
a,b,c,m ∈ Z c = 0 m > 1 mdc(c, m) =
1
ac ≡ bc(mod m)
a ≡ b(mod m)
42 = 6 × 7 ≡ 7 = 1 × 7(mod 5) mdc(5, 7) = 1
6 × 7
7 ≡
1 × 7
7 (mod 5),
6 ≡ 1(mod 5).
a,b,c,d,m ∈ Z
m > 0
a ≡ b(mod m)
c ≡ d(mod m)
a + c ≡ b + d(mod m)
a − c ≡ b − d(mod m)
ac ≡ bd(mod m)
a ≡ b(mod m)
c ≡ d(mod m)
m | (a − b)
m | (c − d)
k
l
km = a − b
lm = c − d
(a + c) − (b + d) = (a − b) + (c − d) = km + lm = (k + l)m
m | [(a + c) − (b + d)]
a + c ≡ b + d(mod m)
(a − c) − (b − d) = (a − b) − (c − d) = km − lm = (k − l)m
m | [(a − c) − (b − d)]
a − c ≡ b − d(mod m)
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ac − bd = ac − bc + bc − bd = c(a − b) + b(c − d) = ckm + blm = m(ck + bl)
m | (ac − bd)
ac ≡ bd(mod m)
13 ≡ 8(mod 5) 7 ≡ 2(mod 5)
20 = 13 + 7 ≡ 8 + 2 ≡ 0(mod 5)
6 = 13 − 7 ≡ 8 − 7 ≡ 1(mod 5)
91 = 13 × 7 ≡ 8 × 2 = 16(mod 5)
{r1, r2, . . . , rm} m
a
mdc(a, m) = 1
{ar1 + b,ar2 + b , . . . , a rm + b}
m
ar1 + b,ar2 + b , . . . , a rm + b
m
ar j + b ≡ ark + b(mod m),
ar j ≡ ark(mod m).
mdc(a, m) = 1
r j ≡ rk(mod m).
r j ≡ rk(mod m) j = k j = k
m
m
m
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a,b,k
m
n > 0, m > 0
a ≡ b(mod m)
an ≡ bn(mod m)
n ∈ N n = 1
n ∈ N
n + 1 ∈ N
an ≡ bn(mod m)
a ≡ b(mod m)
ana ≡ bnb(mod m)
an+1 ≡
bn+1(mod m)
7 ≡ 2(mod 5)
343 = 73 ≡ 23 = 8(mod 5).
a ≡ b(mod m1), a ≡ b(mod m2), . . . , a ≡ b(mod mk) a,b,m1, m−
2, . . . mk m1, m2, . . . , mk
a ≡ b(mod mmc(m1, m2, . . . , mk)).
a ≡ b(mod m1), a ≡ b(mod m2), . . . , a ≡ b(mod mk)
m1 | (a − b), m2 | (a − b), . . . , mk | (a − b)
mmc(m1, m2, . . . , mk) | (a − b)
a ≡ b(mod mmc(m1, m2, . . . , mk))
a ≡ b(mod m1), a ≡ b(mod m2), . . . , a ≡ b(mod mk)
a
b
m1, m2, . . . mk
a ≡ b(mod m1 × m2 × · · · × mk),
m1, m2, . . . , mk
mmc(m1, m2, . . . , mk) = m1 × m2 × · · · × mk
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a ≡ b(mod m1 × m2 × · · · × mk),
ax ≡ b(mod m),
x ∈ Z
x = x0 ax ≡ b(mod m) x1 ≡
x0(mod m) ax1 ≡ ax0 ≡ b(mod m) x1
m
a,b,m ∈ Z
m > 0
mdc(a, m) = d
d |b
ax ≡ b(mod m)
d|b
ax ≡ b(mod m)
d
m
ax ≡ b(mod m)
ax − my = b x ax ≡ b(mod m)
y ∈ Z
ax − my = b
d |b
d|b
ax − my = b
x = x0 + m
d t, y = y0 + a
dt,
x = x0 y = y0 x
x = x0 + m
d t,
x1 =
x0 + m
d t1 x2 = x0 +
m
d t2 m
x0 + m
d t1 ≡ x0 +
m
d t2(mod m).
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x0
m
d t1 ≡
m
d t2(mod m).
md
|m
mdc(m, m
d ) =
md
t1 ≡ t2(mod d).
x = x0 + (m/d)t t
d
x = x0 + (m/d)t t = 0, 1, 2, . . . , d − 1
8x ≡ 4(mod 12)
mdc(8, 12) = 4 4|4.
12/4 = 3
8x−12y = 4
x0 = 2
x = 2 + 3 × 0 = 2,
x = 2 + 3 × 1 = 5,
x = 2 + 3 × 2 = 8,
x = 2 + 3 × 3 = 11.
ax ≡ 1(mod m).
mdc(a, m) = 1
m
a
mdc(a, m) = 1
ax ≡ 1(mod m)
a
m
7x ≡ 1(mod 31)
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mdc(7, 31) = 1.
7x − 31y = 1,
x0 = 9 7x ≡
1(mod 31)
x ≡ 9(mod 31),
a m
ax ≡ b(mod m).
ā
a
m
aā ≡ 1(mod m)
ax ≡ b(mod m),
ā
ā(ax) ≡ āb(mod m),
x ≡ āb(mod m).
7x ≡ 22(mod31)
7x ≡ 22(mod 31).
7̄ = 9
9 × 7x ≡ 9 × 22(mod 31).
x ≡ 198 ≡ 12(mod 31).
mdc(7, 31) = 1
7x ≡ 22(mod 31)
31
p
a
p
a ≡ 1(mod p)
a ≡ −1(mod p)
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2 × 2
a ≡ 1(mod p) a ≡ −1(mod p) a2 ≡ 1(mod p) a
p
a
p
a2 = a.a ≡ 1(mod p)
p|(a2 − 1)
a2 − 1 = (a − 1)(a + 1)
p|(a − 1)
p|(a + 1)
a ≡ 1(mod p)
a ≡ −1(mod p)
2 × 2
a,b,c,d,e,f,m ∈ Z
m > 0
mdc(∆, m) = 1
∆ = ad − bc ax + by ≡ e(mod m)cx + dy ≡ f (mod m)
m
x ≡ ∆̄(de − bf )(mod m)
y ≡ ¯∆(af − ce)(mod m),
∆̄
∆
m
d
b
adx + bdy ≡ de(mod m)bcx + bdy ≡ bf (mod m),
(ad − bc)x ≡ de − bf (mod m),
∆x ≡ de − bf (mod m).
∆̄
∆
m
x ≡ ∆̄(de − bf )(mod m).
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2 × 2
c
a
acx + bcy ≡ ce(mod m)
acx + ady ≡ af (mod m).
(ad − bc)y ≡ af − ce(mod m),
∆y ≡ af − ce(mod m).
∆̄
y ≡ ∆̄(af − ce)(mod m).
(x, y)
x ≡
∆̄(de − bf )(mod m)
y ≡ ∆̄(af − ce)(mod m),
ax + by ≡ a∆̄(de − bf ) + b∆̄(af − ce)≡ ∆̄(ade − abf − abf − bce)
≡ ∆̄(ad − bc)e
≡ e(mod m)
cx + dy ≡ c∆̄(de − bf ) + d∆̄(af − ce)
≡ ∆̄(cde − bcf − adf − cde)
≡ ∆̄(ad − bc)f
≡ f (mod m).
3x + 4y ≡ 5(mod 13)2x + 5y ≡ 7(mod 13) .
∆ = ad − bc = 3 × 5 − 4 × 2 = 7
mdc(∆, m) = mdc(7, 13) = 1
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2 × 2
x ≡ 2 × (de − bf ) = 2 × (5.5 − 4.7) = −6 ≡ 7(mod 13)
y ≡ 2 × (af − ce) = 2 × (3.7 − 2.5) = 22 ≡ 9(mod 13).
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•
•
•
•
•
•
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216 = 65536
m
f
P
C
f
f −1
f
P f −→ C
f −1
−−→ P
0, 1, . . . , 25
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x
y
{0, 1, 2, . . . , 26}
27x + y ∈ {0, 1, 2, . . . , 728} .
27 × 4 + 20 = 128.
729x + 27y + z ∈
{0, 1, 2, . . . , 19682} k
m
0
mk − 1
m
0, 1, 2, . . . , m−1
m
{0, 1, 2, . . . m − 1}
m
m
Zm
A , B , . . . Z
0, 1, . . . 25
P ∈ {0, 1, 2, . . . 25}
f
{0, 1, 2, . . . 25}
f (P ) =
P + 3,
x
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18 11 22 03 19 17 20 03 21 ,
.
.
m
0, 1, . . . , m − 1
b ∈ N
f
C = f (P ) ≡ P + b(mod m).
m = 26
b = 3
C ∈ {0, 1, . . . , m − 1}
P = f −1(C ) ≡ C − b(mod m).
A − Z
0 − 25
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b
C ≡ P + b(mod N )
P ≡ C − b(mod N )
b
.
b
20 ≡ 4 + b(mod 26),
b = 16
.
QHGKY CUTUI = 17 08 07 11 25 03 21 20 21 09
→ 01 18 17 21 09 13 05 04 05 19 = ARQUIMEDES
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b
f
C ≡ aP + b(mod m),
a
b
a = 7
b = 12
C ≡ 7P + 12(mod 26),
ARQUIMEDES = 00 17 16 20 08 12 04 03 04 18
→ 12 01 20 22 16 18 14 07 18 08 = M BUW QSOHOI
C ≡ aP + b(mod m)
P
C
P ≡ āC + −āb(mod m),
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ā
a
m
b
a−1b
mdc(a, m) = 1
mdc(a, m) > 1
a = 1
b = 0
m
a
b
m = 27
P
C
4a + b ≡ 12(mod 27)0a + b ≡ 8(mod 27)
a b
∆ =
4 × 1 − 1 × 0 = 4 mdc(∆, m) = mdc(4, 27) = 1
∆
∆̄ = 7
a ≡ 7(1 × 3 − 1 × 0)(mod 27)
b ≡ 7(12 × 0 − 8 × 3)(mod 27)=⇒
a ≡ 1(mod 27)
b ≡ 8(mod 27)
C ≡ P + 8(mod 27),
P ≡ C − 8(mod 27).
m = 28
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26a + b ≡ 27(mod 28)4a + b ≡ 01(mod 28)
∆ = 26×1−1×4 = 22
mdc(∆, m) = mdc(22, 28) = 1
22a ≡ 26(mod 28),
a = 5
b = 9
a = 19
b = 9
C ≡ 5P + 9(mod 28) C ≡ 19P + 9(mod 28),
P ≡ 11C + 13(mod 28)
P ≡ 3C + 1(mod 28).
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n
n
A
B
n × k Z A
B
m
aij ≡ bij(mod m) 1 ≤ i ≤ n 1 ≤ j ≤ k A
B
m
A ≡ B(mod m),
A ≡ B(mod m).
16 3
8 13
≡
5 3
−3 2
(mod 11).
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1 4 2
2 0 3
3 1 4 ≡
1 4 7
2 5 8
3 6 9 (mod 5).
A
B
n × k
A ≡ B(mod m)
C
k × p D
p × n
Z
AC ≡ BC (mod m) DA ≡ DB(mod m).
aij bij A B 1 ≤ i ≤ n
1 ≤ j ≤ k cij C 1 ≤ i ≤ k 1 ≤ j ≤ p
i
j
AC
BC
nt=1
(aitctj)
nt=1
(bitctj)
A ≡ B (mod m) ait ≡ bit(mod m) i k
n
t=1(aitctj) ≡
n
t=1(bitctj)(mod m).
AC ≡ BC (mod m)
DA ≡ DB(mod m)
a11x1 + a12x2 + . . . a1nxn ≡ b1(mod m)
a21x1 + a22x2 + . . . a2nxn ≡ b2(mod m)
. . . . . . . . . . . .
. . . . . . . . . . . .
an1x1 + an2x2 + . . . annxn ≡ bn(mod m)
n
AX ≡ B(mod m),
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A =
a11 a12 . . . a1n
a21 a22 . . . a2n
: : : :
an1 an2 . . . ann
, X =
x1
X 2
:
xn
B =
b1
b2
:
bn
.
6x + 8y ≡ 5(mod 11)4x + 10y ≡ 7(mod 11)
6 84 10
xy 5
7
≡ (mod 11)
A
Ā
n × n
A Ā ≡ ĀA ≡ I (mod m),
I =
1 0 . . . 0
0 1 . . . 0
. . .
0 0 . . . 1
n
Ā
A
Ā
A
m
B ≡ Ā(mod m)
B
A
m
B1 B2 A B1 ≡ B2(mod m)
B1A ≡ B2A ≡ I (mod m)
B1AB1 ≡ B2AB1(mod m) AB1 ≡ I (mod m)
B1 ≡ B2(mod m)
1 3
2 4
3 4
1 2
=
6 10
10 16
≡
1 0
0 1
(mod 5)
3 4
1 2
1 3
2 4
=
11 25
5 11
≡
1 0
0 1
(mod 5)
3 41 2
11 25
5 11
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2 × 2
m
A = a b
c d ∆ =
detA = ad − bc m
Ā = ∆̄
d −b
−c a
∆̄
∆
m
Ā
A
m
A Ā ≡ ĀA ≡ I (mod m)
A Ā ≡
a bc d
∆̄
d −b−c a
≡ ∆̄
ad − bc 0
0 −bc + ad
≡ ∆̄
∆ 0
0 ∆
≡
∆̄∆ 0
0 ∆∆̄
≡
1 0
0 1
= I (mod m)
ĀA ≡ ∆̄
d −b
−c a
a b
c d
≡ ∆̄
ad − bc 0
0 −bc + ad
≡ ∆̄
∆ 00 ∆
≡
∆̄∆ 00 ∆̄∆
≡
1 00 1
= I (mod m),
∆̄
∆
m
mdc(∆, m) = 1
3 4
4 6
detA = 2
Ā ≡ 7
6 −4
−4 3
≡
42 −28
−28 21
≡
3 11
11 8
(mod 13).
3 11
11 8
3 4
4 6
=
1 0
0 1
(mod 13).
2 37 8
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detA = −5 ≡ 21(mod 26)
Ā ≡ 5
8 −3
−7 2
≡
40 −15
−35 10
≡
14 11
17 10
(mod 26).
n × n
n × n
(i, j)
C ji C ij (−1)
i+ j
A
A
adj(A)
n×n
A n × n det(A) = 0
A−1
= 1
det(A) adj(A)
A
n × n
Z
m ∈ Z
mdc(∆, m) = 1
∆
Ā = ∆̄Adj(A)
A
m
∆̄
∆
m
mdc(det(A), m) = 1
det(A) = 0
1
∆adj(A) = A−1.
A
1∆
Aadj(A) = I .
Aadj(A) = ∆I.
mdc(det(A), m) = 1
∆̄
∆
m
A∆̄Adj(A) ≡ AAdj(A)∆̄ ≡ ∆∆̄I ≡ I (mod m)
∆̄A ≡ ∆̄Adj(A)A ≡ ∆̄∆I ≡ I (mod m).
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Ā = ∆̄Adj(A)
A
m
A =
2 5 62 0 21 2 3
∆ = −5 mdc(∆, 7) = 1
∆ = −5
∆̄ = 4
Ā = 4(adj(A)) = 4
−2 −3 5
−5 0 10
4 1 −10
=
−8 −12 20
−20 0 40
0 4 −40
≡
≡
−8 −12 20−20 0 400 4 −40
≡ 6 2 61 0 5
2 4 2
(mod 7).
A
m
AX ≡ B(mod m),
mdc(∆, m) = 1
Ā
A
Ā(AX ) ≡ ĀB(mod m)
( ĀA)X ≡ ĀB(mod m)
X ≡ ĀB(mod m)
X
ĀB(mod m)
2x + 3y ≡ 1(mod 26)7x + 8y ≡ 2(mod 26)
A =
2 3
7 8
,
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n
Ā =
2 3
7 8
,
X ≡ ĀB =
2 3
7 8
1
2
≡
10
11
(mod 26).
n
n
E n×n
0, 1, 2, . . . m − 1
m
n
n
m
E n×n m E
0 m − 1
n
n
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n
pi n × 1
p1, p2, . . . , pn
n × 1
E p1, p2, . . . , pt
c1, c2, . . . , ct n × 1
c1 = Ep1, c2 = Ep2, c1 = Ep3, . . . , ct = E pt
c1, c2, . . . , ct
V n×t p1, p2, . . . , pt
P = [ p1, p2, . . . , pt] ⇒ C = E P = [c1, c2, . . . , ct]
E =
2 3
7 8
.
PASCAL.
15 00 18 02 00 11.
15 00 18 02 00 11.
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n
P 2×3
P =
15 18 00
0 02 11
.
E
P
C = EP ≡
2 3
7 8
.
15 18 00
0 02 11
=
30 42 33
105 142 88
=
4 16 7
1 12 10
(mod 26).
4 1 16 12 7 10 −→ EBQMHK.
n = 3
E =
2 3 15
5 8 12
1 13 4
.
n = 3
E
det(E )(mod 26) = 11
E
MATEMATICALEGAL.
12 00 19 04 12 00 19 08 02 00 11 04 06 00 11.
12 00 19 04 12 00 19 08 02 00 11 04 06 00 11.
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n
P =
12 4 19 0 6
0 12 8 11 0
19 0 2 4 11
.
C = E.P =
2 3 15
5 8 12
1 13 4
12 4 19 0 6
0 12 8 11 0
19 0 2 4 11
=
23 18 14 15 21
2 12 1 6 6
10 4 1 3 24
C
23 02 10 18 12 04 14 01 01 15 06 03 21 06 24 −→ XCKSM EOBBPGDV GY.
E
E −1
E
D = E −1(mod m)
C
P = DC
P
2 3 15
5 8 121 13 4
−1
(mod 26) =
10 19 16
4 23 717 5 19
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n
C
AJXGTRJXDGKKIXL
00 09 23 06 19 17 09 23 03 06 10 10 08 23 11
C =
00 06 09 03 08
09 19 23 06 23
23 17 03 10 11
P = DC
M = DC = 10 19 164 23 7
17 5 19
00 06 09 03 0809 19 23 06 2323 17 03 10 11
= 19 17 03 18 1704 08 14 12 1414 0 13 11 18
M
19 04 14 17 08 00 03 14 13 18 12 11 17 14 18
TEORIADOSNUMEROS.
.
C = EP
P
C
P
n × n
m
C = EP
E = C P̄
E
m
D
n
E
P
C
n
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n
CRIPTOGRAFIAELEGAL.
P
C
E
D
E
CRIPTOGRAFIAELEGAL.
n = 2
n = 3
02 17 08 15 19 14 06 17 00 05 08 00 04 11 04 06 00 11.
19 08 21 11 05 06 11 10 19 08 11 05 23 00 07 21 06 25
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n
2
17
19
08
,
08
15
21
11
,
19
14
05
06
, . .
2 × 2
P =
02 08
17 15
det(P ) = −106
M
P
P =
02 19
17 14
det(M ) = −295
19 5
08 6
= E
02 19
17 14
E =
19 5
08 6
02 19
17 14
−1
(mod 26) =
3 13
22 4
CRIPTOGRAFIAELEGAL.
02 17 08 15 19 14 06 17 00 05 08 00 04 11 04 06 00 11.
P =
02 08 19 06 00 08 04 04 00
17 15 14 17 05 00 11 06 01
,
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n
C = E P =
3 13
22 4
02 08 19 06 00 08 04 04 00
17 15 14 17 05 00 11 06 01
C =
19 11 5 5 13 24 25 12 13
8 2 6 18 20 20 2 8 4
.
E 3×3
C = E P
19 11 11 08 23 21
08 05 10 11 00 06
21 06 19 05 07 25
= E 02 15 06 05 04 06
17 19 17 08 11 00
08 14 00 00 14 11
.
3 × 3
P =
02 15 06
17 19 17
08 14 00
,
P =
02 15 06
17 19 00
08 14 11
.
det(P )(mod 26) = 1
19 11 21
08 05 06
21 06 25
= E
02 15 06
17 19 00
08 14 11
.
E =
19 11 21
08 05 06
21 06 25
02 15 06
17 19 00
08 14 11
−1
(mod 26) =
2 3 15
5 8 12
9 1 21
.
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19 11 11 08 23 21
08 05 10 11 00 06
21 06 19 05 07 25
= 2 3 15
5 8 12
9 1 21
02 15 06 05 04 06
17 19 17 08 11 00
08 14 00 00 14 11
.
m
2 7
13 9
E
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mdc(∆, m)
∆̄
26
E
26
26
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E
−→ 25 04 17 14
P
25 17
04 14
E
P
C
0 23 2 9 −→
−→ 00 09 12 06
Ē
C
P
3 14 8 18 −→
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,
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1 2 3
4 5 6
7 8 9
M 1
{1, 2, 3}
A,M,M 1
M 2
c