Dissertação - Gilberto Tenani

8/16/2019 Dissertação - Gilberto Tenani

1/62


2/62


3/62


4/62


5/62


6/62


7/62


8/62


9/62

2 × 2

n


10/62


11/62

a

b

a

b

m

a ≡ b(mod m)

m|(a − b)

m

Zm = {0, 1, 2, . . . , m − 1}

m

a

m


12/62


13/62

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


14/62

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


15/62

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


16/62

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


17/62

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)


18/62

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


19/62

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)


20/62

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)


21/62

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


22/62

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


23/62

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).


24/62

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)


25/62

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)


26/62

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).


27/62

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


28/62

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).


29/62


30/62

•

•

•

•

•

•


31/62


32/62

216 = 65536

m

f

P

C

f

f −1

f

P f −→ C

f −1

−−→ P

0, 1, . . . , 25


33/62

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


34/62

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


35/62

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


36/62

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),


37/62

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


38/62

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).


39/62

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).


40/62

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),


41/62

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


42/62

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


43/62

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).


44/62

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

,


45/62

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


46/62

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.


47/62

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.


48/62

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


49/62

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


50/62

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


51/62

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

,


52/62

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

.


53/62

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


54/62

mdc(∆, m)

∆̄

26

E

26

26


55/62

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 −→


56/62


57/62


58/62


59/62

,


60/62


61/62

1 2 3

4 5 6

7 8 9

M 1

{1, 2, 3}

A,M,M 1

M 2

c

Documents

Dissertação - Gilberto Tenani