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M ∈ {m1,m2, · · · ,mL} C ∈ {c1, c2, · · · , cL} K ∈ {k1, k2, · · · , kL}

PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

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1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0K = 1 0 0 1 1 0 1 0 0C = 1 1 0 0 1 0 0 1 0

1

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

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K0 K1 K10

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PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

K0 K1 K10

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

K0 K1 K10

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

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p a ap−1 ≡ 1 (mod p)

M C K KP KS

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��p a ap−1 ≡ 1 (mod p)

M C K KP KS

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�p a 1 ≤ a ≤ p − 1 ap−1 ≡ 1 (mod p)

M C K KP KS

1

��

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

��p = 5

14 = 4 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

25

��


M C K KP KS

1

��p a ap−1 ≡ 1 (mod p)

M C K KP KS

1

�p a 1 ≤ a ≤ p − 1 ap−1 ≡ 1 (mod p)

M C K KP KS

1

��

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

�

a1 = a2 b1 = b2 a1 = b1 (mod p) a2 = b2 (mod p)

p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

��

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

26

��

��p a ap−1 ≡ 1 (mod p)

M C K KP KS

1

� ��

��

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ(n) = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5φ(15) = 2 · 4 = 8

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

1

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 15)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 15)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(2)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

3

27

��

�� p a ap−1 ≡ 1 (mod p)

M C K KP KS

1

� ��

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ(n) = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

��

n

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

�� n

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

n

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

n

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

n

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

n

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

28

��&�$%��!�#�� !�"��

, �+��+p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ(n) = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

+��,'''''''''''''*�)(��n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��, p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

+

��, n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

+n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��,

�,

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��,M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

�,M = Cd (mod n) M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

Cd = (Me)d = Med = Mkφ(n)+1 = M (mod n)

M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

29

��

��#

��#

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��#M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��#M = Cd (mod n) M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

�#!!!! "��

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5φ(15) = 2 · 4 = 8

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

1

M = C2/Cx1 (mod p) e = 3

Cx1 = grx = gxr = yr (mod p)

C1 = gr (mod p) C2 = Myr (mod p) y = gx (mod p)

x = logg y r

φ(n) g gx = y (mod p) x y 1 ≤ x, y ≤ p− 1


M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

1

M = C15 = 32768 = 2184 × 15 + 8 = 8 (mod 15)

M = 8 C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)e × d = 45 = 4 × 11 + 1 = 1 (mod 11)

M = C2/Cx1 (mod p) e = 3 d = 15



x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

1

M = C15 = 470184984576 = 31345665638×15+6 = 6(mod 15)

M = 6 C = M 3 = 216 = 14× 15 + 6 = 6 (mod 15)

M = C15 = 32768 = 2184 × 15 + 8 = 8 (mod 15)

M = 8 C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)e × d = 45 = 4 × 11 + 1 = 1 (mod 11)

M = C2/Cx1 (mod p) e = 3 d = 15



x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

1

d = 3

= {0, 1, 2, · · · , P − 1}

K0 K1 K10

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

d = 3 e× d = 9 = 8 + 1 = 1 (mod 8)

= {0, 1, 2, · · · , P − 1}

K0 K1 K10

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)

d = 3 e× d = 9 = 8 + 1 = 1 (mod 8)

= {0, 1, 2, · · · , P − 1}

K0 K1 K10

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

M = C3 = 8 (mod 15)

C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)

d = 3 e× d = 9 = 8 + 1 = 1 (mod 8)

= {0, 1, 2, · · · , P − 1}

K0 K1 K10

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

M = C3 = 8 (mod 15)

C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)

C = M 3 = 216 = 14× 15 + 6 = 6 (mod 15)

d = 3 e× d = 9 = 8 + 1 = 1 (mod 8)

= {0, 1, 2, · · · , P − 1}

K0 K1 K10

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

M = 0 1 0 1 0 0 1 1 0 · · ·K = 1 0 0 1 1 0 1 0 0 · · ·C = 1 1 0 0 1 0 0 1 0 · · ·

1

M = C3 = 8 (mod 15)

C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)

C = M 3 = 216 = 14× 15 + 6 = 6 (mod 15)

M = C3 = 216 = 14× 15 + 6 = 6 (mod 15)

d = 3 e× d = 9 = 8 + 1 = 1 (mod 8)

= {0, 1, 2, · · · , P − 1}

K0 K1 K10

0⊕ 0 = 0

0⊕ 1 = 1

1⊕ 0 = 1

1⊕ 1 = 0


PM(m) PC(c)

1

30

!�� &-$+,�!%"(&*��#$'(")��

3�2��2p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ(n) = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

2��3.............1�0/�n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��3 p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

2

��3 n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

2n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��3

��3

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��3M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

3�

3��

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

φ(n)


M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��

M = Cd (mod n) M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

31

��" !#��

%�$��$p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ(n) = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

$��%��n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��% p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

$

��% n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

$n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

p, q n = pq φ = (p − 1)(q − 1)

a×1 = b1 (mod p)

a×2 = b2 (mod p)...

a×p − 1 = bp−1 (mod p)

(1)

ap−1(1×2×· · ·×(p−1)) = 1×2×· · ·×(p−1) (mod p)

a1 − a2 = b1 − b2 = 0 (mod p) b1 = b2


p = 5

14 = 1 = 1 (mod 5)

24 = 16 = 1 (mod 5)

34 = 81 = 1 (mod 5)

44 = 256 = 1 (mod 5)

p a 1 ≤ a ≤ p − 1 ap−1 = 1 (mod p)

M C K KP KS

1

%�

%��

��%

��%

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��%M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��%

Ce = (Md)e = Med = Mkπ(n)+1 = M (mod n)

C = Md (mod n) M = Me (mod n)

M = C2/yr (C1, C2) ZN

k k − l H(S)

(5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13))

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

Ce = (Md)e = Med = Mkπ(n)+1 = M (mod n)

C = Md (mod n) M = Ce (mod n)

M = C2/yr (C1, C2) ZN

k k − l H(S)

(5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13))

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

Ce = (Md)e = Med = Mkφ(n)+1 = M (mod n)

C = Md (mod n) M = Ce (mod n)

M = C2/yr (C1, C2) ZN

k k − l H(S)

(5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13))

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

32

�� p a ap−1 ≡ 1 (mod p)

M C K KP KS

1

��φ(n) g


M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

1

φ(n) g gx = y (mod p) x, y 1 ≤ x, y ≤ p − 1


M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

�

��

��

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p) x = logg y



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

��

��

p = 13 g = 2

21 = 2 = 2 (mod 13)

22 = 4 = 4 (mod 13)

23 = 8 = 8 (mod 13)

24 = 16 = 3 (mod 13)

25 = 32 = 6 (mod 13)

26 = 64 = 12 (mod 13)

27 = 128 = 11 (mod 13)

28 = 256 = 9 (mod 13)

29 = 512 = 5 (mod 13)

210 = 1024 = 10 (mod 13)

211 = 2048 = 7 (mod 13)

212 = 4096 = 1 (mod 13)

(1)

M = C15 = 470184984576 = 31345665638×15+6 = 6(mod 15)

M = 6 C = M 3 = 216 = 14× 15 + 6 = 6 (mod 15)

M = C15 = 32768 = 2184 × 15 + 8 = 8 (mod 15)

M = 8 C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)

1

p = 13 g = 2

21 = 2 = 2 (mod 13)

22 = 4 = 4 (mod 13)

23 = 8 = 8 (mod 13)

24 = 16 = 3 (mod 13)

25 = 32 = 6 (mod 13)

26 = 64 = 12 (mod 13)

27 = 128 = 11 (mod 13)

28 = 256 = 9 (mod 13)

29 = 512 = 5 (mod 13)

210 = 1024 = 10 (mod 13)

211 = 2048 = 7 (mod 13)

212 = 4096 = 1 (mod 13)

(1)

M = C15 = 470184984576 = 31345665638×15+6 = 6(mod 15)

M = 6 C = M 3 = 216 = 14× 15 + 6 = 6 (mod 15)

M = C15 = 32768 = 2184 × 15 + 8 = 8 (mod 15)

M = 8 C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)

1

p = 13 g = 2

21 = 2 = 2 (mod 13)

22 = 4 = 4 (mod 13)

23 = 8 = 8 (mod 13)

24 = 16 = 3 (mod 13)

25 = 32 = 6 (mod 13)

26 = 64 = 12 (mod 13)

27 = 128 = 11 (mod 13)

28 = 256 = 9 (mod 13)

29 = 512 = 5 (mod 13)

210 = 1024 = 10 (mod 13)

211 = 2048 = 7 (mod 13)

212 = 4096 = 1 (mod 13)

(1)

M = C15 = 470184984576 = 31345665638×15+6 = 6(mod 15)

M = 6 C = M 3 = 216 = 14× 15 + 6 = 6 (mod 15)

M = C15 = 32768 = 2184 × 15 + 8 = 8 (mod 15)

M = 8 C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)

1

��

33

"%#$&$%��! ��

��p a ap−1 ≡ 1 (mod p)

M C K KP KS

1

)��('')��φ(n) g


M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

1



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

)

)�

)��

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

��)

��)

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1


M C K KP KS

1

(φ(n) g


M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

1

(

�)

��)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��)y = gx (mod p) x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

C1 = gr (mod p) C2 = myr (mod p) y = gx (mod p)

x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1


x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1

�� ) C1 = gr (mod p) C2 = myr (mod p) y = gx (mod p)

x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1


x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1



x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

1

M = C2/Cx1 (mod p)



x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

1

34

��

�!

��!

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

��!y = gx (mod p) x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1


x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1


x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1

�� ! C1 = gr (mod p) C2 = myr (mod p) y = gx (mod p)

x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1


x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1



x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

1

M = C2/Cx1 (mod p)



x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

1

p = 13 g = 2

21 = 2 = 2 (mod 13)

22 = 4 = 4 (mod 13)

23 = 8 = 8 (mod 13)

24 = 16 = 3 (mod 13)

25 = 32 = 6 (mod 13)

26 = 64 = 12 (mod 13)

27 = 128 = 11 (mod 13)

28 = 256 = 9 (mod 13)

29 = 512 = 5 (mod 13)

210 = 1024 = 10 (mod 13)

211 = 2048 = 7 (mod 13)

212 = 4096 = 1 (mod 13)

(1)

M = C15 = 470184984576 = 31345665638×15+6 = 6(mod 15)

M = 6 C = M 3 = 216 = 14× 15 + 6 = 6 (mod 15)

M = C15 = 32768 = 2184 × 15 + 8 = 8 (mod 15)

M = 8 C = M 3 = 512 = 34× 15 + 2 = 2 (mod 15)

1

�!��

C2 = 8 = 32 = 6 (mod 13)

M = 8 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

21 = 2 = 2 (mod 13)

22 = 4 = 4 (mod 13)

23 = 8 = 8 (mod 13)

24 = 16 = 3 (mod 13)

25 = 32 = 6 (mod 13)

26 = 64 = 12 (mod 13)

27 = 128 = 11 (mod 13)

28 = 256 = 9 (mod 13)

29 = 512 = 5 (mod 13)

210 = 1024 = 10 (mod 13)

211 = 2048 = 7 (mod 13)

212 = 4096 = 1 (mod 13)

(1)

M = C15 = 470184984576 = 31345665638×15+6 = 6(mod 15)

M = 6 C = M 3 = 216 = 14× 15 + 6 = 6 (mod 15)

1

��!

��!

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1


M C K KP KS

1

φ(n) g


M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

1

��

C2 = 8 = 32 = 6 (mod 13)

M = 8 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

21 = 2 = 2 (mod 13)

22 = 4 = 4 (mod 13)

23 = 8 = 8 (mod 13)

24 = 16 = 3 (mod 13)

25 = 32 = 6 (mod 13)

26 = 64 = 12 (mod 13)

27 = 128 = 11 (mod 13)

28 = 256 = 9 (mod 13)

29 = 512 = 5 (mod 13)

210 = 1024 = 10 (mod 13)

211 = 2048 = 7 (mod 13)

212 = 4096 = 1 (mod 13)

(1)

M = C15 = 470184984576 = 31345665638×15+6 = 6(mod 15)

M = 6 C = M 3 = 216 = 14× 15 + 6 = 6 (mod 15)

1

Cx1 = 69 = 10077696 = 5

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

21 = 2 = 2 (mod 13)

22 = 4 = 4 (mod 13)

23 = 8 = 8 (mod 13)

24 = 16 = 3 (mod 13)

25 = 32 = 6 (mod 13)

26 = 64 = 12 (mod 13)

27 = 128 = 11 (mod 13)

28 = 256 = 9 (mod 13)

29 = 512 = 5 (mod 13)

210 = 1024 = 10 (mod 13)

211 = 2048 = 7 (mod 13)

212 = 4096 = 1 (mod 13)

(1)

M = C15 = 470184984576 = 31345665638×15+6 = 6

1

Cx1 = 69 = 10077696 = 5

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

21 = 2 = 2 (mod 13)

22 = 4 = 4 (mod 13)

23 = 8 = 8 (mod 13)

24 = 16 = 3 (mod 13)

25 = 32 = 6 (mod 13)

26 = 64 = 12 (mod 13)

27 = 128 = 11 (mod 13)

28 = 256 = 9 (mod 13)

29 = 512 = 5 (mod 13)

210 = 1024 = 10 (mod 13)

211 = 2048 = 7 (mod 13)

212 = 4096 = 1 (mod 13)

(1)

M = C15 = 470184984576 = 31345665638×15+6 = 6

1

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

21 = 2 = 2 (mod 13)

22 = 4 = 4 (mod 13)

23 = 8 = 8 (mod 13)

24 = 16 = 3 (mod 13)

25 = 32 = 6 (mod 13)

26 = 64 = 12 (mod 13)

27 = 128 = 11 (mod 13)

28 = 256 = 9 (mod 13)

29 = 512 = 5 (mod 13)

210 = 1024 = 10 (mod 13)

211 = 2048 = 7 (mod 13)

212 = 4096 = 1 (mod 13)

(1)

M = C15 = 470184984576 = 31345665638×15+6 = 6

1

5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13)

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

(5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13))

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

35

"%#$&$%��! ��

�/

��/

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

� /y = gx (mod p) x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1


x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1


x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1

�� / C1 = gr (mod p) C2 = myr (mod p) y = gx (mod p)

x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1


x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1



x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

1

M = C2/Cx1 (mod p)



x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

1

��/

��/

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1

y = gx (mod p)



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

1


M C K KP KS

1

.φ(n) g


M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

1

.


x = logg y r



M = Cd (mod n)

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

1

k k − l H(S)

(5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13))

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

M = C2/yr

k k − l H(S)

(5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13))

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

�� '�(-

M C C = Me (mod n)

n e d e × d = 1 (mod φ(n))

B = {b1, b2, · · · , bφ(n)}

A = {ab1, ab2, · · · , abφ(n)}

C = b1 × b2 × · · ·× bφ(n)

aφ(n)C = C (mod n)

18 =1 = 1 (mod 15)

28 =256 = 15 × 17 + 1 = 1 (mod 15)

48 =65536 = 15 × 4369 + 1 × 1 = 1 (mod 5)

78 =5764801 = 15 × 384320 + 1 = 1 (mod 5)...

aφ(n) = 1 (mod n)

p, q n = pq φ(n) = (p− 1)(q − 1) n = 15 = 3 · 5

φ(15) = 2 · 4 = 8

1

'�)*,M = C2/yr (C1, C2)

k k − l H(S)

(5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13))

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

'��+-�

��

36

��#�"!��$%&'

��

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N

Pe ≡ 1KEDH(SK, SK)

P (yz|x)

SK

XNSK Y N ZN SK

1

��

XN Y N

Pe ≡ Pr{SK = SK}Rs ≥ H(S) Rk ≥ h

Rs = log MsK Rk = log Ms

K

Ws ∈ {0, 1, · · · ,Ms − 1} Wk ∈ {0, 1, · · · ,Mk − 1}1KH(SK|Ws)

1KH(SK|Ws) ≥ h 0 ≤ h ≤ H(S)

1nI(W1; W2) = rPa ≈ 2−nr

H(Wi)/H(Sn) → 1 + rH(S)

H(Un)/H(Sn) → 1 + rH(S)

W1 = F1(Sn, Un)W2 = F2(Sn, Un)Pa → 0H(Wi)/H(S) → ∞Pa

Wi

W1 = RW2 = S + RWn

1 Wn2 Wn

1 Wn2

W n1 W n

2

(2, 2)

送信可能な情報量u ≤ h − t(h, h − t, n)

1

XN Y N



K

Ws ∈ {0, 1, · · · ,Ms − 1} Wk ∈ {0, 1, · · · ,Mk − 1}1KH(SK|Ws)

1KH(SK|Ws) ≥ h 0 ≤ h ≤ H(S)

1nI(W1; W2) = rPa ≈ 2−nr

H(Wi)/H(Sn) → 1 + rH(S)

H(Un)/H(Sn) → 1 + rH(S)


Wi

W1 = RW2 = S + RWn

1 Wn2 Wn

1 Wn2

W n1 W n

2

(2, 2)


1



K

Ws ∈ {0, 1, · · · ,Ms − 1} Wk ∈ {0, 1, · · · ,Mk − 1}1KH(SK|Ws)

1KH(SK|Ws) ≥ h 0 ≤ h ≤ H(S)

1nI(W1; W2) = rPa ≈ 2−nr

H(Wi)/H(Sn) → 1 + rH(S)

H(Un)/H(Sn) → 1 + rH(S)


Wi

W1 = RW2 = S + RWn

1 Wn2 Wn

1 Wn2

W n1 W n

2

(2, 2)


1

Alice Bob

��

Ws

Pe < ε P (Z)e > 1 − ε

CZ

CZ ≡ maxX

I(X ; Z)

I(X ; Y ) =!

x

!

y

P (x, y) logP (x, y)

P (x)P (y)

H(S) ≡!

s

−P (s) log P (s)

R < CY R > CZ

CY ≡ maxX

I(X ; Y )

"W1"W1 W1

#S

W1 W2 S R

NK = H(S)

I(X;Y )

Pe ≤ ε1KH(SK|Ws) ≥ H(S) − ε

XN Y N

Pe ≡ Pr{SK = SK} ≤ εRs ≥ H(S) Rk ≥ h

1

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

Alice Bob Eve

��

XN Y N



K

Ws ∈ {0, 1, · · · ,Ms − 1} Wk ∈ {0, 1, · · · ,Mk − 1}1KH(SK|Ws)

1KH(SK|Ws) ≥ h 0 ≤ h ≤ H(S)

1nI(W1; W2) = rPa ≈ 2−nr

H(Wi)/H(Sn) → 1 + rH(S)

H(Un)/H(Sn) → 1 + rH(S)


Wi

W1 = RW2 = S + RWn

1 Wn2 Wn

1 Wn2

W n1 W n

2

(2, 2)


1

XN Y N



K

Ws ∈ {0, 1, · · · ,Ms − 1} Wk ∈ {0, 1, · · · ,Mk − 1}1KH(SK|Ws)

1KH(SK|Ws) ≥ h 0 ≤ h ≤ H(S)

1nI(W1; W2) = rPa ≈ 2−nr

H(Wi)/H(Sn) → 1 + rH(S)

H(Un)/H(Sn) → 1 + rH(S)


Wi

W1 = RW2 = S + RWn

1 Wn2 Wn

1 Wn2

W n1 W n

2

(2, 2)


1

M = C2/yr (C1, C2) ZN

k k − l H(S)

(5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13))

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

M = C2/yr (C1, C2) ZN

k k − l H(S)

(5 × 8 = 40 = 3 × 13 + 1 = 1 (mod 13))

M = 6×5−1 = 6×8 = 48 = 13×3+9 = 9 (mod 13)

Cx1 = 69 = 10077696 = 5 (mod 13)

C2 = 9 × 55 = 9 × 3125 = 28125 = 6 (mod 13)

M = 9 r = 5 C1 = 25 = 32 = 6 (mod 13)

p = 13 g = 2 x = 9 y = 29 = 512 = 5 (mod 13)

1

37

��

P (yz|x)S ∈ {0, 1, 2, · · · ,M − 1}Xn

S Y n Zn S

1

��

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

��

��

P (y|x) P (z|x)

K

N≥ max[I(X ; Y ) − I(X ; Z), I(X ; Y ) − I(Y ; Z)]

H(S)K

N≤ min[I(X ; Z), I(X ; Y |Z)]

H(S)

H(X) − H(X|Y ) = I(X ; Y ) R ≥ H(X) ZN

R = log MN R ≥ H(X|Y )

W ∈ {0, 1, · · · ,M − 1}!XN

K

N=

I(X ; Y )

H(S)W = f (x)

S1

KH(SK|Ws) ≥ H(S) − ε

Rk ≥ H(S)P (x) = Q(x) D(P∥Q) ≥ 0

0 ≤ I(X ; Y ) ≤ min{H(X), H(Y )}

H(Y |X) ≤ H(Y )

PX,Y (x, y) = PX(x)PY (y)

D(P∥Q) ="

x∈XP (x) log

P (x)

Q(x)

1

P (y|x) P (z|x)

K


H(S)K

N≤ min[I(X ; Z), I(X ; Y |Z)]

H(S)

H(X) − H(X|Y ) = I(X ; Y ) R ≥ H(X) ZN


W ∈ {0, 1, · · · ,M − 1}!XN

K

N=

I(X ; Y )

H(S)W = f (x)

S1


Rk ≥ H(S)P (x) = Q(x) D(P∥Q) ≥ 0

0 ≤ I(X ; Y ) ≤ min{H(X), H(Y )}

H(Y |X) ≤ H(Y )


D(P∥Q) ="

x∈XP (x) log

P (x)

Q(x)

1

Alice Bob

��

38

��

P (yz|x)S ∈ {0, 1, 2, · · · ,M − 1}Xn

S Y n Zn S

1

��%��

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

1kH(Sk|Zn) 1

KI(Sk; Zn)R ≡ K

N


P (yz|x)

SK

XNSK Y N ZN SK

1

��

��



K

Ws ∈ {0, 1, · · · ,Ms − 1} Wk ∈ {0, 1, · · · ,Mk − 1}1KH(SK|Ws)

1KH(SK|Ws) ≥ h 0 ≤ h ≤ H(S)

1nI(W1; W2) = rPa ≈ 2−nr

H(Wi)/H(Sn) → 1 + rH(S)

H(Un)/H(Sn) → 1 + rH(S)


Wi

W1 = RW2 = S + RWn

1 Wn2 Wn

1 Wn2

W n1 W n

2

(2, 2)


1

Pe < ε Pe > 1 − ε

CZ

CZ ≡ I(X ; Z)

I(X ; Y ) =!

x

!

y


P (x)P (y)

H(S) ≡!

s

−P (s) log P (s)

R < CY

CY ≡ maxX

I(X ; Y )

"W1"W1 W1

#S

W1 W2 S R

NK = H(S)

I(X;Y )


XN Y N


1

Pe < ε P (Z)e > 1 − ε

CZ

CZ ≡ I(X ; Z)

I(X ; Y ) =!

x

!

y


P (x)P (y)

H(S) ≡!

s

−P (s) log P (s)

R < CY R > CZ

CY ≡ maxX

I(X ; Y )

"W1"W1 W1

#S

W1 W2 S R

NK = H(S)

I(X;Y )


XN Y N


1

��-

Pe < ε P (Z)e > 1 − ε

CZ

CZ ≡ maxX

I(X ; Z)

I(X ; Y ) =!

x

!

y


P (x)P (y)

H(S) ≡!

s

−P (s) log P (s)

R < CY R > CZ

CY ≡ maxX

I(X ; Y )

"W1"W1 W1

#S

W1 W2 S R

NK = H(S)

I(X;Y )


XN Y N


1

Pe < ε P (Z)e > 1 − ε

CZ

CZ ≡ maxX

I(X ; Z)

I(X ; Y ) =!

x

!

y


P (x)P (y)

H(S) ≡!

s

−P (s) log P (s)

R < CY R > CZ

CY ≡ maxX

I(X ; Y )

"W1"W1 W1

#S

W1 W2 S R

NK = H(S)

I(X;Y )


XN Y N


1

��),'- R = K/N

Ws

Pe < ε P (Z)e > 1 − ε

CZ

CZ ≡ maxX

I(X ; Z)

I(X ; Y ) =!

x

!

y


P (x)P (y)

H(S) ≡!

s

−P (s) log P (s)

R < CY R > CZ

CY ≡ maxX

I(X ; Y )

"W1"W1 W1

#S

W1 W2 S R

NK = H(S)

I(X;Y )


XN Y N

1

R = K/N H(S)

Ws

Pe < ε P (Z)e > 1 − ε

CZ

CZ ≡ maxX

I(X ; Z)

I(X ; Y ) =!

x

!

y


P (x)P (y)

H(S) ≡!

s

−P (s) log P (s)

R < CY R > CZ

CY ≡ maxX

I(X ; Y )

"W1"W1 W1

#S

W1 W2 S R

NK = H(S)

I(X;Y )


XN Y N

1

��&+'*(,-

R = K/N H(S)

Ws

Pe < ε P (Z)e > 1 − ε

CZ

CZ ≡ maxX

I(X ; Z)

I(X ; Y ) =!

x

!

y


P (x)P (y)

H(S) ≡!

s

−P (s) log P (s)

R < CYH(S) R > CZ

H(S)

CY ≡ maxX

I(X ; Y )

"W1"W1 W1

#S

W1 W2 S R

NK = H(S)

I(X;Y )


XN Y N

1

R = K/N H(S)

Ws

Pe < ε P (Z)e > 1 − ε

CZ

CZ ≡ maxX

I(X ; Z)

I(X ; Y ) =!

x

!

y


P (x)P (y)

H(S) ≡!

s

−P (s) log P (s)

R < CYH(S) R > CZ

H(S)

CY ≡ maxX

I(X ; Y )

"W1"W1 W1

#S

W1 W2 S R

NK = H(S)

I(X;Y )


XN Y N

1

P (y|x) P (z|x)

K


H(S)K

N≤ min[I(X ; Z), I(X ; Y |Z)]

H(S)

H(X) − H(X|Y ) = I(X ; Y ) R ≥ H(X) ZN


W ∈ {0, 1, · · · ,M − 1}!XN

K

N=

I(X ; Y )

H(S)W = f (x)

S1


Rk ≥ H(S)P (x) = Q(x) D(P∥Q) ≥ 0

0 ≤ I(X ; Y ) ≤ min{H(X), H(Y )}

H(Y |X) ≤ H(Y )


D(P∥Q) ="

x∈XP (x) log

P (x)

Q(x)

1

P (y|x) P (z|x)

K


H(S)K

N≤ min[I(X ; Z), I(X ; Y |Z)]

H(S)

H(X) − H(X|Y ) = I(X ; Y ) R ≥ H(X) ZN


W ∈ {0, 1, · · · ,M − 1}!XN

K

N=

I(X ; Y )

H(S)W = f (x)

S1


Rk ≥ H(S)P (x) = Q(x) D(P∥Q) ≥ 0

0 ≤ I(X ; Y ) ≤ min{H(X), H(Y )}

H(Y |X) ≤ H(Y )


D(P∥Q) ="

x∈XP (x) log

P (x)

Q(x)

1

��%��%�#��#$"��

CY > CZ P (y|x) P (z|x)

K


H(S)K

N≤ min[I(X ; Z), I(X ; Y |Z)]

H(S)

H(X) − H(X|Y ) = I(X ; Y ) R ≥ H(X) ZN


W ∈ {0, 1, · · · ,M − 1}!XN

K

N=

I(X ; Y )

H(S)W = f (x)

S1


Rk ≥ H(S)P (x) = Q(x) D(P∥Q) ≥ 0

0 ≤ I(X ; Y ) ≤ min{H(X), H(Y )}

H(Y |X) ≤ H(Y )


D(P∥Q) ="

x∈XP (x) log

P (x)

Q(x)

1

Alice Bob

��! �

39

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