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## Intro to Crypto and Mod

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e.g.1 (Page 4)

- E.g., m = 21 n = 9 21 can be expressed as 9 x 2 + 3 (i.e., nq + r)

r is defined to be 21 mod 9

q = 2

r = 3

21 mod 9 is equal to 3

0 r < n

e.g.2 (Page 9)

- Illustration of [(-m) mod n] = n – [m mod n]
- E.g., m = 4 n = 5E.g., m = 9 n = 5

4 mod 5 = 4

-4 mod 5

= 5 – (4 mod 5)

= 5 – 4 = 1

9 mod 5 = 4

-9 mod 5

= 5 – (9 mod 5)

= 5 – 4 = 1

e.g.3 (Page 10)

2 x 8 mod 9 = 16 mod 9 = 7

(2 mod 9) (8 mod 9) = 2 x 8 = 16

Conclusion: 2 x 8 mod 9 (2 mod 9) (8 mod 9)

(2 + 8) mod 9 = 1

(2 mod 9) + (8 mod 9) = 2 + 8 = 10

Conclusion: (2 + 8) mod 9 (2 mod 9) + (8 mod 9)

e.g.4 (Page 11)

- Illustration of Lemma 2.2
- E.g., i = 1 n = 5

1 mod 5 = 1

(1 + (-1)x5) mod 5 = -4 mod 5 = 1

(1 + 5) mod 5 = 6 mod 5 = 1

(1 + (-2)x5) mod 5 = -9 mod 5 = 1

(1 + 2x5) mod 5 = 11 mod 5 = 1

(1 + (-3)x5) mod 5 = -14 mod 5 = 1

(1 + 3x5) mod 5 = 16 mod 5 = 1

(1 + k.5) mod 5 = 1

e.g.5 (Page 11)

- Illustration of Lemma 2.2
- E.g., i = 11 n = 5

11 mod 5 = 1

(11 + (-1)x5) mod 5 = 6 mod 5 = 1

(11 + 5) mod 5 = 16 mod 5 = 1

(11 + 2x5) mod 5 = 21 mod 5 = 1

(11 + (-2)x5) mod 5 = 1 mod 5 = 1

(11 + 3x5) mod 5 = 26 mod 5 = 1

(11 + (-3)x5) mod 5 = -4 mod 5 = 1

(11 + k.5) mod 5 = 1

e.g.6 (Page 11)

- Prove that 11 mod 5 = (11 + k.5) mod 5for all integers k

Let r = 11 mod 5

By Euclid’s Division Theorem, we can write 11 = 5q + r

where q and r are two unique integers and 0 r < 5

Consider 11 + k.5

= (5q + r) + k.5

= 5q + r + k.5

= 5q + k.5 + r

= 5(q + k) + r

By the definition of Euclid’s division theorem, (11 + k.5) mod 5 = r

e.g.7 (Page 12)

- Illustration of Lemma 2.3
- E.g.,

(2 + 8) mod 9 = (2 + (8 mod 9)) mod 9

= ((2 mod 9) + 8) mod 9

= ((2 mod 9) + (8 mod 9)) mod 9

(2.8) mod 9 = (2 . (8 mod 9)) mod 9

= ((2 mod 9) . 8) mod 9

= ((2 mod 9) . (8 mod 9)) mod 9

e.g.8 (Page 12)

2 x 8 mod 9 = 16 mod 9 = 7

((2 mod 9) (8 mod 9)) mod 9 = 2 x 8 mod 9 = 16 mod 9 = 7

Conclusion: 2 x 8 mod 9 = ((2 mod 9) (8 mod 9)) mod 9

(2 + 8) mod 9 = 1

((2 mod 9) + (8 mod 9)) mod 9 = (2 + 8) mod 9 = 10 mod 9 = 1

Conclusion: (2 + 8) mod 9 = ((2 mod 9) + (8 mod 9)) mod 9

e.g.8

Claim: (20 + 17) mod 9 = ((20 mod 9) + (17 mod 9)) mod 9

(2 + 8) mod 9 = 1

((2 mod 9) + (8 mod 9)) mod 9 = (2 + 8) mod 9 = 10 mod 9 = 1

Conclusion: (2 + 8) mod 9 = ((2 mod 9) + (8 mod 9)) mod 9

e.g.8

Lemma 2.2 (Y + 9k) mod 9 = Y mod 9

Claim: (20 + 17) mod 9 = ((20 mod 9) + (17 mod 9)) mod 9

Why is it correct?

By Euclid’s Division Theorem, we can write 20 as follows. 20 = 9q1 + r1 where q1 and r1 are some unique integers.

= 9q1 + (20 mod 9)

By Euclid’s Division Theorem, we can write 17 as follows. 17 = 9q2 + r2 where q2 and r2 are some unique integers.

= 9q2 + (17 mod 9)

Consider (20 + 17) mod 9

= {[9q1 + (20 mod 9)] + [9q2 + (17 mod 9)]} mod 9

= {9q1 + (20 mod 9) + 9q2 + (17 mod 9)} mod 9

= [ (20 mod 9) + (17 mod 9) + 9q1 + 9q2] mod 9

= [ (20 mod 9) + (17 mod 9) + 9(q1 + q2)] mod 9

= [ (20 mod 9) + (17 mod 9)] mod 9

(by Lemma 2.2)

e.g.10 (Page 15)

- Illustration of Theorem 2.4
- Commutative law
- Associative law
- Distributive law

2 x9 8 = 8 x9 2

2 +9 8 = 8 +9 2

2 x9 (8 x9 1) = (2 x9 8) x9 1

2 +9 (8 +9 1) = (2 +9 8) +9 1

2 x9 (8 +9 1) = (2 x9 8) +9 (2 x9 1)

decryption

I love UST.

kfjEfklje$3

kfjEfklje$3

e.g.11keroro

keroro

I love UST.

Undecipherable

(cannot be decrypted easily)

attacker

sender

receiver

keroro

keroro

e.g.12 (Page 32)

Encryption function

= 5 .12 x

Multiplication modn

encryption

a = 5

n = 12

Is there any

division modn ?

7

Encrypted value

= 5 .12 7

= 5 . 7 mod 12

= 35 mod 12

= 11

e.g.14 (Page 32)

- Examples that an inverse function does not exist

T

S

a

1

2

b

f

3

c

4

S

T

Not a function!

a

1

2

b

No such f-1

3

c

4

Encrypted

0

6

0

3

e.g.15 (Page 34)Case (a): a = 4, n = 12

f4, 12 (x) = 4.x mod 12

The inverse of f4, 12 does not exist

S

T

0

0

1

1

The receiver cannot determine the original number.

2

2

3

3

4

4

f4, 12

5

5

6

6

sender

receiver

7

7

8

8

9

9

10

10

sender

receiver

11

11

Encrypted

6

6

6

2

e.g.16 (Page 35)Case (b): a = 3, n = 12

f3, 12 (x) = 3.x mod 12

The inverse of f3, 12 does not exist

S

T

0

0

1

1

The receiver cannot determine the original number.

2

2

3

3

4

4

f3, 12

5

5

6

6

sender

receiver

7

7

8

8

9

9

10

10

sender

receiver

11

11

Encrypted

5

1

11

7

e.g.17 (Page 36)Case (c): a = 5, n = 12

f5, 12 (x) = 5.x mod 12

The inverse of f5, 12 exists

S

T

0

0

1

1

The receiver can uniquely determine the original number.

2

2

3

3

4

4

f5, 12

5

5

6

6

sender

receiver

7

7

8

8

9

9

10

10

sender

receiver

11

11

Encrypted

Encrypted

1

5

5

1

5

e.g.18 (Page 41)- Private-key cryptosystems

Encrypt this message with this private-key

Decrypt this message with the same private-key

sender

receiver

private-key (e.g., 2)

private-key (e.g., 2)

It should be kept privately at the sender’s side.

It should be kept privately at the receiver’s side.

Encrypted

Encrypted

5

1

5

5

1

e.g.19 (Page 41)- Public-key cryptosystems

Encrypt this message with a public-key

Decrypt this message with a secret-key

It has some relationships with the public key.

sender

receiver

public key (e.g., 2)

secret-key (e.g., 4)

It can be kept publicly.

It should be kept privately at the receiver’s side.

Encrypted

Encrypted

Encrypted

5

1

5

5

1

e.g.19 (Page 41)This directory is accessible to the public.

Raymond

2

Peter

7

- Public-key cryptosystems

Encrypt this message with a public-key

Decrypt this message with a secret-key

Peter

Raymond

sender

receiver

public key (e.g., 2)

secret-key (e.g., 4)

It can be kept publicly.

It should be kept privately at the receiver’s side.

e.g.20 (Page 44)

- An ideal key pair (public key and secret key)
- Given a public key,
- it is difficult for the adversary to deduce the secret key

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