# UTC 501 31/01/2025

## Exercice 7 

```
p = 17, q = 11

n = pq = 187

e = 7 premier avec (p - 1)(q - 1) = 160
```

### 1)

```
1 = u * 7 + v * 160
    |
    V
    d

160 = 22 * 7 + 6
7 = 1 * 6 + 1
6 = 6 * 1 + 0                   d    e
                                |    |
1 = 7 - 1 * 6                   V    V
1 = 7 - 1(160 - 22 * 7) =-160 + 23 * 7
```

### 2)

```
12 00 13 01 20 08
l  _  m  a  t  h

séparer en groupe de 3

120 013 012 008

--------------------------

120 = 

(120²)³ * 120 Mod(187)
120² = 14400 ≡ 1 Mod (187)
1 * 120 ≡ 120 Mod (187)

120

--------------------------

013 =

(13³)² = 2197² ≡ 140² Mod(187)
19600 ≡ 152 Mod (187)
152*13 ≡ 103 Mod (187)

106

--------------------------

012 = 

(12³)² = 1728² ≡ 45² Mod (187)
1764 ≡ 81 Mod (187)
81 * 12 ≡ 37 Mod (187)




---------------

12⁷ (Mod 187) 

35831808 modulo 187 = 177

35831808 ≡ 177 (mod 187)
177

---------------

00⁷ Mod(187)

0 modulo 187 = 0

0 ≡ 0 (mod 187)
0

---------------

13⁷ (Mod 187)

62748517 modulo 187 = 106

62748517 ≡ 106 (mod 187)
106

---------------

01⁷ (Mod 187)

1 modulo 187 = 1

1 ≡ 1 (mod 187)
1

---------------

20⁷ (Mod 187)

1280000000 modulo 187 = 147

1280000000 ≡ 147 (mod 187)
147

---------------

08⁷ Mod(187)

2097152 modulo 187 = 134

2097152 ≡ 134 (mod 187)
134

---------------

Résultat Final 177-000-106-001-147-134

---------------

120


```