Vazirligi mirzo ulug„bek nomidagi

Shifrlash.

Berilgan xabar M harfi bilan belgilanadi. Shifrtekst esa, C bilan belgilanadi. Xabar bit ko‗rinishida yozib olinadi (m₁, m₂, …, m_n):
⎛ ⁿ ⎞
_⎜∑b_i m_{i ⎟}mod q = c_i
⎝ i=1 ⎠
formula yordamida shifrlanadi. Shifrtekst C = {c_1, c₂, …} ko‗rinishida hosil bo‗ladi.

Shifrni ochish.

C = {c_1, c₂, …} ko‗rinishidagi shifrtekstni
(r⁻¹c ) mod q = m
i i
formula yordamida ochiladi. Ochilgan m_i larni yig‗ib chiqib, M hosil qilinadi.
R.Merkl tizimni buzib ocha olgan odamga 100 AQSh dollari mukofoti berishini e‘lon qildi. 1982 yilda Adi Shamir bu mukofotga sazovor bo‗ldi. U mahfiy kalitga teng bo‗lmagan kalit yasab shifrlarni ocha oldi.

Misol.

O‗suvchi ketma-ketlik w = {2, 3, 7, 15}, modul son q=28, q modul bilan o‗zaro tub bo‗lgan r=11, shifrlanadigan matn M=HA so‗zi bo‗lsin.

Shifrlash:

1. 
   HA so‗zini ASCII jadvali yordamida bit ko‗rinishga o‗tkazamiz: 0100100001000001.
   
2. 
   b_i = (w_i * r) mod q formula orqali normal ketma-ketlik hosil qilamiz:
   

b₁=(w₁ ⋅ r) mod q= b₂ =(w₂ ⋅ r) mod q= b₃ =(w₃ ⋅ r) mod q= b₄ =(w₄ ⋅ r) mod q=
(^{2 ⋅11})^{mod 28=22}
(^3⋅11)^{mod 28=5}
₍7 ⋅11₎mod 28=21
(^{15 ⋅11})^{mod 28=25}

3. 
   Ketma-ketlik hadlari 4 ta bo‗lgani uchun matnni 4 bitdan bloklarga
   

M₁ = _{m₁, m₂ , m₃ , m_{4 }} = 0100, M ₂ = _{m₁, m₂ , m₃ , m_{4 }} = 1000,

ajratamiz: _M
= _{m , m , m , m _} = 0100, M = _{m , m , m , m _} = 0001.

3 1 2 3 4 4 1 2 3 4

⎛ ⁴ ⎞

4. 
   _⎜∑b_i m_{i ⎟}mod q = c_i
   

formula yordamida shifrlaymiz:

⎝ i=1 ⎠


_{c =} ⎛ ⁴ ⎞
_{1 ⎜ ∑}b_i m_{i ⎟} mod q = ₍22 ⋅ 0 + 5 ⋅1+ 21⋅ 0 + 25 ⋅ 0₎mod 28 = 5
⎝ i=1 ⎠
_{c =} ⎛ ⁴ ⎞
_{2 ⎜ ∑}b_i m_{i ⎟} mod q = ₍22 ⋅1+ 5 ⋅ 0 + 21⋅ 0 + 25⋅ 0₎mod 28 = 22
⎝ i=1 ⎠
⎛ ⁴ ⎞
c₃ = _{⎜ ∑}b_i m_{i ⎟} mod q = ₍22 ⋅ 0 + 5 ⋅1+ 21⋅ 0 + 25 ⋅ 0₎mod 28 = 5
⎝ i=1 ⎠
⎛ ⁴ ⎞
c₄ = _{⎜ ∑}b_i m_{i ⎟} mod q = ₍22 ⋅ 0 + 5 ⋅ 0 + 21⋅ 0 + 25⋅1₎mod 28 = 25
⎝ i=1 ⎠

5. 
   Shifrtekst hosil bo‗ladi:
   

Shifrni ochish:

C = _{c₁, c₂ , c₃ , c_{4 }} = _{5, 22,5, 25_}.

O‗suvchi ketma-ketlik w = {2, 3, 7, 15}, modul son q=28, q modul bilan o‗zaro tub bo‗lgan r=11, C={5, 22, 5, 25} bizga ma‘lum.

1. 
   r sonining q modul bo‗yicha teskarisini topamiz: 11^-1mod28= 11¹¹mod28=(11(11² mod 28)⁵ )mod 28 = (11*9⁵ )mod 28 = (11*9(9² mod
   

28)² )mod 28 = (11 * 9 * 25² )mod 28 = 23

2. 
   (r⁻¹c ) mod q = m
   

formuladan matnni topamiz:

i

M₁ = (23

i

⋅ 5) mod 28 = 3=2 ⋅ 0 +3⋅1 + 7 ⋅ 0 + 15 ⋅ 0 ^{, bundan M1 = 0100}

ekanligini aniqlaymiz.

3. 
   M₂
   

= (23
⋅ 22) mod 28 = 2=2 ⋅1 +3⋅ 0 + 7 ⋅ 0 + 15⋅ 0 ^{, bundan M2=1000}

ekanligini aniqlaymiz.

4. 
   M₃ = (23 ⋅ 5) mod 28 = 3=2 ⋅ 0 +3⋅1 + 7 ⋅ 0 + 15 ⋅ 0 ^{, bundan M3 = 0100}
   

ekanligini aniqlaymiz.

5. 
   M₄=(23 · 25) mod 28 = 15 = 2·0+3·0+7·0+15·1, bundan M₄=0001
   

ekanligini aniqlaymiz.

6. 
   Barcha M_i larni ketma-ket yozib harflarga o‗tiladi: M=0100100001000001
   

0100 1000 ₂ = 48 ₁₆ = H
0100 0001 ₂ = 41 ₁₆ = A
M = ―HA‖

Mustaqil ish uchun misollar.

1. 
   m=397, n=17, a={2,3,6,12,24,49,100,200}, C={163, 314, 591, 62}, M=?
   
2. 
   m=397, n=17, a={2,3,6,12,24,49,100,200}, C={437, 101, 294, 286}, M=?
   
3. 
   m=397, n=17, a={2,3,6,12,24,49,100,200}, C={275, 213, 90, 490 }, M=?
   

4. 
   m=397, n=17, a={2,3,6,12,24,49,100,200}, C={ 90, 286, 325, 275 }, M=?
   
5. 
   m=397, n=17, a={2,3,6,12,24,49,100,200}, C={398, 518, 163, 275 }, M=?
   
6. 
   m=397, n=17, a={2,3,6,12,24,49,100,200}, C={163, 275, 398, 518 }, M=?
   
7. 
   m=397, n=17, a={2,3,6,12,24,49,100,200}, C={479, 437, 367, 286 }, M=?
   
8. 
   m=397, n=17, a={2,3,6,12,24,49,100,200}, C={367, 437, 325, 275 }, M=?
   
9. 
   m=397, n=17, a={2,3,6,12,24,49,100,200}, C={101, 286, 437, 213 }, M=?
   
10. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={398, 437, 367, 294 }, M=?
    
11. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={426, 437, 591, 294 }, M=?
    
12. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={591, 437, 367, 294 }, M=?
    
13. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={518, 314, 202, 275 }, M=?
    
14. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={426, 437, 101, 202 }, M=?
    
15. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={90, 437, 266, 275 }, M=?
    
16. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={101, 286, 325, 275 }, M=?
    
17. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={294, 286, 367, 314 }, M=?
    
18. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={591, 62, 314, 367 }, M=?
    
19. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={591, 275, 90, 437 }, M=?
    
20. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={398, 437, 101, 275 }, M=?
    
21. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={163, 314, 591, 62}, M=?
    
22. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={437, 101, 294, 286}, M=?
    
23. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={275, 213, 90, 490 }, M=?
    
24. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={90, 286, 325, 275 }, M=?
    
25. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={398, 518, 163, 275 }, M=?
    
26. 
    m=727, n=111, a={2,3,6,12,24,49,100,200}, C={540,662,378,662,
    

296}, M=?

27. 
    m=727, n=111, a={2,3,6,12,24,49,100,200}, C={613,285,378,
    

418,573}, M=?

28. 
    m=727, n=111, a={2,3,6,12,24,49,100,200}, C={8,662,378,236,41}, M=?
    
29. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={296,142,30,296,
    

377}, M=?

30. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={378,377,72, 223,
    

367}, M=?

31. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={82,40, 101, 142,
    

184}, M=?

32. 
    m=397, n=17, a={2,3,6,12,24,49,100,200}, C={ 30, 19, 396, 367,
    

40}, M=?

4.