شرح بالتفصيل أخطر برنامج لاختراق هواتف الأندرويد

Classical Cryptography - Arabic Book

  التشفير الكلاسيكى 

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ﺍﻟﻄﺮﻕ ﺍﻟﻜﻼﺳﻴﻜﻴﺔ Classical Method ﻫﻲ ﻋﺒﺎﺭﺓ ﻋﻦ ﻃﺮﻕ ﺗﺸﻔﻴﺮ ﺍﺳﺘﺨﺪﻣﺖ ﻣﻨﺬ ﺯﻣﻦ ﺑﻌﻴﺪ ﻭﺧﺎﺻﺔ ﻓﻲ ﺃﻳﺎﻡ ﺍﻟﺤﺮﺏ ﺍﻟﻌﺎﻟﻤﻴﺔ ﺍﻷﻭﻟﻰ ﻭﺍﻟﺜﺎﻧﻴﺔ ، ﺣﻴﺚ ﻛﺎﻧﺖ ﺧﻄﻂ ﺍﻟﺤﺮﺏ ﻭﻃﺮﻕ ﺍﻟﻬﺠﻮﻡ ﻋﻠﻰ ﺍﻟﻌﺪﻭ ﺗﺮﺳﻞ ﻋﻦ ﻃﺮﻳﻖ ﺭﺳﺎﺋﻞ ﻋﺎﺩﻳﻪ ﻣﻜﺘﻮﺑﺔ ﺑﺨﻂ ﺍﻟﻴﺪ (ﻓﻲ ﺍﻷﻏﻠﺐ ) ﻭﻟﻜﻨﻬﺎ ﺗﺸﻔﺮ ﺑﺄﺣﺪ ﺍﻟﻄﺮﻕ ، ﺧﻮﻓﺎ ﻣﻦ ﺃﻥ ﺗﻘﻊ ﻓﻲ ﺃﻳﺪﻱ ﺍﻟﻌﺪﻭ ﻭﺑﺎﻟﺘﺎﻟﻲ ﺗﻔﺸﻞ ﺗﻠﻚ ﺍﻟﺨﻄﻂ .

ﺑﺎﻟﺮﻏﻢ ﻣﻦ ﺃﻥ ﺗﻠﻚ ﺍﻟﻄﺮﻕ ﻏﻴﺮ ﻣﺠﺪﻳﻪ ﺃﺑﺪﺍ ﻓﻲ ﺍﻟﻮﻗﺖ ﺍﻟﺤﺎﻟﻲ ، ﻓﺈﻧﻬﺎ ﻣﻮﺿﻮﻉ ﻫﺬﺍ ﺍﻟﻜﺘﻴﺐ. ﺭﺑﻤﺎ ﺗﺘﺴﺎﺀﻝ ﻭﻣﺎ ﺍﻟﺠﺪﻭﻯ ﻣﻦ ﺫﻟﻚ ، ﺍﻟﺠﻮﺍﺏ ﺑﻜﻞ ﺑﺴﺎﻃﻪ ، ﻷﻧﻬﺎ ﺗﻌﺘﺒﺮ ﺍﻷﺳﺎﺱ ﻟﻠﻜﺜﻴﺮ ﻣﻦ ﺍﻟﺸﻔﺮﺍﺕ ﺍﻟﺤﺪﻳﺜﺔ ، ﺍﺿﺎﻓﻪ ﺇﻟﻰ ﺃﻥ ﺩﺭﺍﺳﺘﻬﺎ ﻳﻨﻤﻰ ﺍﻟﻌﻘﻞ ﻋﻠﻰ ﺍﻟﺘﻔﻜﻴﺮ ﻭﺍﻟﺒﺤﺚ ، ﺣﻴﺚ ﺃﻧﻬﺎ ﺗﻌﺘﻤﺪ ﻋﻠﻰ ﻓﻘﻂ ﻋﻠﻰ ﺍﻟﺘﻼﻋﺐ ﺑﺎﻷﺣﺮﻑ (ﺇﻣﺎ ﺗﺒﺪﻳﻞ ﺃﻣﺎﻛﻦ ﺍﻷﺣﺮﻑ Transposition، ﺃﻭ ﺗﺒﺪﻳﻠﻬﺎ ﺑﺄﺣﺮﻑ ﺃﺧﺮﻯ ﺑﻌﺪ ﻋﻤﻠﻴﻪ ﺣﺴﺎﺑﻴﻪ ﻣﺎ ( Substitution ، ﻏﻴﺮ ﺫﻟﻚ ﻓﻌﻤﻠﻴﺔ ﻛﺴﺮ ﻫﺬﻩ ﺍﻟﺸﻔﺮﺍﺕ ﺃﻣﺮ ﻓﻲ ﻏﺎﻳﺔ ﺍﻟﻤﺘﻌﺔ .

ﻳﻨﻘﺴﻢ ﺍﻟﻜﺘﻴﺐ ﺇﻟﻰ ﺃﺭﺑﻌﺔ ﺃﻗﺴﺎﻡ ﺭﺋﻴﺴﻴﻪ :

ﺍﻟﻘﺴﻢ ﺍﻷﻭﻝ ﺍﻷﺳﺎﺳﻴﺎﺕ ﻳﺘﻨﺎﻭﻝ ﻫﺬﺍ ﺍﻟﻘﺴﻢ ﺑﻌﺾ ﻣﻦ ﺍﻟﺨﻮﺍﺭﺯﻣﻴﺎﺕ ﻭﺍﻟﻘﻮﺍﻋﺪ ﺍﻟﺮﻳﺎﺿﻴﺔ ﺍﻟﻀﺮﻭﺭﻳﺔ ﻓﻲ ﻋﺎﻟﻢ ﺍﻟﺘﺸﻔﻴﺮ ﻟﺒﻌﺾ ﺍﻟﻄﺮﻕ ﺍﻟﻜﻼﺳﻴﻜﻴﺔ ، ﻣﺜﻞ ﻧﻈﺮﻳﻪ ﺍﻟﻘﺴﻤﺔ ، ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ ، ﺧﻮﺍﺭﺯﻣﻴﺔ ﺃﻗﻠﻴﺪﺱ ، ﺧﻮﺍﺭﺯﻣﻴﺔ ﺍﻗﻠﻴﺪﺱ ﺍﻟﻤﻤﺘﺪﺓ .

ﺍﻟﻘﺴﻢ ﺍﻟﺜﺎﻧﻲ Introduction to Classical Cipher ﻳﺘﻨﺎﻭﻝ ﺍﻟﺸﻔﺮﺍﺕ ﺍﻟﻜﻼﺳﻴﻜﻴﺔ ﻭﻛﻴﻒ ﻳﻤﻜﻦ ﺍﻟﺘﺸﻔﻴﺮ ﻭﻓﻚ ﺍﻟﺘﺸﻔﻴﺮ ﺑﺘﻠﻚ ﺍﻟﻄﺮﻕ ، ﻭ ﻳﺘﻨﺎﻭﻝ ﻭﺑﺸﻜﻞ ﻣﺒﺴﻂ ﻛﻴﻒ ﻳﻤﻜﻦ ﻛﺴﺮ ﺍﻟﺸﻔﺮﺍﺕ Cryptanalysis ﻟﺒﻌﺾ ﻣﻦ ﻫﺬﻩ ﺍﻟﻄﺮﻕ .

ﺍﻟﻘﺴﻢ ﺍﻟﺜﺎﻟﺚ The Implementation ﻳﺤﺘﻮﻱ ﻋﻠﻰ ﺃﻛﻮﺍﺩ ﺑﺮﻣﺠﻴﻪ (ﻣﻜﺘﻮﺑﺔ ﺑـ ( C++ ﻟﺘﻄﺒﻴﻖ ﺗﻠﻚ ﺍﻟﺸﻔﺮﺍﺕ ﺍﻟﻤﻮﺟﻮﺩﺓ ﺑﺎﻟﻔﺼﻞ ﺍﻟﺜﺎﻧﻲ ، ﻭﻛﻤﺎ ﻳﺤﺘﻮﻱ ﺃﻛﻮﺍﺩ ﻟﺒﻌﺾ ﻣﻦ ﺍﻟﺨﻮﺍﺭﺯﻣﻴﺎﺕ ﺍﻟﺮﻳﺎﺿﻴﺔ ﺍﻟﻤﻮﺟﻮﺩﺓ ﺑﺎﻟﻔﺼﻞ ﺍﻷﻭﻝ .(ﺍﻟﺒﺮﺍﻣﺞ ﻛﺎﻣﻠﺔ

ﺍﻟﻘﺴﻢ ﺍﻟﺮﺍﺑﻊ Introduction To Modern Cryptography ، ﻳﺤﺘﻮﻱ ﻋﻠﻰ ﻣﻘﺪﻣﻪ ﺧﻔﻴﻔﺔ "ﻧﻈﺮﻳﻪ" ﻟﻠﺘﺸﻔﻴﺮ ﺑﺎﻟﻄﺮﻕ ﺍﻟﺤﺪﻳﺜﺔ ، ﻭﻳﻮﺿﺢ ﺍﻟﻜﺜﻴﺮ ﻣﻦ ﺍﻟﻤﺼﻄﻠﺤﺎﺕ ﺍﻟﻤﺴﺘﺨﺪﻣﺔ ﻓﻲ ﻋﺎﻟﻢ ﺍﻟﺘﺸﻔﻴﺮ ، ﻳﻌﺘﺒﺮ ﻗﺴﻢ ﻣﻬﻢ ﻟﻤﻦ ﻳﺮﻳﺪ ﺍﻟﺒﺪﺀ ﻓﻲ ﻋﺎﻟﻢ ﺍﻟﺘﺸﻔﻴﺮ

ﺍﻟﻘﺴﻢ ﺍﻷﻭﻝ : ﺍﻷﺳﺎﺳﻴﺎﺕ 7.................................................................................................................. Preliminaries

ﺧﻮارزﻣﯿﺔ اﻟﻘﺴﻤﺔ 7.................................................................................................. THE DIVISION ALGORITHM

اﻷﻋﺪاد اﻷوﻟﯿﺔ 8....................................................................................................................................... Prime Number

اﻟﻘﺎﺳﻢ اﻟﻤﺸﺘﺮك اﻷﻋﻈﻢ 10................................................................................................... Greatest Common Divisor

ﺧﻮارزﻣﯿﺔ أﻗﻠﯿﺪس 11.................................................................................................................... Euclidean Algorithm

اﻟﺨﻮارزﻣﯿﺔ اﻹﻗﻠﯿﺪﯾﮫ اﻟﻤﻤﺘﺪة 12................................................................................... Extended Euclidean Algorithm

اﻟﻨﻈﺮﯾﺔ اﻷﺳﺎﺳﯿﺔ ﻓﻲ اﻟﺤﺴﺎب 16............................................................ The Fundamental Theorem of Arithmetic

اﻟﻤﻀﺎﻋﻒ اﻟﻤﺸﺘﺮك اﻷﺻﻐﺮ 17............................................................................................... Least Common Multiple

اﻟﻤﻌﺎﻣﻞ 17.................................................................................................................................. XOR – Exclusive-Or

18................................................................................................................................................................ Logarithms

ﺍﻟﻘﺴﻢ ﺍﻟﺜﺎﻧﻲ : ﺍﻟﺸﻔﺮﺍﺕ ﺍﻟﻜﻼﺳﻴﻜﻴﺔ........................................................................................................................................ 20

اﻟﺘﺮﻣﯿﺰ 21.............................................................................................................................................................. Coding

اﻟﺘﺸﻔﯿﺮ ﺑﺎﻟﻄﺮق اﻟﻜﻼﺳﯿﻜﯿﺔ 23................................................................................................................ Classical Method

ﺷﻔﺮات 24..................................................................................................... Monoalphabetic Substitution Cipher

ﺷﻔﺮه ﻗﯿﺼﺮ 24......................................................................................................................................... Caesar Cipher

ﺷﻔﺮه أﺗﺒﺎش29........................................................................................................................................... Atbash Cipher

ﺷﻔﺮه 29 ............................................................................................................................................................... ROT13

ﺷﻔﺮه 30.................................................................................................................................................... Affine Cipher

ﻛﺴﺮ اﻟﺸﻔﺮات ﻣﻦ ﻧﻮع32........................................................................................................... MONOALPHABETIC

ﺷﻔﺮات 34.......................................................................................................... Polyalphabetic substitution cipher

ﺷﻔﺮه 34 .................................................................................................................... Simple Shift Vigenere Cipher

ﻛﺴﺮ ﺷﻔﺮات ﻓﺠﯿﻨﯿﺮ Vigenere اﻟﺒﺴﯿﻄﺔ.............................................................................................................................. 37

ﻃﺮﯾﻘﮫ ﻛﯿﺴﺴﻜﻲ KAISISKI ﻟﻤﻌﺮﻓﮫ ﻋﺪد اﻟﻤﻔﺎﺗﯿﺢ 38......................................................................................... Key Length

ﻃﺮﯾﻘﮫ ﻓﺠﯿﻨﯿﺮ اﻟﻜﺎﻣﻠﺔ42....................................................................................... THE FULL VIGENERE CIPHER

ﺷﻔﺮه ﻓﺠﯿﻨﯿﺮ ﺗﻠﻘﺎﺋﯿﺔ اﻟﻤﻔﺘﺎح 43.................................................................. THE AUTO-KEY VIGENERE CIPHER

ﺷﻔﺮه ﻓﺠﯿﻨﯿﺮ ﻃﻮﯾﻠﺔ اﻟﻤﻔﺘﺎح 44............................................................. THE Running KEY VIGENERE CIPHER

ﺷﻔﺮات 45................................................................................................................ PolyGram Substitution Cipher

ﺷﻔﺮه ﺑﻼﻓﯿﺮ46.................................................................................................................. THE PLAYFAIR CIPHER

ﺷﻔﺮه ھﯿﻞ 48................................................................................................................................................... Hill Cipher

ﻛﺴﺮ اﻟﺨﻮارزﻣﯿﺎت ﻣﻦ ﻧﻮع54...................................................................................... PolyGram Substitution Cipher

أﺳﻄﻮاﻧﺔ ﺟﯿﻔﯿﺮﺳﻮن 54............................................................................................ THE JEFFERSON CYLINDER

اﻟﺘﺸﻔﯿﺮ ﺑﻄﺮﯾﻘﮫ اﻟـ 56................................................................... HOMOPHONIC SUBSTITUTION CIPHERS

اﻟﺘﺸﻔﯿﺮ ﺑﺎﻹﺑﺪال 58...................................................................................................... TRANSPOSITION CIPHERS

اﻟﺸﻔﺮه اﻷﻣﻨﮫ61 ...................................................................................................................... THE ONE-TIME PAD

أﺳﺌﻠﺔ ﺷﺎﻣﻠﮫ ﻟﻜﻞ اﻟﻤﻮﺿﻮع.................................................................................................................................................. 63

ﺍﻟﻘﺴﻢ ﺍﻟﺜﺎﻟﺚ : ﺍﻟﺘﻄﺒﻴﻖ 66........................................................................................................... Implementation

اﻟﺘﻌﺎﻣﻞ ﻣﻊ اﻟﺤﺮوف 67...................................................................................................................................... characters

إﯾﺠﺎد اﻟﻘﺎﺳﻢ اﻟﻤﺸﺘﺮك اﻷﻋﻈﻢ 70........................................................................................... Greatest Common Divisor

ﺧﻮارزﻣﯿﺔ اﻗﻠﯿﺪس اﻟﻤﻤﺘﺪة..................................................................................................................................................... 71

اﺧﺘﺒﺎر أوﻟﯿﮫ اﻟﻌﺪد ﺑﺎﺳﺘﺨﺪام 72...................................................................................................................... Trial Division

اﻟﺘﻌﺎﻣﻞ ﻣﻊ اﻟﻠﻮﻏﺮﯾﺘﻢ........................................................................................................................................................... 72

73........................................................................................................................ Fast Exponentiation Algorithms

ﺷﻔﺮة ﻗﯿﺼﺮ 74......................................................................................................................................... Caesar Cipher

ﺷﻔﺮة 76 ............................................................................................................................................................... ROT13

اﻟﺘﺸﻔﯿﺮ ﺑﻄﺮﯾﻘﮫ 77....................................................................................................................................... Affine Cipher

ﺷﻔﺮة ﻓﯿﺠﻨﯿﺮ اﻟﺒﺴﯿﻄﺔ 78................................................................................................ Simple Shift Vigenere Cipher

ﺷﻔﺮه ﻓﯿﺠﯿﻨﺮ اﻟﻜﺎﻣﻠﺔ 79................................................................................................................. Full Vigenere Cipher

ﺷﻔﺮه ﻓﺠﯿﻨﯿﺮ ﺗﻠﻘﺎﺋﯿﺔ اﻟﻤﻔﺘﺎح 81.............................................................................................. Auto Key Vigenere Cipher

ﺷﻔﺮه ﻓﯿﺠﯿﻨﺮ ﻃﻮﯾﻠﺔ اﻟﻤﻔﺘﺎح 82 .............................................................................. the Running key Vigenere Cipher

ﺷﻔﺮة ﺑﻼﻓﯿﺮ 83....................................................................................................................................... Playfair Cipher

ﺷﻔﺮه 83................................................................................................................................................. Reverse Cipher

ﺍﻟﻔﺼﻞ ﺍﻟﺮﺍﺑﻊ : ﻣﻘﺪﻣﻪ ﻓﻲ ﺍﻟﺘﺸﻔﻴﺮ ﺑﺎﻟﻄﺮﻕ ﺍﻟﺤﺪﻳﺜﺔ.................................................................................................................. 84

ﻟﻤﺎذا ﺑﺎﻟﺘﺸﻔﯿﺮ 85.............................................................................................................................. Why Cryptography

اﻟﺘﺸﻔﯿﺮ ﺑﺎﻟﻤﻔﺘﺎح اﻟﻤﺘﻨﺎﻇﺮ 87.......................................................................................... Symmetric key Cryptography

ﻣﺎ ھﻮ اﻟﻤﻔﺘﺎح ، وﻣﺎ ھﻲ أھﻤﯿﺘﮫ............................................................................................................................................. 91

ﺗﻮﻟﯿﺪ اﻟﻤﻔﺘﺎح 94...................................................................................................................................... Key Generation

ﺷﻔﺮات اﻟﻜﺘﻞ100......................................................................................................................................... Block Cipher

ﺷﻔﺮه اﻟﺘﺪﻓﻖ 102.................................................................................................................................... Stream Ciphers

Block VS Stream ﻣﻦ اﻷﻓﻀﻞ................................................................................................................................... 103

103 ............................................................................................................................................................. Triple DES

105...................................................................................................................... Advanced Encryption Standard

اداره اﻟﻤﻔﺘﺎح اﻟﻤﺘﻨﺎﻇﺮ 105............................................................................................ Symmetric-Key Management

أﺟﮭﺰه ﺣﻔﻆ اﻟﻤﻔﺎﺗﯿﺢ 109.............................................................................................. Hardware-Based Key Storage

110............................................................................................................................................. Crypto Accelerators

ﺍﻟﻘﺴﻢ ﺍﻷﻭﻝ : ﺍﻷﺳﺎﺳﻴﺎﺕ

Preliminaries

ﻟﻔﮭﻢ أﻏﻠﺐ اﻟﺸﻔﺮات اﻟﺤﺪﯾﺜﺔ ﯾﻠﺰم ﻓﮭﻢ اﻟﻜﺜﯿﺮ ﻣﻦ ﻧﻈﺮﯾﮫ اﻷﻋﺪاد Number Theory ، وﻟﻜﻦ ﺑﻤﺎ أن اﻟﻜﺘﯿﺐ ﯾﺮﻛﺰ ﻋﻠﻰ اﻟﺸﻔﺮات ﺑﺎﻟﻄﺮق اﻟﻜﻼﺳﯿﻜﯿﺔ ﻓﺴﻮف ﻧﺘﻨﺎول ﻣﺎ ﯾﮭﻤﻨﺎ ﻓﻘﻂ ﻓﻲ اﻟﻮﻗﺖ اﻟﺤﺎﻟﻲ . وﺳﻮف ﺗﻜﻮن اﻟﻘﻮاﻋﺪ ﻣﻜﺘﻮﺑﺔ وﻟﻜﻦ ﻣﻦ ﻏﯿﺮ إﺛﺒﺎت رﯾﺎﺿﻲ ﻟﮭﺎ Prove ، ﯾﻤﻜﻨﻚ ﺗﺠﺎوز ھﺬا اﻟﻘﺴﻢ واﻟﻌﻮدة إﻟﯿﮫ ﻻﺣﻘﺎ ﻋﻨﺪﻣﺎ ﺗﺤﺘﺎﺟﮫ ﻓﻲ ﺷﻔﺮه Affine Cipher ، وﻟﻜﻨﻲ ﻻ أﻓﻀﻞ ذﻟﻚ

اذا ﻛﺎن ﻟﺪﯾﻨﺎ ﻋﺪدﯾﻦ ﺻﺤﯿﺤﯿﻦ a و b ، وﻛﺎن a !=0 (ﻻ ﯾﺴﺎوي . ( 0 ﻧﻘﻮل ﻋﻦ a ﯾﻘﺴﻢ b اذا 

 ﻛﺎن ﻟﺪﯾﻨﺎ ﻋﺪد ﺛﺎﻟﺚ c ﺑﺤﯿﺚ . b = a*c اذا ﻛﺎن a ﯾﻘﺴﻢ b ﻧﺸﯿﺮ إﻟﯿﮫ ﺑﺎﻟﺮﻣﺰ . 

a|b

ﻣﺜﺎل ﺑﺴﯿﻂ :

3|27 ﺻﺤﯿﺢ ، ﻻن                           .  27 = 9*3

5|32 ﻏﯿﺮ ﺻﺤﯿﺤﺔ ، ﻷن 32 = 4*c وﻻ ﺗﻮﺟﺪ ﻋﺪد ﺻﺤﯿﺢ ﯾﺤﻞ ﻣﻜﺎن . c

 

اذا ﻛﺎن ﻟﺪﯾﻨﺎ ﺛﻼﺛﺔ أﻋﺪاد ﺻﺤﯿﺤﺔ x,y,z ، وﻛﺎن x|y و y|z                                 اذا . x|z

 

ﻣﺜﺎل :

ﻟﺪﯾﻨﺎ 3|9 ، و 9|72 ، اذا 3 ﺗﻘﺴﻢ 3|27 = 72

 

ﺧﻮﺍﺭﺯﻣﻴﺔ ﺍﻟﻘﺴﻤﺔ THE DIVISION ALGORITHM

 

وھﻲ أﺣﺪ اﻟﺨﻮارزﻣﯿﺎت اﻟﻤﮭﻤﺔ ﺟﺪا ، ﺣﯿﺚ ﺗﻘﻮل اﻧﮫ ﯾﻤﻜﻨﻨﺎ أن ﻧﻤﺜﻞ أي ﻋﺪد ﺻﺤﯿﺢ ، وذﻟﻚ ﺑﻮاﺳﻄﺔ ﺿﺮب ﻋﺪد ﺻﺤﯿﺢ b ﻣﻊ اﺿﺎﻓﮫ ﺑﺎﻗﻲ r ﺑﺤﯿﺚ ﯾﻜﻮن اﻟﺒﺎﻗﻲ ﻋﺪد ﻣﻮﺟﺐ وأﻗﻞ ﻣﻦ اﻟﻌﺪد . b

اذا ﻛﺎن ﻟﺪﯾﻨﺎ ﻋﺪدﯾﻦ ﺻﺤﯿﺢ y,b ، وﻛﺎن b أﻛﺒﺮ ﻣﻦ ﺻﻔﺮ ، اذا ﺳﯿﻜﻮن ﻟﺪﯾﻨﺎ ﻋﺪدﯾﻦ q,r ﺑﺤﯿﺚ :

Y = b*q + r

  q ھﻮ ﺣﺎﺻﻞ اﻟﻘﺴﻤﺔ r . Quotient ھﻮ اﻟﺒﺎﻗﻲ b . remainder ھﻮ اﻟﻤﻘﺴﻮم Divisor

ﻣﺜﺎل ﻟﺪﯾﻨﺎ اﻟﻤﻌﺎدﻟﺔ :

65 = 3*q + r

ﻗﯿﻤﮫ ال q ھﻲ 21 (وذﻟﻚ ﺑﻘﺴﻤﮫ 65 ﻋﻠﻰ (3 ، واﻟﺒﺎﻗﻲ r ھﻮ . 2

3*21+2 وﺗﺴﺎوي . 65

 

مثال اخر

-21 = 5*q + r

ﺑﻘﺴﻤﮫ -21 ﻋﻠﻰ 5 ، ﺳﻮف ﻧﺤﺼﻞ ﻋﻠﻰ ﺣﺎﺻﻞ اﻟﻘﺴﻤﺔ -4 ، واﻟﺒﺎﻗﻲ ﺳﻮف ﯾﻜﻮن 1- ، وﻟﻜﻦ ﻛﻤﺎ ذﻛﺮﻧﺎ ﻓﻲ اﻟﻘﺎﻋﺪة اﻟﺴﺎﺑﻘﺔ أن اﻟﺒﺎﻗﻲ r داﺋﻤﺎ ﯾﻜﻮن ﻣﻮﺟﺐ ، ﻟﺬﻟﻚ ﻧﻘﻮم ﺑﺠﻤﻊ 5 ﻓﻲ اﻟﺒﺎﻗﻲ ، وﻧﻄﺮح 1 ﻣﻦ ﺣﺎﺻﻞ اﻟﻘﺴﻤﺔ

   

ﻟﯿﻜﻮن ﻟﺪﯾﻨﺎ q = -5 ، . r = 4

ﻧﻄﺒﻘﮭﺎ ﻓﻲ اﻟﻤﻌﺎدﻟﺔ :

-21 = 5*-5+4 وھﻮ ﺻﺤﯿﺢ

ﺍﻷﻋﺪﺍﺩ ﺍﻷﻭﻟﻴﺔ : Prime Number

ﺗﻠﻌﺐ اﻷﻋﺪاد اﻷوﻟﯿﺔ ﺗﻠﻌﺐ دورا ﻛﺒﯿﺮا ﺟﺪا ﻓﻲ اﻟﺘﺸﻔﯿﺮ وﺧﺎﺻﺔ ﻓﻲ اﻟﻄﺮق اﻟﺤﺪﯾﺜﺔ ، وﺗﻌﺮﯾﻔﮭﺎ ﻛﺎﻟﺘﺎﻟﻲ 

اﻟﻌﺪد اﻷوﻟﻲ ، ھﻮ اﻟﻌﺪد اﻟﺼﺤﯿﺢ integer أﻛﺒﺮ ﻣﻦ 1 ،وﻻ ﯾﻘﺒﻞ اﻟﻘﺴﻤﺔ إﻻ ﻋﻠﻰ ﻧﻔﺴﮫ وﻋﻠﻰ . 1 ﺑﺎﻗﻲ اﻷﻋﺪاد اﻟﺘﻲ أﻛﺒﺮ ﻣﻦ 1 وﻏﯿﺮ أوﻟﯿﮫ ﺗﺴﻤﻰ أﻋﺪاد ﻣﺮﻛﺒﮫ Composite Number

ﻣﺜﺎل ﻋﻠﻰ أﻋﺪاد أوﻟﯿﮫ :

2و 3 و 7 و 23 و29 و 163 واﻟﻜﺜﯿﺮ ﻏﯿﺮھﺎ .

ﻣﺜﺎل ﻋﻠﻰ أﻋﺪاد ﻣﺮﻛﺒﮫ :

4 (ﺣﯿﺚ أﻧﮭﺎ ﺗﻘﺒﻞ اﻟﻘﺴﻤﺔ ﻋﻠﻰ ( 2

100 (ﺗﻘﺒﻞ اﻟﻘﺴﻤﺔ ﻋﻠﻰ 2 و . (5

ﻣﺜﺎل ﻋﻠﻰ أﻋﺪاد ﻏﯿﺮ أوﻟﯿﮫ وﻏﯿﺮ ﻣﺮﻛﺒﮫ

0 و 1 وﺟﻤﯿﻊ اﻷﻋﺪاد اﻟﺴﺎﻟﺒﺔ ، ﻣﺜﻞ . 21

ﺟﻤﯿﻊ اﻷﻋﺪاد اﻟﺼﺤﯿﺤﺔ اﻟﺘﻲ أﻛﺒﺮ ﻣﻦ 1 ، ﻟﮭﺎ ﻗﺎﺳﻢ أوﻟﻲ .

اﻷﻋﺪاد اﻷوﻟﯿﺔ ﻏﯿﺮ ﻣﻨﺘﮭﯿﺔ .

اذا ﻛﺎن ﻟﺪﯾﻨﺎ ﻋﺪد ﺻﺤﯿﺢ ﻣﺮﻛﺐ N ، اذا ﯾﻜﻮن ﻟﺪى n ﻗﺎﺳﻢ أوﻟﻲ ﻻ ﯾﺘﻌﺪى اﻟﺠﺬر اﻟﺘﺮﺑﯿﻌﻲ ل. N

وھﺬا ﻣﻌﻨﺎه اذا أردﻧﺎ أن ﻧﻌﺮف ﻋﻠﻰ اﻟﻌﺪد x ھﻮ أوﻟﻲ أم ﻻ ، ﺳﻮف ﻧﺒﺤﺚ ﻣﻦ اﻟﺒﺪاﯾﺔ 2 (ﻻن واﺣﺪ ﻟﯿﺲ أوﻟﻲ) إﻟﻰ أن ﻧﺼﻞ إﻟﻰ ﺟﺬر اﻟﻌﺪد . x وﻧﺨﺘﺒﺮ ﻛﻞ ﻋﺪد ﻣﻦ ھﺬه اﻷﻋﺪاد ، ھﻞ ﯾﻘﺒﻞ x اﻟﻘﺴﻤﺔ ﻋﻠﯿﮭﺎ ، ﻓﻲ ﺣﺎل ﺗﺤﻘﻖ ذﻟﻚ ، ﻧﻜﻮن ﻗﺪ ﻋﺮﻓﻨﺎ أن اﻟﻌﺪد ﻟﯿﺲ أوﻟﻲ ، وإذا ﻟﻢ ﯾﺘﺤﻘﻖ ﻓﯿﻜﻮن اﻟﻌﺪد أوﻟﻲ

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