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.................................................................................................................. Preliminariesﺧﻮارزﻣﯿﺔ 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 CIPHERSاﻟﺸﻔﺮه 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 aو b ، وﻛﺎن a !=0(ﻻ ﯾﺴﺎوي . ( 0 ﻧﻘﻮل ﻋﻦ aﯾﻘﺴﻢ b اذا
ﻛﺎن ﻟﺪﯾﻨﺎ ﻋﺪد ﺛﺎﻟﺚ c ﺑﺤﯿﺚ . b = a*c اذا ﻛﺎن aﯾﻘﺴﻢ 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 bﻣﻊ 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
ﻣﺜﺎل ﻟﺪﯾﻨﺎ اﻟﻤﻌﺎدﻟﺔ :
65 = 3*q + r
ﻗﯿﻤﮫ ال q ھﻲ 21 (وذﻟﻚ ﺑﻘﺴﻤﮫ 65ﻋﻠﻰ (3 ، واﻟﺒﺎﻗﻲ r ھﻮ . 2
3*21+2 وﺗﺴﺎوي . 65
مثال اخر
-21 = 5*q + r
ﺑﻘﺴﻤﮫ -21ﻋﻠﻰ 5 ، ﺳﻮف ﻧﺤﺼﻞ ﻋﻠﻰ ﺣﺎﺻﻞ اﻟﻘﺴﻤﺔ -4 ، واﻟﺒﺎﻗﻲ ﺳﻮف ﯾﻜﻮن 1-، وﻟﻜﻦ ﻛﻤﺎ ذﻛﺮﻧﺎ ﻓﻲ اﻟﻘﺎﻋﺪة اﻟﺴﺎﺑﻘﺔ أن اﻟﺒﺎﻗﻲ rداﺋﻤﺎ 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
اﻟﻌﺪد اﻷوﻟﻲ ، ھﻮ اﻟﻌﺪد اﻟﺼﺤﯿﺢinteger أﻛﺒﺮ ﻣﻦ1 ،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
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 ﯾﺘﻌﺪىاﻟﺠﺬر 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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