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Math Help - how to do arithmetic operations in galois field

  1. #1
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    how to do arithmetic operations in galois field

    could any one please help in galois field



    is primitive in GF(2)[x].
    Let be a root of .
    Addition:


    α2 means 0 1 0 0
    α5 means 0 1 1 1
    - ----------------
    when we add the both 1 0 1 1


    but in galois field answer is α3 means that 0 0 1 1 from the table
    please explain how it is

    Multiplication:
    in multiplication when we multiply two 4 bit numbers we got 8 bit number
    in galois field we got 4 bit number only



    please explain these two questions
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  2. #2
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    Quote Originally Posted by t s k r View Post
    but in galois field answer is α3 means that 0 0 1 1 from the table
    Because \alpha is a root of x^3+x+1.
    Therefore \alpha^3 + \alpha + 1 = 0 \implies \alpha^3 = - \alpha - 1.
    But this field has charachteristic two i.e. x=-x.
    Thus, \alpha^3 = -\alpha - 1 = \alpha + 1.
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