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Math Help - Multiplying in GF(2)

  1. #1
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    Multiplying in GF(2)

    We are working in GF(2). Consider following polynomial:
    <br />
 A: x_1+x_2+1=0<br />

    Now multiply it by all monomials, that mean  x_1, x_2, x_3, x_4 to get:
    <br />
A_1: x_1x_2=0 <br />
    <br />
A_2: x_1x_3+x_2x_3+x_3=0<br />
    <br />
A_3: x_2x_4+x_2x_4+x_4=0<br />

    I know that A_2:=A*x_3, A_3:=A*x_4 but from where we have A_1? We should multiply A by x_1 and then by x_2. From this two new equations we have A_1 but how we could get it?
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  2. #2
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    I found the solution. Many thanks for tonio
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  3. #3
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    Quote Originally Posted by Migotek84 View Post
    We are working in GF(2). Consider following polynomial:
    <br />
A: x_1+x_2+1=0<br />

    Now multiply it by all monomials, that mean  x_1, x_2, x_3, x_4 to get:
    <br />
A_1: x_1x_2=0 <br />
    <br />
A_2: x_1x_3+x_2x_3+x_3=0<br />
    <br />
A_3: x_2x_4+x_2x_4+x_4=0<br />

    I know that A_2:=A*x_3, A_3:=A*x_4 but from where we have A_1? We should multiply A by x_1 and then by x_2. From this two new equations we have A_1 but how we could get it?


    This is the second time you ask this question and this is the second time I don't understand what you mean: you call x_1+x_2+1 = 0 "A polynomial", which it is not since it is equalled to zero. It would then be an equality.

    Next, you want this equality to be multiplied "by the monomials x_1,x_2,x_3,x_4 "....so you're working in the ring of polynomials in 4 indeterminates ?!

    Anyway, you get 4 weird expressions that you call A_1, A_2, A_3...and I don't have the faintest idea from where did you get these expressions!. For example , multiplying the forementioned equality by the monomial x_1 we get:

    x_1(x_1+x_2+1) = x_1\cdot 0\Longrightarrow x_i^2+x_1x_2+x_1=0....so ?? Of course , if you already know that x_1^2+x_1=0 no matter what VALUE in \mathbb{F}_2 we choose for x_1 then you can EVALUATE and get x_1x_2=0...both AS POLYNOMIALS this is not so!

    I think you must first understand well what you want, then decide how to ask it in a clear way and then post again.

    Tonio
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  4. #4
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    I agree with tonio.
    These are equations not polynomials.
    I think that this is some part of Yours previous post http://www.mathhelpforum.com/math-he...imination.html - 4 variables over GF(2).
    So You consider equation A and then You multiplied it by 4 monomials:
    x_1(x_1+x_2+1)=0 => x_1x_2=0
    x_2(x_1+x_2+1)=0 => x_1x_2=0
    x_3(x_1+x_2+1)=0 => x_1x_3+x_2x_3+x_3=0
    x_4(x_1+x_2+1)=0 => x_1x_4+x_2x_4+x_4=0
    Am I right?

    In last part You used word 'equations' - good!

    PS. I knew that this problem/example I saw before. This is a toy example from http://math.uc.edu/~aac/pqcrypto2008...008mohamed.pdf
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  5. #5
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    I'm sorry for my mistake
    Of course it is not a polynomial but polynomial equation.
    Yes Arczi1984, it is from this paper.
    Many thanks for tonio beacuse You help me very much.
    Once more I'm so sorry for this mess.
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