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Given the problem of computing the GCD of two given elements over any finite field with characteristic 2.

$$ r_1 = q_1r_2 + r_3 \\ r_2 = q_2r_3 + r_4 \\ r_3 = q_3r_4 + r_5 \\ \vdots \\ r_{k-1} = q_{k-1}r_{k} + r_{k+1}\\ r_k = q_kr_{k+1}$$

Exist any way of find the aproximate value of $r_{k+1}$, for $k > 3$ without know the $q_{k-2}$ value?

wiskundeliefhebber
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juaninf
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    Do you mean the gcd of two polynomials? The non-zero elements of a field, finite or not, are units. – André Nicolas Mar 26 '13 at 18:45
  • GCD = greatest common divisor – juaninf Mar 26 '13 at 18:52
  • approximate mean any polynomial such that permit advance in the iterative euclidean algorithm and this easiest to calculate than calculate the $r_{k+1}$ normally. – juaninf Mar 26 '13 at 18:57
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    @juaninf: The same problem as in your previous question. I'm sorry that I have to put it bluntly. Either you don't understand what a finite field is, or you don't understand what Euclidean algorithm is. If all the elements $r_i,q_i$ are, indeed, elements of a finite field, then $r_3=0$. Always. No exceptions. This is what André is also saying. For your previous question I asked you to write down an example, but you didn't oblige. Care to do it now??! – Jyrki Lahtonen Mar 27 '13 at 13:30

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