Solve a set of non linear Equations on Galois Field M 1 = y 1...

Ellen Chang

Ellen Chang

Answered

2022-07-02

Solve a set of non linear Equations on Galois Field
M 1 = y 1 y 0 x 1 x 0
M 2 = y 2 y 0 x 2 x 0
M 1 , M 2 , x 1 , y 1 , x 2 , y 2 ,, are known and they are chosen from a G F ( 2 m ) .. I want to find x 0 , y 0 I ll restate my question. Someone chose three distinct x 0 , x 1 , x 2 , as well as y 0 , y 1 , y 2 , then computed M 1 , M 2 , and finally revealed M 1 , M 2 , x 1 , y 1 , x 2 , y 2 , but not x 0 , y 0 . to us.All the variables are chosen from a Galois Field.
I want to recover the unknown x 0 , y 0 .. Is it possible to accomplish that?
If a set of nonlinear equations have been constructed with the aforementioned procedure e.g.
M 1 = k 1 ( y 0 + ( y 1 y 0 x 1 x 0 ) ( l 1 x 0 ) ) ( l 1 x 0 ) ( l 1 x 1 )
M 2 = k 2 ( y 0 + ( y 1 y 0 x 1 x 0 ) ( l 2 x 0 ) ) ( l 2 x 0 ) ( l 2 x 1 )
M 3 = k 3 ( y 0 + ( y 1 y 0 x 1 x 0 ) ( l 3 x 0 ) ) ( l 3 x 0 ) ( l 3 x 1 )
M 4 = k 4 ( y 0 + ( y 1 y 0 x 1 x 0 ) ( l 4 x 0 ) ) ( l 4 x 0 ) ( l 4 x 1 )
where x 0 , y 0 x 1 , y 1 are the unknown GF elements. Can I recover the unknown elements?
My question was if the fact that the set of equations is defined on a Galois Field imposes any difficulties to find its solution.
If not I suppose that the set can be solved. Is this true?

Answer & Explanation

Ordettyreomqu

Ordettyreomqu

Expert

2022-07-03Added 22 answers

The first system can be solved in the usual way, provided the "slopes" M i i are distinct. Solve each for the knowns y k , k = 1 , 2 and subtract. You can then get to
x 0 = M 2 x 2 M 1 x 1 y 2 + y 1 M 2 M 1 ,
and then use one of the equations you already formed with this x 0 plugged in to get y 0 .. Since this method only uses addition/subtraction multiplication/(nonzero)division it works in any field, in particular in your Galois field.

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