1

I'm having troubles to understand how to perform simple operations such as addition and multiplication with matrices that contain hex values over the ${\mathbb{F}_{16}} = {\mathbb{F}_2}[x]/\left\langle {{x^4} + x + 1} \right\rangle $ field.

Suppose we have a $2 \times 2$ matrix $M = \left( {\begin{array}{*{20}{c}} 4&8 \\ 5&f \end{array}} \right)$ and a vectors $a = (7,c)$ and $b = (4,3)$.

Now i want to do:

  1. Find the inverse of $M$
  2. Perform the following operation: $M \cdot a + b$

My attempts:

  1. First i tried to convert the hex values to decimal to simplify the calculations.

    $\begin{gathered} {M_{16}} = \left( {\begin{array}{*{20}{c}} 4&8 \\ 5&f \end{array}} \right) \Leftrightarrow {M_{10}} = \left( {\begin{array}{*{20}{c}} 4&8 \\ 5&{15} \end{array}} \right) \\ {M_{16}}^{ - 1} = \frac{1}{{\det \,{M_{10}}}} \cdot C_{10}^*\,(\bmod \,16)\\ C_{10}^* = \left( {\begin{array}{*{20}{c}} 4&8 \\ 5&{15} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {15}&{ - 8} \\ { - 5}&4 \end{array}} \right) \\ \det \,{M_{10}} = 4 \cdot 15 - 5 \cdot 8 = 20 \\ {M_{16}}^{ - 1} = \frac{1}{{20}} \cdot \left( {\begin{array}{*{20}{c}} {15}&{ - 8} \\ { - 5}&4 \end{array}} \right)\,(\bmod \,16) \\ \end{gathered} $

No solutions from the last step...Does it mean that i cannot convert hex to base 10 as i did? For the second part if i convert matrix and vector to base 10 and carry out the calculations i get wrong results. Can anyone explain me how to deal with hex values to get the correct result for part 1 and 2?

Remark:

I'm not sure about the correct name of the field. But here is the given condition: $(M,b) \in GL(2,{\mathbb{F}_{16}}) \times {\mathbb{F}^2}$ and i need to perform the following operations: $M \cdot a + b$ and ${M^{ - 1}} \cdot c + b$ for some vector $c$

user_777
  • 529
  • Hexadecimals do not form a field. Are you referring to the convention of expressing elements in $GF(2^n)$ as binary numbers. i.e. ${4} = x^2$, ${8} = x^3$, ${5} = x^2+1$ and ${f} = x^3+x^2+x+1$??? In that case, people usually surround the number by braces to indicate it is not ordinary hexadecimal numbers. In any event, for this sort of "hex" numbers, the rules of addition and multiplication are different. You cannot evaluate them by casting them to decimal numbers. – achille hui Sep 29 '14 at 16:26
  • I'm not sure about the correct name of the field. But here is the given condition: $(M,b) \in GL(2,{\mathbb{F}_{16}}) \times {\mathbb{F}^2}$ and i need to perform the following operations: $M \cdot a + b$ and ${M^{ - 1}} \cdot c + b$ for some vector $c$ – user_777 Sep 29 '14 at 16:38
  • In that case, numbers like $4$ in your $M$ are not ordinary numbers but the short hand I given above. e.g.

    $$\det M = {4}{f} - {5}{8} = x^2 (x^3 + x^2 + x + 1 ) - (x^2+1)(x^3)\ = x^4 + x^2 = x^2 + x + 1 = {5} \pmod{x^4+x+1 \wedge 2 = 0} $$ Addition/multiplication of other pair of numbers follow the same pattern. It is sort of tedious and I won't reproduce it here.

     – achille hui Sep 29 '14 at 17:04
  • Thanx! It makes sense now) – user_777 Sep 29 '14 at 17:53

0 Answers0