Let $P=X^{10} +5X^{5}+1$ and $Q=5X^{8}+6X^{3}$ in $\mathbb{F}{7}[X]$. How we can prove this strange relation of Euclid division $$U{2i}P-X^{4}U_{2i-1}^{7}=4.3^{2i-1}Q$$ $$U_{2i+1}P-X^{-4}U_{2i}^{7}=4.3^{2i}Q/X$$ with $U_{1}=X^{2}$?Note that $U_{2i}=[X^{4}U_{2i-1}^{7}/P]$ and $U_{2i+1}=[X^{-4}U_{2i}^{7}/P]$.
Asked
Active
Viewed 75 times
0
-
3You may want to post this in [math stackexchange](https://math.stackexchange.com/) . – rcgldr Feb 07 '21 at 21:44
-
For $i=1$, $4.3^{2i-1}Q=5Q=4X^{8}+2X^{3}$ – Oussema Feb 15 '21 at 09:35
-
For $i=2$, $4.3^{2i-1}Q=3Q=X^{8}+4X^{3}$ – Oussema Feb 15 '21 at 09:36
-
I'm not sure what is meant by `4.3`. Again, it would be better to post this at [math stackexchange](https://math.stackexchange.com/questions/tagged/finite-fields) . – rcgldr Feb 15 '21 at 15:52
-
I wanted to do it but they answer me: Sorry, we are no longer accepting questions from this account. – Oussema Feb 16 '21 at 16:37
-
I'm not familiar with this type of finite field. I don't understand how 4.3^(1) == 5 or 4.3^(3) == 3. – rcgldr Feb 16 '21 at 16:41
-
In some cases, 4.3 is an alternate way to express 4^3, but (4^3)%7 == 1 for both integer and binary (xor) math, not 5. – rcgldr Mar 12 '21 at 22:42