## More on ONB1 identity element

“Application of Finite Fields”  page 98:

For a type I optimal normal basis, its minimal polynomial is obviously x $x^n + ... + x + 1$, which is irreducible over $F_q$ if and only if $n + 1$ is a prime and $q$ is primitive in $Z_{n+1}$

For n = 4,
$x^4+x^3+x^2+x = 1$
$x^5 = 1$
$x^6 = x$
$x^7 = x^2$
$x^8 = x^3$
$x^9 = x^4$

Ex :
Let’s prove that 1111 is the identity element of ONB1 n = 4.

1010 x 1111 = ($x^8+x^2$) x ($x^8+x^4+x^2+x$)
= $x^{16}+x^{12}+x^{10}+x^9+x^{10}+x^6+x^4+x^3$
= $x^{16}+x^{12}+x^9+x^6+x^4+x^3$
= $x + x^2 + x^4 + x + x^4 + x^3$
= $x^2 + x^3$
= $x^2 + x^8$
= $x^8 + x^2$
= 1010

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