The canonization theorem says that for given $m,n$ for some $m^{*}$ (the first one is called $ER(n;m)$) we have for every function $f$ with domain ${\left[1,\cdots ,m^{*}\right]}^n$, for some $A\in {\left[1,\cdots ,m^{*}\right]}^m$, the question of when the equality $f\left(i_1,\cdots ,i_n\right)=f\left(j_1,\cdots ,j_n\right)$ (where $i_1<\cdots <i_n$ and $j_1<\cdots j_n$ are from $A$) holds has the simplest answer: for some $v\subseteq \{1,\cdots ,n\}$ the equality holds iff ${\bigwedge }_{\ell \in v}{i}_{\ell }=j_{\ell }$. We improve the bound on $ER(n,m)$ so that fixing $n$ the number of exponentiation needed to calculate $ER(n,m)$ is best possible.

For $a$ generator of the multiplicative group of the field $GF(p,k)$, the discrete logarithm of an element $b$ of the field to the base $a$, $b\ne 0$ is that integer $z:1\le z\le p^k-1$, $b=a^z$. The $p$-ary digits which represent $z$ can be described with extremely simple polynomial forms.