The cubic sieve is an algorithm for computing discrete logarithm.
The initial step is to find a solution to
$x^3\equiv y^2z\pmod p$ such that $x^3\ne y^2z$ with $x,y,z$ of order $p^\alpha$.
But what does it mean to have $\frac{1}{3}\le\alpha\le\frac{1}{2}$? $\frac{1}{3}$ of what? Also, does it change anything for the variant of the algorithm working with finite fields having dimension >2 ?
In the question, $\alpha$ is a dimensionless real number in the interval $[1/3,1/2]$, e.g. 0.42…
In the context, of order means having logarithm similar to that of (as in order of magnitude). That is, it's asked that the size of $x$ is about a third to a half of the size of $p$, with size measured as the number of digits in base 2 (or some higher base). Same for $y$ and for $z$.
I don't have the foggiest idea about how to adapt that for solving the DLP in a field $\mathbb F_{p^k}$ with $k>1$. The question's reference only considers a prime field $\mathbb F_p$.
