On construction of high order elements in general finite fields

Gao [6] conjectured that if a general finite field has q
ⁿ
elements, and m is the smallest power of q greater or equal to n, then there exists a polynomial g (x) ϵ F
_q
[x] of degree at most 2 ⌈log
₂
n⌉ such that x
^m
- g (x) has an irreducible factor f (x) of degree n. If the conjecture holds, this guarantees existence of high order elements that are used in the known cryptographic primitives. The conjecture was verified in [6] for q = 2 and n ≤ 300. This paper verifies Gao's conjecture for q = 2 and 300 <; n ≤ 400, q = 3 and n ≤ 300, q = 5 and n ≤ 100.