Computing the Characteristic Polynomials of a Class of Hyperelliptic Curves for Cryptographic Applications

Table 2

Some cases in which 𝐶_𝑞 have reducible characteristic polynomials.

𝑛	𝑎	𝑏
	−1	0
1	1	∈𝔽𝑝
	2< ord𝔽^∗𝑝(𝑎)<𝑝−1	∈𝔽𝑝
	−1	0
	1	∈𝔽𝑝
2	a primitive root of 𝑝	∈𝔽𝑝2
	2< ord𝔽^∗𝑝(𝑎)<𝑝−1	∈𝔽𝑝2
	𝑎∉𝔽𝑝, a nonprimitive root of 𝑝^2	∈𝔽𝑝2
	𝑎∉𝔽𝑝,𝑎^𝑝+1=1	0 or 𝑎=𝑏^1−𝑝