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

Table 1

The characteristic polynomials of the curve 𝑣²=𝑢^𝑝+𝑎𝑢+𝑏 over 𝔽_𝑝𝑛.

𝑛	𝑎	𝑏	𝑃(𝑡) over 𝔽𝑝𝑛	Notes
1	−1	𝑏=0	(𝑡^2−𝑝)^(𝑝−1)/2
	1	𝑏∈𝔽𝑝	(𝑡^2+𝑝)^(𝑝−1)/2
	a primitive root modulo 𝑝	𝑏∈𝔽𝑝	𝑡^𝑝−1+𝑝^(𝑝−1)/2
	ord𝔽^∗𝑝(𝑎)=𝑚	𝑏∈𝔽𝑝	(𝑡^𝑚+(−1/𝑝)(−𝑝)^𝑚/2)^(𝑝−1)/𝑚	𝑚 is even
	2<𝑚<𝑝−1		(𝑡^2𝑚+𝑝^𝑚)^(𝑝−1)/2𝑚	𝑚 is odd
2	−1	𝑏=0	(𝑡−𝑝)^𝑝−1
		𝑏(≠0)∈𝔽𝑝2	(𝑡^𝑝−𝑝^𝑝)/(𝑡−𝑝)
	1	𝑏∈𝔽𝑝	(𝑡+𝑝)^𝑝−1
		𝑏∉𝔽𝑝	(𝑡^𝑝+𝑝^𝑝)/(𝑡+𝑝)
	a primitive root of 𝑝	𝑏∈𝔽𝑝2	(𝑡^(𝑝−1)/2+𝑝^(𝑝−1)/2)^2
	ord𝔽^∗𝑝(𝑎)=𝑚	𝑏∈𝔽𝑝2	(𝑡^𝑚/2+(−1/𝑝)(−𝑝)^𝑚/2)^2(𝑝−1)/𝑚	𝑚 is even
	2<𝑚<𝑝−1		(𝑡^𝑚+𝑝^𝑚)^(𝑝−1)/𝑚	𝑚 is odd
	𝑎∉𝔽𝑝, 𝑚=ord𝔽^∗𝑝2(𝑎)	𝑏∈𝔽𝑝2	(𝑡^𝑒+(−1)^𝑒(𝑝+1)/𝑚+𝑒−1𝑝^𝑒)^(𝑝−1)/𝑒	𝑒(>1) is the smallest s.t. 𝑚∣𝑒(𝑝+1).
	𝑎∉𝔽𝑝, 𝑎^𝑝+1=1	𝑏=0, or 𝑎=𝑏^1−𝑝	(𝑡+𝑎^(𝑝+1)/2𝑝)^𝑝−1
		𝑏≠0, 𝑎≠𝑏^1−𝑝	(𝑡^𝑝+𝑝^𝑝)/(𝑡+𝑝)