Hyperelliptic curves - Mathematics of Public Key Cryptography - page 184

10.1.2 Uniformisers on hyperelliptic curves

The aim of this section is to determine uniformisers for all points on hyperelliptic curves. We
begin in Lemma 10.1.16 by determining uniformisers for the affine points of a hyperelliptic
curve.

Lemma 10.1.16 Let P = ( x P ,y P ) ∈ C ( k ) be a point on a hyperelliptic curve. If P = ι ( P )

then ( y − y P ) is a uniformiser at P (and v P ( x − x P ) = 2 ). If P = ι ( P ) then ( x − x P ) is a

uniformiser at P.

Proof We have

y 2 ( y P +

( y − y P )( y + y P + H ( x P )) = + H ( x P ) y − H ( x P ) y P )

= F ( x ) + y ( H ( x P ) − H ( x )) − F ( x P ) .

Now, use the general fact for any polynomial that F ( x ) = F ( x P ) + ( x − x P ) F ( x P )

(mod ( x − x P ) 2 ). Hence, the above expression is congruent modulo ( x − x P ) 2 to

x P )( F ( x P ) yH ( x P )) (mod ( x x P ) 2 ) . ( x − − −

y P ) 2 . Note also that F ( x P ) When P = ι ( P ) then ( y − y P )( y + ( y P + H ( x P ))) = ( y − −

y P H ( x P ) is not zero since 2 y P +

H ( x P ) = 0 and yet C is not singular. Writing G ( x,y ) =

y P ) 2 / ( x

( y − − x P ) ∈ k [ x,y ]wehave G ( x P ,y P ) = 0 and

1
G ( x,y ) .

y P ) 2 x − x P = ( y −

Hence, a uniformiser at P is ( y − y P ) and v P ( x − x P ) = 2.

For the case P = ι ( P ) note that v P ( y − y P ) > 0 and v P ( y + y P + H ( x P )) = 0. It fol-

− ≥ −

lows that v P ( y y P ) v P ( x x P ).

We now consider uniformisers at infinity on a hyperelliptic curve C over k . The easiest

way to proceed is to use the curve C † of equation ( 10.2 ).

C † be as in equation ( 10.2 ). Lemma 10.1.17 LetC be a hyperelliptic curve and letρ : C →

∞ + ) (0 ,α + ) C † ( ∞ + ) =∞ + (i.e., if there is one point at infin-

Let P = ρ ( = ∈ k ) .Ifι (

α + is a uniformiser at P on C † and so ( y/x d ) α + is a uniformiser at

ity) then Y − −

∞ + on C.Ifι ( ∞ + ) =∞ + then Z is a uniformiser at P on C † (i.e., 1 /x is a uniformiser

∞ + on C). at

∞ + ) =∞ + then ι ( P ) ∞ + ) =∞ + then ι ( P ) Proof Note that if ι ( = P and if ι ( = P .It

α + or Z is a uniformiser at P on
C † . Lemma 8.1.13 ,Exercise 8.2.8 and Lemma 8.2.6 show that for any f

immediately follows from Lemma 10.1.16 that Y −

( C † ) and

∈ k

α + )

P ∈ C ( k ), v P ( f ◦ ρ ) = v ρ ( P ) ( f ). Hence, uniformisers at infinity on C are ( Y − ◦ ρ =

( y/x d ) α + or Z − ◦ ρ = 1 /x .