Elliptic Curves over Finite Fields - Elliptic Curves: Number Theory and Cryptography - page 141

(a) Show that the points ( b, b √ a +2 b )and( b √ a − b, − − 2 b )haveor-

der 4.
(b) Show that at least one of a +2 b , a 2 b , a 2 4 b 2 is a square in F q . − −

(c) Show that if a 2 4 b 2 is a square in F q ,then E [2] − ⊆ E ( F q ).

(d) (Suyama) Show that # E ( F q ) is a multiple of 4.
(e) Let E be defined by y 2 = x 3 2 ax 2 +( a 2 4 b 2 ) x . Show that

E [2] ⊆ E ( F q ). Conclude that # E ( F q ) is a multiple of 4.

− −

The curve E is isogenous to E via

( x ,y )=( y 2 /x 2 ,y ( b 2 x 2 ) /x 2 ) −

(see the end of Section 8.6 and also Chapter 12). It can be shown that
this implies that # E ( F q )=# E ( F q ). This gives another proof of the
result of part (d). The curve E has been used in certain elliptic curve
factorization implementations (see [19]).

4.12 Let p be a prime and let E be a supersingular elliptic curve over the
finite field F p .Let φ p be the Frobenius endomorphism. Show that some
power of φ p is an integer. ( Note : This is easy when p ≥ 5. The cases
p =2 , 3 can be done by a case-by-case calculation.)

4.13 Let E be an elliptic curve over F q . Show that Hasse's theorem can be
restated as

− √ q ≤

# E ( F q )

1 .

4.14 Let E be an elliptic curve over F q . Assume that q = r 2 for some integer
r . Suppose that # E ( F q )=( r 1) 2 .Let φ = φ q be the q th power −

Frobenius endomorphism.

(a) Show that ( φ − r ) 2 =0.
(b) Show th at φ−r =0. ( Hint: A nonzero endomorphism is surjective
on E ( F q ) by Theorem 2.22.)
(c) Show that ( r − 1) E ( F q )=0.
(d) Show that E ( F q ) Z r− 1 .

(e) Now suppose E is an elliptic curve over F q with # E ( F q )=( r +1) 2
(where q = r 2 ). Show that E ( F q )

Z r− 1 ⊕

Z r +1 ⊕ Z r +1 .