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elliptic curve given by y 2 = x3 в€’ kx over Fp .

4.9 Let E be an elliptic curve over Fq with q = p2m . Suppose that #E(Fq ) =
в€љ
q + 1 в€’ 2 q.

(a) Let П†q be the Frobenius endomorphism. Show that (П†q в€’pm )2 = 0.
(b) Show that П†q в€’ pm = 0 (Hint: Theorem 2.22).
(c) Show that П†q acts as the identity on E[pm в€’ 1], and therefore that
E[pm в€’ 1] вЉ† E(Fq ).
Zpm в€’1 вЉ• Zpm в€’1 .
(d) Show that E(Fq )

4.10 Let E be an elliptic curve over Fq with q odd. Write #E(Fq ) = q +1в€’a.
Let d в€€ FГ— and let E (d) be the twist of E, as in Exercise 2.23. Show
q
that
d
#E (d) (Fq ) = q + 1 в€’ a.
Fq
(Hint: Use Exercise 2.23(c) and Theorem 4.14.)

4.11 Let Fq be a п¬Ѓnite п¬Ѓeld of odd characteristic and let a, b в€€ Fq with
a = В±2b and b = 0. Deп¬Ѓne the elliptic curve E by

y 2 = x3 + ax2 + b2 x.

В© 2008 by Taylor & Francis Group, LLC
142 CHAPTER 4 ELLIPTIC CURVES OVER FINITE FIELDS
в€љ в€љ
(a) Show that the points (b, b a + 2b) and (в€’b, в€’b a в€’ 2b) have or-
der 4.
(b) Show that at least one of a + 2b, a в€’ 2b, a2 в€’ 4b2 is a square in Fq .
(c) Show that if a2 в€’ 4b2 is a square in Fq , then E вЉ† E(Fq ).
(d) (Suyama) Show that #E(Fq ) is a multiple of 4.
2 3 2
(e) Let E be deп¬Ѓned by y = x в€’ 2ax + (a2 в€’ 4b2 )x . Show that
E  вЉ† E (Fq ). Conclude that #E (Fq ) is a multiple of 4.
The curve E is isogenous to E via

(x , y ) = (y 2 /x2 , y(b2 в€’ x2 )/x2 )

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

4.12 Let p be a prime and let E be a supersingular elliptic curve over the
п¬Ѓnite п¬Ѓeld Fp . Let П†p be the Frobenius endomorphism. Show that some
power of П†p is an integer. (Note: This is easy when p в‰Ґ 5. The cases
p = 2, 3 can be done by a case-by-case calculation.)
4.13 Let E be an elliptic curve over Fq . Show that HasseвЂ™s theorem can be
restated as
в€љ
#E(Fq ) в€’ q в‰¤ 1.

4.14 Let E be an elliptic curve over Fq . Assume that q = r2 for some integer
r. Suppose that #E(Fq ) = (r в€’ 1)2 . Let П† = П†q be the qth power
Frobenius endomorphism.
2
(a) Show that (П† в€’ r) = 0.
(b) Show that П† в€’ r = 0. (Hint: A nonzero endomorphism is surjective
on E(Fq ) by Theorem 2.22.)
(c) Show that (r в€’ 1)E(Fq ) = 0.
Zrв€’1 вЉ• Zrв€’1 .
(d) Show that E(Fq )
(e) Now suppose E is an elliptic curve over Fq with #E (Fq ) = (r+1)2
(where q = r2 ). Show that E (Fq ) Zr+1 вЉ• Zr+1 .

В© 2008 by Taylor & Francis Group, LLC

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