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

4.9 Let E be an elliptic curve over Fq with q = p2m . Suppose that #E(Fq ) =

q + 1 ’ 2 q.

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

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

4.11 Let Fq be a ¬nite ¬eld of odd characteristic and let a, b ∈ Fq with
a = ±2b and b = 0. De¬ne the elliptic curve E by

y 2 = x3 + ax2 + b2 x.




© 2008 by Taylor & Francis Group, LLC
142 CHAPTER 4 ELLIPTIC CURVES OVER FINITE FIELDS
√ √
(a) Show that the points (b, b a + 2b) and (’b, ’b a ’ 2b) have or-
der 4.
(b) Show that at least one of a + 2b, a ’ 2b, a2 ’ 4b2 is a square in Fq .
(c) Show that if a2 ’ 4b2 is a square in Fq , then E[2] ⊆ E(Fq ).
(d) (Suyama) Show that #E(Fq ) is a multiple of 4.
2 3 2
(e) Let E be de¬ned by y = x ’ 2ax + (a2 ’ 4b2 )x . Show that
E [2] ⊆ 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 ’ 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 [19]).

4.12 Let p be a prime and let E be a supersingular elliptic curve over the
¬nite ¬eld Fp . Let φp be the Frobenius endomorphism. Show that some
power of φp 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 Fq . Show that Hasse™s theorem can be
restated as

#E(Fq ) ’ q ¤ 1.

4.14 Let E be an elliptic curve over Fq . Assume that q = r2 for some integer
r. Suppose that #E(Fq ) = (r ’ 1)2 . Let φ = φq be the qth power
Frobenius endomorphism.
2
(a) Show that (φ ’ r) = 0.
(b) Show that φ ’ r = 0. (Hint: A nonzero endomorphism is surjective
on E(Fq ) by Theorem 2.22.)
(c) Show that (r ’ 1)E(Fq ) = 0.
Zr’1 • Zr’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 • Zr+1 .




© 2008 by Taylor & Francis Group, LLC

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