ON REMARKS OF LIFTING PROBLEMS FOR ELLIPTIC CURVES 1 ...

6 HWAN JOON KIM, JUNG HEE CHEON, AND SANG GEUN HAHN
In the above three cases, the possible maximum of ĥ(R) is smaller than 3ĥ(P 3)/4,
so we have proved the inequality (2).
Moreover, if
(3/4) ⌊n/3⌋ max (ĥ(P i)) < ĥE,
i
then (2) means that P (n)
1 is a torsion-point. By solving this inequality for n, we
have proved the theorem.
□
Remark 2.4. As we have seen in the proof, n(E, P 1 , · · · , P 3 ) is very rough upper
bound for the running-time of Descent method. In fact, the heights of R’s decrease
more rapidly.
Remark 2.5. Even in the case of r ≥ 4, this algorithm can be applied. First, we
consider the case of r = 4. In Step 3, we get
2R = ɛ 1 P 1 + ɛ 2 P 2 + ɛ 3 P 3 + ɛ 4 P 4 .
If one of ɛ i ’s are 0, then it is satisfied that
ĥ(R) < (3/4) max (ĥ(P i)).
i
Moreover, even in the case that all ɛ i ’s are non-zero,
ĥ(R) = average of ĥ(P i) < max (ĥ(P i)).
i
This means that the heights of R’s decrease, so the algorithm terminates successfully.
Remark 2.6. Although the heights of R’s may not decrease in the case that r is
larger, the heights of them are willing to be bounded since we choose the points R’s
with small heights. Therefore, we can expect that some point appears twice in this
procedure. When this happens, it is not hard to construct the dependence equation
as long as we keep track of the points R’s and ɛ i ’s [13].
2.3. ECDLP on F 2 m. In general, we cannot lift an elliptic curve Ẽ over F p m to
an elliptic curve over Q for n > 1, so we should lift it to an elliptic curve over a
number field or an elliptic curve over a function field F p (T ). Here, we show that
the 2-descent method can be applied not only to the case of the rational field but
also to the case of the function field, so that the lifting problem generalized to the
function fields means the ECDLP on F p m for m > 1. For simplicity, we consider
only the case of p = 2. 5
It is easy to generalize the definition 2.1 to the function fields and to see that
each step of the algorithm to solve the ECDLP in the section 2 holds even in the
case of the function field except the 2-descent procedure because there is no known
algorithm to compute the canonical heights on the function fields. 6 However, the
following theorem show that if the difference between the absolute heights of two
points are sufficiently large, it is easy to compare between the canonical heights of
them. Hence we can easily make the sequence of the points whose canonical heights
are decreasing because the absolute heights are easily computable.
5 Our algorithm may be applied to any finite fields with small characteristic p.
6 For the definitions of absolute heights and canonical heights over the function fields, see [10].