A Mean Value Formula for Elliptic Curves
It is proved in this paper that, for any point on an elliptic curve, the mean value of -coordinates of its -division points is the same as its -coordinate and that of -coordinates of its -division points is times that of its -coordinate.
Let be a field with and let be the algebraic closure of . Every elliptic curve over can be written as a classical Weierstrass equation as follows:
with coefficients . A point on is said to be smooth (or nonsingular) if , where . The point multiplication is the operation of computing
for any point and a positive integer . The multiplication-by- map
is an isogeny of degree . For a point , any element of is called an -division point of . Assume that . In this paper, the following result on the mean value of the -coordinates of all the -division points of any smooth point on an elliptic curve is proved.
Theorem 1. Let be an elliptic curve defined over and let be a point with . Set
According to Theorem 1, let , , be all the points such that and let be the slope of the line through and ; then . Therefore,
Thus we have where , , , and are the average values of the variables , , , and , respectively. Therefore,
Remark 2. The discrete logarithm problem in elliptic curve is to find by given with . The above theorem gives some information on the integer .
2. Proof of Theorem 1
It can be checked easily by induction that the ’s are polynomials. Moreover, when is odd, and when is even. Define the polynomial
for . Then . Since , replacing by , one has . So we can denote it by . Note that if and have the same parity. Furthermore, the division polynomials have the following properties.
Lemma 3. Consider
when is odd and
when is even.
Proof. We prove the result by induction on . It is true for . Assume that it holds for all with . We give the proof only for the case for odd . The case for even can be proved similarly. Now let be odd, where . If is even, then by induction
Substituting by , we have
The case when is odd can be proved similarly.
The following corollary follows immediately from Lemma 3.
Corollary 4. Consider
Proof of Theorem 1. Define as
Then for any , we have ()
If , then . Therefore, for any , the -coordinate of satisfies the equation . From Corollary 4, we have that
Since , every root of is the -coordinate of some . Therefore,
by Vitae’s theorem.
Now we prove the mean value formula for -coordinates. Let be the complex number field first and let and be complex numbers which are linearly independent over . Define the lattice
and the Weierstrass -function by
For integers , define the Eisenstein series by
Set and ; then
Let be the elliptic curve given by . Then the map
is an isomorphism of groups and . Conversely, it is well known  that, for any elliptic curve over defined by , there is a lattice such that and there is an isomorphism between groups and given by and . Therefore, for any point , we have and for some .
Let for a . Then for any , , there exist integers with , such that
which comes from . Differentiate with respect to , we have
Secondly, let be a field of characteristic and let be the elliptic curve over given by the equation . Then all of the equations describing the group law are defined over . Since is algebraically closed and has infinite transcendence degree over , can be considered as a subfield of . Therefore we can regard as an elliptic curve defined over . Thus the result follows.
At last assume that is a field of characteristic . Then the elliptic curve can be viewed as one defined over some finite field , where for some integer . Without loss of generality, let for convenience. Let be an unramified extension of the -adic numbers of degree , and let be an elliptic curve over which is a lift of . Since , the natural reduction map is an isomorphism. Now for any point with , we have a point such that the reduction point is . For any point with , its lifted point satisfies and whenever . Thus,
since is a field of characteristic 0. Therefore the formula holds by the reduction from to .
Remark 5. (1)The result for -coordinate of Theorem 1 holds also for the elliptic curve defined by the general Weierstrass equation .(2)The mean value formula for -coordinates was given in the first version of this paper  with a slightly complicated proof. The formula for -coordinates was conjectured by Feng and Wu based on  and numerical examples in a personal email communication with Moody (June 1, 2010).(3)Recently, some mean value formulae for twisted Edwards curves [3, 4] and other alternate models of elliptic curves were given by [5, 6].
3. An Application
Let be an elliptic curve over given by the Weierstrass equation . Then we have a nonzero invariant differential . Let be a nonzero endomorphism. Then for some , since the space of differential forms on is a -dimensional -vector space. Since and , we have
Hence has neither zeros nor poles and . Let and be two nonzero endomorphisms; then
Therefore, . For any nonzero endomorphism , set , where and are rational functions. Then where is the differential of . In particular, for any positive integer , the map on is an endomorphism. Set . From and , we have
For any and any
we have . Therefore, Theorem 1 gives
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (no. 11101002, no. 61370187, and no. 11271129) and Beijing Natural Science Foundation (no. 1132009).
D. Moody, Divison Polynomials for Alternate Models of Elliptic Curves, http://eprint.iacr.org/2010/630.pdf.