#### Abstract

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.

#### 1. Introduction

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
**
Then
*

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

To prove Theorem 1, define division polynomials [1] on an elliptic curve inductively as follows:

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

Therefore,

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 ([1])

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 [1] 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

Thus,

which comes from . Differentiate with respect to , we have

That is,

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 [2] with a slightly complicated proof. The formula for -coordinates was conjectured by Feng and Wu based on [2] 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

Thus,

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

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).