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. 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