The Number of Rational Points on Conics Cp,k : x2 − ky2 = 1 over Finite Fields Fp

Authors: Ahmet Tekcan

Abstract:

Let p be a prime number, Fp be a finite field, and let k ∈ F*p. In this paper, we consider the number of rational points onconics Cp,k: x2 − ky2 = 1 over Fp. We proved that the order of Cp,k over Fp is p-1 if k is a quadratic residue mod p and is p + 1 if k is not a quadratic residue mod p. Later we derive some resultsconcerning the sums ΣC[x]p,k(Fp) and ΣC[y]p,k(Fp), the sum of x- and y-coordinates of all points (x, y) on Cp,k, respectively.

Keywords: Elliptic curve, conic, rational points.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1328960

References:

[1] A.O.L. Atkin and F. Moralin. Elliptic Curves and Primality Proving.Math. Comp. 61(203)(1993), 29?68.
[2] S. Goldwasser and J. Kilian. Almost all Primes Can be Quickly Certified,In Proc. 18th STOC (Berkeley, May 28-30, 1986). ACM, New York,(1986), 316?329.
[3] N. Koblitz. Elliptic Curve Cryptosystems. Math. Comp. 48(177)(1987),203?209.
[4] H.W.Jr. Lenstra. Factoring Integers with Elliptic Curves. Annals of Maths.126(3)(1987), 649?673.
[5] V.S. Miller. Use of Elliptic Curves in Cryptography, in Advances inCryptology?CRYPTO?85. Lect. Notes in Comp. Sci. 218, Springer-Verlag,Berlin (1986), 417?426.
[6] R.A. Mollin. An Introduction to Cryptography. Chapman&Hall/CRC,2001.
[7] L.J. Mordell. On the Rational Solutions of the Indeterminate Equationsof the Third and Fourth Degrees. Proc. Cambridge Philos. Soc. 21(1922),179?192.
[8] J.H. Silverman. The Arithmetic of Elliptic Curves. Springer-Verlag, 1986.
[9] J.H. Silverman and J. Tate. Rational Points on Elliptic Curves. Under-graduate Texts in Mathematics, Springer, 1992.
[10] L.C. Washington. Elliptic Curves, Number Theory and Cryptography.Chapman & Hall/CRC, Boca London, New York, Washington DC, 2003.
[11] A. Wiles. Modular Elliptic Curves and Fermat?s Last Theorem. Annalsof Maths. 141(3)(1995), 443?551.