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THE NUMBER OF POINTS ON ELLIPTIC CURVES y2 = x3 + Ax AND y2 = x3 + B3 MOD 24

  • Received : 2012.09.11
  • Published : 2013.07.31

Abstract

In this paper, we calculate the number of points on elliptic curves $y^2=x^3+Ax$ over $F_{p^r}$ modulo 24. This is a generalization of [8], [9] and [16].

References

  1. I. F. Blake, G. Seroussi, and N. P. Smart, Elliptic Curves in Cryptography, Reprint of the 1999 original. London Mathematical Society Lecture Note Series, 265. Cambridge University Press, Cambridge, 2000.
  2. B. M. Brewer, On certain character sums, Trans. Amer. Math. Soc. 99 (1961), 241-245. https://doi.org/10.1090/S0002-9947-1961-0120202-1
  3. M. Demirci, G. Soydan, and I. N. Cangul, Rational points on elliptic curves E : $y^2=x^3+a^3$ in $\mathbb{F}_p$ where $p{\equiv}1$ (mod 6) is prime, Rocky Mountain J. Math. 37 (2007), no. 5, 1483-1491. https://doi.org/10.1216/rmjm/1194275930
  4. I. Inam, O. Bizim, and I. N. Cangul, Rational points on Frey elliptic curves E : $y^2=x^3-n^2x$, Adv. Stud. Contemp. Math. (Kyungshang) 14 (2007), no. 1, 69-76.
  5. I. Inam, G. Soydan, M. Demirci, O. Bizim, and I. N. Cangul, Corrigendum on "The number of points on elliptic curves E : $y^2=x^3$ +cx over $\mathbb{F}_p$ mod 8", Commun. Korean Math. Soc. 22 (2007), no. 2, 207-208. https://doi.org/10.4134/CKMS.2007.22.2.207
  6. K. Ireland and M. Rosen A Classical Introduction to Mordern Number Theory, Springer-Verlag, 1981.
  7. A. W. Knapp, Elliptic Curves, Princeton Uinversity Press, New Jersey 1992.
  8. H. Park, D. Kim, and E. Lee The number of points on elliptic curves E : $y^2=x^3$ + cx over $\mathbb{F}_p$ mod 8, Commun. Korean Math. Soc. 18 (2003), no. 1, 31-37. https://doi.org/10.4134/CKMS.2003.18.1.031
  9. H. Park, S. You, H. Park, D. Kim, and H. Kim The number of points on elliptic curves $E_A^0$ : $y^2=x^3$ + Ax over $\mathbb{F}_p$ mod 24, Honam Math. J. 34 (2012), no.1, 93-101. https://doi.org/10.5831/HMJ.2012.34.1.93
  10. A. R. Rajwade, A note on the number of solutions $N_p$ of the congruence $y^2{\equiv}x^3$-Dx (mod p), Proc. Cambfidge Philos. Soc. 67 (1970), 603-605. https://doi.org/10.1017/S0305004100045916
  11. R. Schoof, Counting points on elliptic curves over finite fields, J. Theor. Nombres Bordeaux 7 (1995), no. 1, 219-254. https://doi.org/10.5802/jtnb.142
  12. J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986.
  13. Z. H. Sun, Supplements to the theory of quartic residues, Acta Arith. 97 (2001), no. 4, 361-377. https://doi.org/10.4064/aa97-4-5
  14. B. A. Venkov, Elementary Number Theory, translated form the Russian and edited by H. Alderson, Wolters-Noordhoff, Groningen, 1970.
  15. A. Weil, Sur les courbes algebriques et les varietes qui s'en deduisent, Hermann, Paris, 1948.
  16. S. You, H. Park, and H. Kim The Number of points on elliptic curves $E_0^a\;^3$ : $y^2=x^3+a^3b$ over $\mathbb{F}_p$ mod 24, Honam Math. J. 31 (2009), no. 3, 437-449. https://doi.org/10.5831/HMJ.2009.31.3.437