Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30172
Positive Definite Quadratic Forms, Elliptic Curves and Cubic Congruences

Authors: Ahmet Tekcan

Abstract:

Let F(x, y) = ax2 + bxy + cy2 be a positive definite binary quadratic form with discriminant Δ whose base points lie on the line x = -1/m for an integer m ≥ 2, let p be a prime number and let Fp be a finite field. Let EF : y2 = ax3 + bx2 + cx be an elliptic curve over Fp and let CF : ax3 + bx2 + cx ≡ 0(mod p) be the cubic congruence corresponding to F. In this work we consider some properties of positive definite quadratic forms, elliptic curves and cubic congruences.

Keywords: Binary quadratic form, elliptic curves, cubic congruence.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1072571

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1149

References:


[1] A.O.L. Atkin and F. Moralin. Elliptic Curves and Primality Proving. Math. Comp. 61(1993), 29-68.
[2] J. Buchmann and U. Vollmer. Binary Quadratic Forms: An Algorithmic Approach. Springer-Verlag, Berlin, Heidelberg, 2007.
[3] J. Buchmann. A generalization of Voronoi-s unit algorithm I,II. J. Number Theory 20(1985), 177-191, 192-209.
[4] D.A. Buell. Binary Quadratic Forms, Clasical Theory and Modern Computations. Springer-Verlag, New York, 1989.
[5] B.N. Delone and D.K. Faddeev. The Theory of Irrationalities of the Third Degree. Transl. Math. Monographs 10, Amer. Math. Soc., Providence, Rhode Island 1964.
[6] R. Dietmann. Small Solutions of Additive Cubic Congruences. Archiv der Mathematik 75(3)(2000), 195-197.
[7] D.E. Flath. Introduction to Number Theory. Wiley, 1989.
[8] B. Gezer, H.O¨ zden, A.Tekcan, O.Bizim. The Number of Rational Points on Elliptic Curves y2 = x3+b2 over Finite Fields. International Journal of Computational and Mathematical Sciences 1(3)(2007), 178-184.
[9] 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.
[10] N. Koblitz. A Course in Number Theory and Cryptography. Springer- Verlag, 1994.
[11] H.W.Jr. Lenstra. Factoring Integers with Elliptic Curves. Annals of Maths. 126(2)(1987), 649-673.
[12] Y.I. Manin. On a Cubic Congrunce to a Prime Modules. Amer. Math. Soc. Transl. 13(1960), 1-7.
[13] V.S. Miller. Use of Elliptic Curves in Cryptography, in Advances in Cryptology-CRYPTO-85. Lect. Notes in Comp. Sci. 218, Springer- Verlag, Berlin (1986), 417-426.
[14] R.A. Mollin. An Introduction to Cryptography. Chapman&Hall/CRC, 2001.
[15] R.A. Mollin. Advanced Number Theory with Applications. CRC Press, Taylor and Francis Group, Boca Raton, London, New York, 2009.
[16] L.J. Mordell. On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees. Proc. Cambridge Philos. Soc. 21(1922), 179-192.
[17] L.J. Mordell. On the Congruence ax3+by3+cz3+dxyz ≡ n(mod p). Duke Math. J. 31(1)(1964), 123-126.
[18] J.H. Silverman. The Arithmetic of Elliptic Curves. Springer-Verlag, 1986.
[19] B.K. Spearman and K. Williams. The Cubic Congrunce x3 + Ax2 + Bx + C ≡ 0(mod p) and Binary Quadratic Forms II. J. of London Math. Soc. 64(2)(2001), 273-274.
[20] A. Tekcan and O. Bizim. The Connection between Quadratic Forms and the Extended Modular Group. Mathematica Bohemica 128(3)(2003), 225-236.
[21] A. Tekcan. The Cubic Congruence x3+ax2+bx+c ≡ 0(mod p) and Binary Quadratic Forms F(x, y) = ax2+bxy+cy2. Ars Combinatoria 85(2007), 257-269.
[22] A. Tekcan. The Cubic Congruence x3+ax2+bx+c ≡ 0(mod p) and Binary Quadratic Forms F(x, y) = ax2 + bxy + cy2 II. Accepted for publication to Acta Universitatis Apulensis.
[23] A .Tekcan. The Elliptic Curves y2 = x3 − t2x over Fp. International Journal of Comp. and Maths. Sci. 1(3)(2007), 165-171.
[24] A. Tekcan, A. O┬¿ zkoc┬©, B. Gezer, O. Bizim. Elliptic Curves, Conics and Cubic Congruencies associated with Indefinite Binary Quadratic Forms. Novi Sad Journal of Mathematics 38(2)(2008), 71-81.
[25] A. Tekcan. The Number of Rational Points on Singular Curves y2 = x(x − a)2 over Finite Fields Fp. Int. Journal of Comp.and Math.Sci. 3(1)(2009), 14-17.
[26] A. Tekcan. The Elliptic Curves y2 = x3 − 1728x over Finite Fields. Journal of Algebra, Number Theory: Advances and Applications 1(1)(2009), 61-74.
[27] A. Tekcan. The Elliptic Curves y2 = x(x − 1)(x − λ). Accepted for publication to Ars Combinatoria.
[28] G.F. Voronoi. On a Generalization of the Algorithm of Continued Fractions. (in Russian). Phd Dissertation, Warsaw, 1896.
[29] L.C. Washington. Elliptic Curves, Number Theory and Cryptography. Chapman&Hall /CRC, Boca London, New York, Washington DC, 2003.
[30] A. Wiles. Modular Elliptic Curves and Fermat-s Last Theorem. Annals of Maths. 141(3)(1995), 443-551.
[31] H.C. Williams and C.R. Zarnke. Some Algoritms for Solving a Cubic Congruence modulo p. Utilitas Mathematica 6(1974), 285-306.