# Some remarks on arithmetical properties of recursive sequences on elliptic curves over a finite field

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## Abstract

In connection with problems of information theory, we study arithmetical progressions constructed at the points of elliptic curves over a finite field. For certain types of such curves, we establish the distribution of the quadratic residues at the *x*-coordinates of the sequence of points corresponding to progressions if the elliptic curves is defined over a simple field. A description of the set of all progressions on elliptic curves over a finite field is also given.

## Key words

Weierstrass normal form elliptic curve arithmetical progression finite field generator of pseudorandom numbers Sylow subgroup## Preview

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## References

- 1.G. Gong, T. Berson, and D. Stinson, “Elliptic curve pseudorandom sequences generator,” in
*Selected Areas in Cryptography, Lecture Notes in Comput. Sci., Kingston, ON, 1999*(Springer-Verlag, Berlin, 2000), Vol. 1758, pp. 34–48.Google Scholar - 2.G. Gong and C. Lam, “Linear recursive sequences over elliptic curves,” in
*Sequences and Their Applications, Discrete Math. Theor. Comput. Sci. (Lond.), Bergen, 2001*(Springer-Verlag, London, 2002), pp. 182–196.Google Scholar - 3.C. P. Xing, “Constructions of sequences from algebraic curves over finite fields,” in
*Sequences and Their Applications, Discrete Math. Theor. Comput. Sci. (Lond.), Bergen, 2001*(Springer-Verlag, London, 2002), pp. 88–100.Google Scholar - 4.N. Koblitz,
*A Course in Number Theory and Cryptography*, Graduate Texts in Mathematics, (Springer-Verlag, New York, 1987), Vol. 114.MATHGoogle Scholar - 5.V. E. Tarakanov, “Linear recursive sequences on elliptic curves and their applications in cryptography,” in
*Works in Discrete Mathematics*(Fizmatlit, Moscow, 2006), Vol. 9, pp. 340–356 [in Russian].Google Scholar - 6.I. R. Shafarevich,
*Foundations of Algebraic Geometry*(Nauka, Moscow, 1972) [in Russian].Google Scholar - 7.J. H. Silverman and J. Tate,
*Rational Points on Elliptic Curves*, Undergraduate Texts in Math. (Springer-Verlag, New York, 1992).MATHGoogle Scholar - 8.J. H. Silverman,
*The Arithmetic of Elliptic Curves*, Graduate Texts in Mathematics (Springer-Verlag, New York, 1986), Vol. 106.MATHGoogle Scholar - 9.V. E. Tarakanov, “Divisibility properties of the points of elliptic curves over a finite field,” in
*Works in Discrete Mathematics*(Fizmatlit, Moscow, 2001), Vol. 4, 243–258 [in Russian].Google Scholar - 10.R. Schoof, “Nonsingular plane curves over finite fields,” J. Combin. Theory Ser. A
**46**(2), 183–211 (1987).MATHCrossRefMathSciNetGoogle Scholar - 11.J. Miret, R. Moreno, A. Rao, and M. Valis, “Determining the 2-Sylow subgroup of an elliptic curve over a finite field,” Math. Comp.
**74**(249), 411–427 (2005).MATHCrossRefMathSciNetGoogle Scholar

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