A technique based on the euclidean algorithm and its applications to cryptography and nonlinear diophantine equations

Authors

  • Luis A. Cortés Vega Universidad de Antofagasta.
  • Daniza E. Rojas Castro Universidad de Antofagasta.
  • Yolanda S. Santiago Ayala Universidad Nacional Mayor de San Marcos.
  • Santiago C. Rojas Romero Universidad Nacional Mayor de San Marcos, Lima.

DOI:

https://doi.org/10.4067/S0716-09172007000300007

Keywords:

Algorithmic matrix function, Euclidean algorithm, Non-linear Diophantine equations, Message codification and decoding, Gauβ’s congruence module p.

Abstract

The main objective of this work is to build, based on the Euclidean algorithm, a “matrix of algorithms ”
formula1.JPG where formula2.JPG is a fixed matrix on formula3.JPG". The function ΦB is called the algorithmic matrix function. Here we show its properties and some applications to Cryptography and nonlinear Diophantine equations. The case n = m = 1 has particular interest. On this way we show equivalences between ΦB and the Carl Friedrich Gauβ’s congruence module p.

Author Biographies

Luis A. Cortés Vega, Universidad de Antofagasta.

Departamento de Matemáticas, Facultad de Ciencias Básicas.

Daniza E. Rojas Castro, Universidad de Antofagasta.

Departamento de Matemáticas, Facultad de Ciencias Básicas.

Yolanda S. Santiago Ayala, Universidad Nacional Mayor de San Marcos.

Facultad de Ciencias Matemáticas.

Santiago C. Rojas Romero, Universidad Nacional Mayor de San Marcos, Lima.

Facultad de Ciencias Matemáticas.

References

[1] R. D, Mauldin, A Generalization of Fermat’s Last Theorem: The Beal’s Conjeture and Prize Problem. Notices of the AMS, 44, No 11, pp, 1436—1437, (1997).

[2] A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography, CRC Press, (1996).

[3] C. Mercado, Historia de las Matemáticas, Ed. Universitaria, Santiago de Chile, (1972).

[4] S. Lang, Old and new conjetured diophantine inequalities, Bull. Amer. Math. Soc, 23, pp, 37—75, (1990)

[5] A. Walis, Modular elliptic curves and Fermat‘s Last Theorem, Ann. Math. 141 (1995), pp. 443—551, (1995).

[6] S. Singh, O ´ultimo Teorema de Fermat, Ed. Record, Rio de Janeiro, (1999).

[7] N. L. Biggs, Matemática Discreta, Ed. Vicens—Vives, (1994).

[8] L. A. Cortés Vega, D. E. Rojas Castro, Y. S. Santiago Ayala and S. C. Rojas Romero, Entre la congruencia de Gauβ y el Algoritmo de Euclides: Una Observación, Pre-Print, (2007).

[9] L. A. Cortés—Vega, D. E. Rojas—Castro, The Quotient Function and its applications to some problems yielding in Discrete Mathematics and Artificial Intelligence, Pre-print, (2007).

[10] B. Kolman, R. S. Busby y S. C. Ross, Estructura de matemáticas discretas para la computación, Prentice Hall, (1887).

[11] R. Graham, D. E. Knuth y O. Patashnik, Concrete Mathematics: A fondation for Computer Science, Addison-Wesley, (1994).

[12] D. Burton, Elementary Number Theory, Ed. Allyn—Bacon, (1980).

[13] A. Hodges, Alan Turing: The Enigma of Intelligence, Ed. Unwin— Paperbacks, (1983).

[14] R. P. Grimaldi, Discrete and Combinatorial Mathematics. An Applied Introduction, Addison-Wesley, (1994).

Published

2017-04-12

How to Cite

[1]
L. A. Cortés Vega, D. E. Rojas Castro, Y. S. Santiago Ayala, and S. C. Rojas Romero, “A technique based on the euclidean algorithm and its applications to cryptography and nonlinear diophantine equations”, Proyecciones (Antofagasta, On line), vol. 26, no. 3, pp. 309-339, Apr. 2017.

Issue

Section

Artículos