SciELO - Scientific Electronic Library Online

 
vol.26 issue3THE MODES OF POSTERIOR DISTRIBUTIONS FOR MIXED LINEAR MODELSON THE LOCAL HYPERCENTER OF A GROUP author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

Share


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.26 no.3 Antofagasta Dec. 2007

http://dx.doi.org/10.4067/S0716-09172007000300007 

 

Proyecciones Journal of Mathematics
Vol. 26, Nº 3, pp. 309-339, December 2007.
Universidad Católica del Norte
Antofagasta - Chile


A TECHNIQUE BASED ON THE EUCLIDEAN ALGORITHM AND ITS APPLICATIONS TO CRYPTOGRAPHY AND NONLINEAR DIOPHANTINE EQUATIONS


LUIS A. CORTÉS VEGA 1
DANIZA E. ROJAS CASTRO 1

YOLANDA S. SANTIAGO AYALA 2
SANTIAGO C. ROJAS ROMERO 2


1 Universidad de Antofagasta, Chile
2 Universidad Nacional Mayor de San Marcos, Perú

Correspondencia a:



Abstract
The main objective of this work is to build, based on the Euclidean algorithm, a “matrix of algorithms”

Where is a fixed matrix onThe function B 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 B and the Carl Friedrich Gauߒs congruence module p.


Key words : Algorithmic matrix function, Euclidean algorithm, non-linear Diophantine equations, message codification and decoding, Gauߒs congruence module p

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,         [ Links ](1997).
[2] A. Menezes, P. van Oorschot, and S. Vanstone, Handbook of Applied Cryptography, CRC Press,         [ Links ](1996).
[3] C. Mercado, Historia de las Matemáticas, Ed. Universitaria, Santiago de Chile,         [ Links ](1972).
[4] S. Lang, Old and new conjetured diophantine inequalities, Bull. Amer. Math. Soc, 23, pp, 37—75,         [ Links ](1990).
[5] A. Walis, Modular elliptic curves and Fermat‘s Last Theorem, Ann. Math. 141 (1995), pp. 443—551,         [ Links ](1995).
[6] S. Singh, O último Teorema de Fermat, Ed. Record, Rio de Janeiro,         [ Links ](1999).
[7] N. L. Biggs, Matemática Discreta, Ed. Vicens—Vives,         [ Links ](1994).
[8] L. A. Cortés Vega, D. E. Rojas Castro, Y. S. Santiago Ayala and S. C. Rojas Romero, Entre la congruencia de Gauß y el Algoritmo de Euclides: Una Observación, Pre-Print,         [ Links ](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,         [ Links ](2007).
[10] B. Kolman, R. S. Busby y S. C. Ross, Estructura de matemáticas discretas para la computación, Prentice Hall,         [ Links ](1887).
[11] R. Graham, D. E. Knuth y O. Patashnik, Concrete Mathematics: A fondation for Computer Science, Addison-Wesley,         [ Links ](1994).
[12] D. Burton, Elementary Number Theory, Ed. Allyn—Bacon,         [ Links ](1980).
[13] A. Hodges, Alan Turing: The Enigma of Intelligence, Ed. Unwin— Paperbacks,         [ Links ](1983).
[14] R. P. Grimaldi, Discrete and Combinatorial Mathematics. An Applied Introduction, Addison-Wesley, (1994).
        [ Links ]

LUIS A. CORTÉS VEGA
Departamento de matemáticas
Facultad de Ciencias Básicas
Universidad de Antofagasta
Casilla 170
Chile
e-mail : mailto:lcortes@uantof.cl

DANIZA E. ROJAS CASTRO
Departamento de matemáticas
Facultad de Ciencias Básicas
Universidad de Antofagasta
Casilla 170
Chile
e-mail : mailto:drojas@uantof.cl

YOLANDA S. SANTIAGO AYALA
Facultad de Ciencias Matemáticas
Universidad Nacional Mayor de San Marcos
Lima
Perú
e-mail : mailto:ysantiagoa@unmsm.edu.pe

SANTIAGO C. ROJAS ROMERO
Facultad de Ciencias Matemáticas
Univesidad Nacional Mayor de San Marcos
Lima
Perú
e-mail : mailto:srojasr@unmsm.edu.pe

Received : November 2006. Accepted : March 2007

 

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License