A technique based on the euclidean algorithm and its applications to cryptography and nonlinear diophantine equations
DOI:
https://doi.org/10.4067/S0716-09172007000300007Keywords:
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 ”
where is a fixed matrix on . 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.
References
[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
How to Cite
Issue
Section
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.