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Computación y Sistemas

Print version ISSN 1405-5546

Comp. y Sist. vol.12 n.3 México Jan./Mar. 2009

 

Artículos

 

Nontrivial Solutions to the Cubic Sieve Congruence Problem: x3 y2z mod p

 

Soluciones no Triviales al Problema de Congruencia de Criba Cúbica: x3 y2z mod p

 

Subhamoy Maitra1, Y. V. Subba Rao2, Pantelimon Stanica3 and Sugata Gangopadhyay4

 

1 Applied Statistics Unit, Indian Statistical Institute, 203, B T Road, Calcutta 700 108, INDIA. E–mail: subho@isical.ac.in

2 School of Maths, and Comp./Info. Sciences, University of Hyderabad, P. O. Central University, Gachbowli, Hyderabad 500 046, INDIA. E–mail: yvsrcs@uohyd.ernet.in

3 Applied Mathematics Department Graduate School of Engineering & Applied Sciences (GSEAS) Naval Postgraduate School, Monterey, CA 93943, USA. E–mail: pstanica@nps.edu

4 Mathematics Department, Indian Institute of Technology, Roorkee, Haridwar, Uttaranchal, INDIA. E–mail: sugata70@rediffmail.com

 

Article received on March 1, 2008
Accepted on June 14, 2008

 

Abstract

In this paper we discuss the problem of finding nontrivial solutions to the Cubic Sieve Congruence problem, that is, solutions of x3 = y2z (mod p), where x,y,z < and x3 y2z. The solutions to this problem are useful in solving the Discrete Log Problem or factorization by index calculus method. Apart from the cryptographic interest, this problem is motivating by itself from a number theoretic point of view. Though we could not solve the problem completely, we could identify certain sub classes of primes where the problem can be solved in time polynomial in log p. Further we could extend the idea of Reyneri's sieve and identify some cases in it where the problem can even be solved in constant time. Designers of cryptosystems should avoid all primes contained in our detected cases.

Keywords: Cubic Sieve Congruence, Discrete Log Problem, Prime Numbers.

 

Resumen

En este artículo se discute el problema de cómo encontrar soluciones no triviales al problema de congruencia de la criba cúbica, esto es, soluciones a la ecuación: x3 = y2z (mod p), donde x,y, z < y x3 y2z. Las soluciones a este problema resultan útiles para resolver el problema del logaritmo discreto o el de factorización entera cuando se utiliza el método de index calculus. Además del evidente interés criptográfico, este problema tiene también relevancia desde el punto de vista de la teoría elemental de números. Aunque no logramos resolver totalmente el problema, sí pudimos identificar ciertas subclases de primos donde el problema puede ser resuelto en tiempo polinomial en logp. Asimismo, extendimos la idea de cribado de Reyneri e identificamos algunas clases en donde el problema puede ser resuelto en tiempo constante. Los diseñadores de cripto–esquemas deben evitar utilizar cualquiera de los primos contenidos en los casos aquí detectados.

Palabras Claves: Congruencia de criba cúbica, problema del logaritmo discreto, números primos.

 

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References

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3. A. Das and C. E. Veni Madhavan. On the cubic sieve method for computing discrete logarithms over prime fields. International Journal of Computer Mathematics, 82(12): 1481–1495, December, 2005.        [ Links ]

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