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Computación y Sistemas

Print version ISSN 1405-5546

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

 

Artículos

 

Searching Prime Numbers with Short Binary Signed Representation

 

Búsqueda de Números Primos con Representaciones Signadas Cortas

 

Jose de Jesús Angel Angel1 and Guillermo Morales–Luna1

 

1 Computer Science Department CINVESTAV–IPN, Mexico: E–mails: jjangel@computacion.cs.cinvestav.mx ; gmorales@cs.cinvestav.mx

 

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

 

Abstract

Modular arithmetic with prime moduli has been crucial in present day cryptography. The primes of Mersenne, Solinas, Crandall and the so called IKE–MODP primes have been widely used in efficient implementations. In this paper we study the density of primes with binary signed representation involving a small number of non–zero ±1–digits, and its repercussion in the generation of those primes.

Keywords: Pairing cryptography, prime numbers, signed representation.

 

Resumen

La aritmetica de residuos con números primos es crucial en la criptografía actual. Los números primos de Mersenne, Solinas, Crandall y los llamados IKE–MODP han sido extensamente utilizados en diversas implementaciones. Estudiamos aquí la densidad de los primos con representaciones signadas que involucran sólo un número pequeño de dígitos no–nulos ±1, así como su impacto en la generacion de tales primos.

Palabras Claves: Criptografía de emparejamientos, números primos, representaciones signadas.

 

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References

1. Chung, J. and A. Hasan (2003, April). More generalized mersenne number. Technical Report CORR–2003–17, Dept. of Computer Science, University of Waterloo.        [ Links ]

2. Crandall, R. E. (1994). Method and apparatus for public key exchange in a cryptographic system. Technical Report         [ Links ]5463690, U.S. Patents.

3. (FIPS), F. I. P. S. (2000). Digital signature standard. Technical Report 186–2, National Institute of Standards and Technology (NIST).        [ Links ]

4. Knuth, D. E. (1997, November). Art of Computer Programming, Volume 2: Seminumerical Algorithms (3rd Edition). Addison–Wesley Professional.        [ Links ]

5. Solinas, J. (1999). Generalized Mersenne numbers. Technical Report CORR 1999–39, University of Waterloo.        [ Links ]

6. Wagstaff, S. S. (2000). Prime numbers with a fixed number of one bits and zero bits in their binary representation. Experimental Mathematics 10(2), 267–273.        [ Links ]

7. Yie, I., S. Lim, S. Kim, and D. Kim (2003). Prime numbers of diffie–hellman groups for ike–modp. In T. Johansson and S. Maitra (Eds.), INDOCRYPT, Volume 2904 of Lecture Notes in Computer Science, pp. 228–234. Springer.        [ Links ]

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