The index calculus method using non-smooth polynomials

Authors:
Theodoulos Garefalakis and Daniel Panario

Journal:
Math. Comp. **70** (2001), 1253-1264

MSC (2000):
Primary 11Y16, 12E05; Secondary 11T71, 68P25, 68Q25, 94A60

Published electronically:
March 7, 2001

MathSciNet review:
1826581

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We study a generalized version of the index calculus method for the discrete logarithm problem in , when , is a small prime and . The database consists of the logarithms of all irreducible polynomials of degree between given bounds; the original version of the algorithm uses lower bound equal to one. We show theoretically that the algorithm has the same asymptotic running time as the original version. The analysis shows that the best upper limit for the interval coincides with the one for the original version. The lower limit for the interval remains a free variable of the process. We provide experimental results that indicate practical values for that bound. We also give heuristic arguments for the running time of the Waterloo variant and of the Coppersmith method with our generalized database.

**1.**I. F. Blake, R. Fuji-Hara, R. C. Mullin, and S. A. Vanstone,*Computing logarithms in finite fields of characteristic two*, SIAM J. Algebraic Discrete Methods**5**(1984), no. 2, 276–285. MR**745447**, 10.1137/0605029**2.**Manuel Blum and Silvio Micali,*How to generate cryptographically strong sequences of pseudorandom bits*, SIAM J. Comput.**13**(1984), no. 4, 850–864. MR**764183**, 10.1137/0213053**3.**A. J. Menezes and S. A. Vanstone (eds.),*Advances in cryptology—CRYPTO ’90*, Lecture Notes in Computer Science, vol. 537, Springer-Verlag, Berlin, 1991. MR**1232866****4.**Don Coppersmith,*Fast evaluation of logarithms in fields of characteristic two*, IEEE Trans. Inform. Theory**30**(1984), no. 4, 587–594. MR**755785**, 10.1109/TIT.1984.1056941**5.**Whitfield Diffie and Martin E. Hellman,*New directions in cryptography*, IEEE Trans. Information Theory**IT-22**(1976), no. 6, 644–654. MR**0437208****6.**Peter B. Busschbach, Michiel G. L. Gerretzen, and Henk C. A. van Tilborg,*On the covering radius of binary, linear codes meeting the Griesmer bound*, IEEE Trans. Inform. Theory**31**(1985), no. 4, 465–468. MR**798551**, 10.1109/TIT.1985.1057073**7.**Shuhong Gao, Jason Howell, and Daniel Panario,*Irreducible polynomials of given forms*, Finite fields: theory, applications, and algorithms (Waterloo, ON, 1997), Contemp. Math., vol. 225, Amer. Math. Soc., Providence, RI, 1999, pp. 43–54. MR**1650605**, 10.1090/conm/225/03208**8.**Shuhong Gao, Joachim von zur Gathen, and Daniel Panario,*Gauss periods: orders and cryptographical applications*, Math. Comp.**67**(1998), no. 221, 343–352. With microfiche supplement. MR**1458221**, 10.1090/S0025-5718-98-00935-1**9.**T. Garefalakis and D. Panario.

Polynomials over finite fields free from large and small degree irreducible factors. Submitted, 1999.**10.**J. von zur Gathen and D. Panario.

A survey on factoring polynomials over finite fields.

To appear in*J. Symb. Comp.*, 2000.**11.**Rudolf Lidl and Harald Niederreiter,*Finite fields*, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR**746963****12.**R. Lovorn.*Rigourous, subexponential algorithms for discrete logarithms over finite fields*.

PhD thesis, University of Georgia, 1992.**13.**Renet Lovorn Bender and Carl Pomerance,*Rigorous discrete logarithm computations in finite fields via smooth polynomials*, Computational perspectives on number theory (Chicago, IL, 1995) AMS/IP Stud. Adv. Math., vol. 7, Amer. Math. Soc., Providence, RI, 1998, pp. 221–232. MR**1486839****14.**Alfred Menezes,*Elliptic curve public key cryptosystems*, The Kluwer International Series in Engineering and Computer Science, 234, Kluwer Academic Publishers, Boston, MA, 1993. With a foreword by Neal Koblitz; Communications and Information Theory. MR**1700718****15.**Volker Müller, Andreas Stein, and Christoph Thiel,*Computing discrete logarithms in real quadratic congruence function fields of large genus*, Math. Comp.**68**(1999), no. 226, 807–822. MR**1620235**, 10.1090/S0025-5718-99-01040-6**16.**A. M. Odlyzko,*Discrete logarithms in finite fields and their cryptographic significance*, Advances in cryptology (Paris, 1984) Lecture Notes in Comput. Sci., vol. 209, Springer, Berlin, 1985, pp. 224–314. MR**825593**, 10.1007/3-540-39757-4_20**17.**A. M. Odlyzko,*Discrete logarithms and smooth polynomials*, Finite fields: theory, applications, and algorithms (Las Vegas, NV, 1993), Contemp. Math., vol. 168, Amer. Math. Soc., Providence, RI, 1994, pp. 269–278. MR**1291435**, 10.1090/conm/168/01706**18.**Douglas H. Wiedemann,*Solving sparse linear equations over finite fields*, IEEE Trans. Inform. Theory**32**(1986), no. 1, 54–62. MR**831560**, 10.1109/TIT.1986.1057137

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Additional Information

**Theodoulos Garefalakis**

Affiliation:
Department of Computer Science, University of Toronto, Toronto, M5S 3G4, Canada

Email:
theo@cs.toronto.edu

**Daniel Panario**

Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, K1S 5B6, Canada

Email:
daniel@math.carleton.ca

DOI:
https://doi.org/10.1090/S0025-5718-01-01298-4

Keywords:
Finite fields,
discrete logarithm problem,
cryptography,
smooth polynomials

Received by editor(s):
May 24, 1999

Published electronically:
March 7, 2001

Additional Notes:
Work done while the second author was with the Department of Computer Science, University of Toronto.

Article copyright:
© Copyright 2001
American Mathematical Society