## An algorithm for evaluation of discrete logarithms in some nonprime finite fields

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- by Igor A. Semaev PDF
- Math. Comp.
**67**(1998), 1679-1689 Request permission

## Abstract:

In this paper we propose an algorithm for evaluation of logarithms in the finite fields $F_{p^n}$, where the number $p^n-1$ has a small primitive factor $r$. The heuristic estimate of the complexity of the algorithm is equal to $\exp ((c+o(1))(\log p r\log ^2r)^{1/3})$, where $n$ grows to $\infty$, and $p$ is limited by a polynomial in $n$. The evaluation of logarithms is founded on a new congruence of the kind of D. Coppersmith, $C(x)^k\equiv D(x)$, which has a great deal of solutions—pairs of polynomials $C(x),D(x)$ of small degrees.## References

- Whitfield Diffie and Martin E. Hellman,
*New directions in cryptography*, IEEE Trans. Inform. Theory**IT-22**(1976), no. 6, 644–654. MR**437208**, DOI 10.1109/tit.1976.1055638 - Stephen C. Pohlig and Martin E. Hellman,
*An improved algorithm for computing logarithms over $\textrm {GF}(p)$ and its cryptographic significance*, IEEE Trans. Inform. Theory**IT-24**(1978), no. 1, 106–110. MR**484737**, DOI 10.1109/tit.1978.1055817 *20th Annual Symposium on Foundations of Computer Science*, IEEE Computer Society, New York, 1979. Held in San Juan, Puerto Rico, October 29–31, 1979. MR**598097**- 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**, DOI 10.1007/3-540-39757-4_{2}0 - Don Coppersmith,
*Fast evaluation of logarithms in fields of characteristic two*, IEEE Trans. Inform. Theory**30**(1984), no. 4, 587–594. MR**755785**, DOI 10.1109/TIT.1984.1056941 - I. A. Semaev,
*On the number of small solutions of a linear homogeneous congruence*, Mat. Zametki**50**(1991), no. 4, 102–107, 159 (Russian); English transl., Math. Notes**50**(1991), no. 3-4, 1055–1058 (1992). MR**1162918**, DOI 10.1007/BF01137738 - O. Schirokauer, D. Weber and T. Denny,
*Discrete logarithms: the effectiveness of the index calculus method*, Algorithmic number theory, Lecture notes in computer science; vol. 1122, Springer, Berlin and New York, 1996, pp. 337–361.

## Additional Information

**Igor A. Semaev**- Affiliation: 43-2 Profsoyuznaya Street, Apartment #723, 117420 Moscow, Russia
- Received by editor(s): March 30, 1993
- Received by editor(s) in revised form: August 30, 1995
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp.
**67**(1998), 1679-1689 - MSC (1991): Primary 11T71, 11Y16, 94A60
- DOI: https://doi.org/10.1090/S0025-5718-98-00969-7
- MathSciNet review: 1474656