Computing small discrete logarithms faster

Daniel J.  Bernstein; Tanja Lange

Paper 2012/458

Computing small discrete logarithms faster

Daniel J. Bernstein and Tanja Lange

Abstract

Computations of small discrete logarithms are feasible even in "secure" groups, and are used as subroutines in several cryptographic protocols in the literature. For example, the Boneh--Goh--Nissim degree-2-homomorphic public-key encryption system uses generic square-root discrete-logarithm methods for decryption. This paper shows how to use a small group-specific table to accelerate these subroutines. The cost of setting up the table grows with the table size, but the acceleration also grows with the table size. This paper shows experimentally that computing a discrete logarithm in an interval of order l takes only 1.93*l^{1/3} multiplications on average using a table of size l^{1/3} precomputed with 1.21*l^{2/3} multiplications, and computing a discrete logarithm in a group of order l takes only 1.77*l^{1/3} multiplications on average using a table of size l^{1/3} precomputed with 1.24*l^{2/3} multiplications.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Published elsewhere. Unknown where it was published
Keywords: discrete logarithms random walks precomputation
Contact author(s): tanja @ hyperelliptic org
History: 2012-09-20: revised; 2012-08-13: received; See all versions
Short URL: https://ia.cr/2012/458
License: CC BY

BibTeX

@misc{cryptoeprint:2012/458,
      author = {Daniel J.  Bernstein and Tanja Lange},
      title = {Computing small discrete logarithms faster},
      howpublished = {Cryptology {ePrint} Archive, Paper 2012/458},
      year = {2012},
      url = {https://eprint.iacr.org/2012/458}
}