## Square-Root Algorithms For The Discrete Logarithm Problem (a Survey) (2001)

Venue: | In Public Key Cryptography and Computational Number Theory, Walter de Gruyter |

Citations: | 32 - 0 self |

### BibTeX

@INPROCEEDINGS{Teske01square-rootalgorithms,

author = {Edlyn Teske},

title = {Square-Root Algorithms For The Discrete Logarithm Problem (a Survey)},

booktitle = {In Public Key Cryptography and Computational Number Theory, Walter de Gruyter},

year = {2001},

pages = {283--301}

}

### Years of Citing Articles

### OpenURL

### Abstract

The best algorithms to compute discrete logarithms in arbitrary groups (of prime order) are the baby-step giant-step method, the rho method and the kangaroo method. The first two have (expected) running time O( p n) group operations (n denoting the group order), thereby matching Shoup's lower bounds. While the baby-step giant-step method is deterministic but with large memory requirements, the rho and the kangaroo method are probabilistic but can be implemented very space efficiently, and they can be parallelized with linear speed-up. In this paper, we present the state of the art in these methods.

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Citation Context ...n key exchange protocol, the ElGamal encryption and signature schemes, the U.S. Government's Digital Signature Algorithm (DSA), and its elliptic curve analogue (ECDSA), rely on the diculty of the DLP =-=[MvOV96]-=-. These and other systems are being deployed on the Internet (to enable secure electronic mail, home banking, and Internet browsers), in thesnancial services industry (in electronic cash applications,... |

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Citation Context ...ric square-root algorithms for the DLP known to date are based on only a few methods: the baby-step giant-step method due to Shanks [Sha71], and the rho method and the kangaroo method, due to Pollard =-=[Pol-=-78]. The baby-step giant-step method is a deterministic method that uses a time-memory trade-o and takes const p ord g group operations and has to store const p ord g group elements. The rho method ... |

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Citation Context ...ge number of 1:37 p p 1 iterations. In other groups, or for other ways of partitioning (Z=pZ) , they occur even later, after an (experimental) average of 1:56 p n iterations, where n = ord g (see [T=-=es00]-=-). 4.1. Better random walks. Better walks for the rho method have been introduced that yield the same performance as we expect from a random mapping, which in the case of an arbitrary group means a sp... |

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1 |
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Citation Context ...elative to the discrete logarithm rather than relative to an upper bound on it. We discuss two such approaches, which work with increasing the giant-step width in the course of the algorithm. Thesrst =-=[BJT97-=-] works with doubling the step width at certain intervals. We describe a simplied version, which is based on the following statement: Lemma 3.1. [BJT97, Lemma 2.1] For every positive integer x there a... |

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Citation Context ...algorithm thesrst match is found after an expected number of 1:97 p n iterations. Modications of Brent's algorithm that require slightly more storage but less iterations can be found in [SL84] and [T=-=es98a]-=-. Pollard's original application can easily be generalized to any other group. All we need is a rule how to partition the group into 3 disjoint sets of equally large size, which can be done 10 EDLYN T... |