Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group
In 2004, Muzereau, Smart and Vercauteren [A. Muzereau, N. P. Smart and F. Vercauteren, The equivalence between the DHP and DLP for elliptic curves used in practical applications, LMS J. Comput. Math. 7 2004, 50–72] showed how to use a reduction algorithm of the discrete logarithm problem to Diffie–H...
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doaj-583f6b6989354bc580f3e3f653b085f62021-09-06T19:40:45ZengDe GruyterJournal of Mathematical Cryptology1862-29761862-29842018-06-0112210111810.1515/jmc-2017-0053Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary groupKushwaha Prabhat0SEAL Lab, Computer Science and Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721302, IndiaIn 2004, Muzereau, Smart and Vercauteren [A. Muzereau, N. P. Smart and F. Vercauteren, The equivalence between the DHP and DLP for elliptic curves used in practical applications, LMS J. Comput. Math. 7 2004, 50–72] showed how to use a reduction algorithm of the discrete logarithm problem to Diffie–Hellman problem in order to estimate lower bound for the Diffie–Hellman problem on elliptic curves. They presented their estimates on various elliptic curves that are used in practical applications. In this paper, we show that a much tighter lower bound for the Diffie–Hellman problem on those curves can be achieved if one uses the multiplicative group of a finite field as auxiliary group. The improved lower bound estimates of the Diffie–Hellman problem on those recommended curves are also presented. Moreover, we have also extended our idea by presenting similar estimates of DHP on some more recommended curves which were not covered before. These estimates of DHP on these curves are currently the tightest which lead us towards the equivalence of the Diffie–Hellman problem and the discrete logarithm problem on these recommended elliptic curves.https://doi.org/10.1515/jmc-2017-0053discrete logarithm problemlower bound for the diffie–hellman problemelliptic curves used in practical applications94a60 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Kushwaha Prabhat |
spellingShingle |
Kushwaha Prabhat Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group Journal of Mathematical Cryptology discrete logarithm problem lower bound for the diffie–hellman problem elliptic curves used in practical applications 94a60 |
author_facet |
Kushwaha Prabhat |
author_sort |
Kushwaha Prabhat |
title |
Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group |
title_short |
Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group |
title_full |
Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group |
title_fullStr |
Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group |
title_full_unstemmed |
Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group |
title_sort |
improved lower bound for diffie–hellman problem using multiplicative group of a finite field as auxiliary group |
publisher |
De Gruyter |
series |
Journal of Mathematical Cryptology |
issn |
1862-2976 1862-2984 |
publishDate |
2018-06-01 |
description |
In 2004, Muzereau, Smart and Vercauteren [A. Muzereau, N. P. Smart and F. Vercauteren,
The equivalence between the DHP and DLP for elliptic curves used in practical applications,
LMS J. Comput. Math. 7 2004, 50–72] showed how to use a reduction algorithm of the discrete logarithm problem to Diffie–Hellman problem in order to estimate lower bound for the Diffie–Hellman problem on elliptic curves. They presented their
estimates on various elliptic curves that are used in practical applications. In this paper, we show that a much tighter
lower bound for the Diffie–Hellman problem on those curves can be achieved if one uses the multiplicative group of a finite
field as auxiliary group. The improved lower bound estimates of the Diffie–Hellman problem on those recommended curves
are also presented. Moreover, we have also extended our idea by presenting similar
estimates of DHP on some more recommended curves which were not covered before. These estimates of DHP on these curves are
currently the tightest which lead us towards the equivalence of the Diffie–Hellman problem and the discrete
logarithm problem on these recommended elliptic curves. |
topic |
discrete logarithm problem lower bound for the diffie–hellman problem elliptic curves used in practical applications 94a60 |
url |
https://doi.org/10.1515/jmc-2017-0053 |
work_keys_str_mv |
AT kushwahaprabhat improvedlowerboundfordiffiehellmanproblemusingmultiplicativegroupofafinitefieldasauxiliarygroup |
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1717767899322515456 |