Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation?
We give a 4-list algorithm for solving the Elliptic Curve Discrete Logarithm (ECDLP) over some quadratic field 𝔽p2. Using the representation technique, we reduce ECDLP to a multivariate polynomial zero testing problem. Our solution of this problem using bivariate polynomial multi-evaluation yields a...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2020-08-01
|
| Series: | Journal of Mathematical Cryptology |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/jmc-2019-0025 |
| id |
doaj-531f0a379b2642f8b6e328a9af927de1 |
|---|---|
| record_format |
Article |
| spelling |
doaj-531f0a379b2642f8b6e328a9af927de12021-09-06T19:40:45ZengDe GruyterJournal of Mathematical Cryptology1862-29761862-29842020-08-0114129330610.1515/jmc-2019-0025jmc-2019-0025Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation?Delaplace Claire0May Alexander1Horst Görtz Institute for IT Security, Ruhr University Bochum, Bochum, GermanyHorst Görtz Institute for IT Security, Ruhr University Bochum, Bochum, GermanyWe give a 4-list algorithm for solving the Elliptic Curve Discrete Logarithm (ECDLP) over some quadratic field 𝔽p2. Using the representation technique, we reduce ECDLP to a multivariate polynomial zero testing problem. Our solution of this problem using bivariate polynomial multi-evaluation yields a p1.314-algorithm for ECDLP. While this is inferior to Pollard’s Rho algorithm with square root (in the field size) complexity 𝓞(p), it still has the potential to open a path to an o(p)-algorithm for ECDLP, since all involved lists are of size as small as p34,$\begin{array}{} p^{\frac 3 4}, \end{array}$ only their computation is yet too costly.https://doi.org/10.1515/jmc-2019-0025ecdlprepresentationsmultivariate polynomial zero-testing11y9911t0611t71 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Delaplace Claire May Alexander |
| spellingShingle |
Delaplace Claire May Alexander Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation? Journal of Mathematical Cryptology ecdlp representations multivariate polynomial zero-testing 11y99 11t06 11t71 |
| author_facet |
Delaplace Claire May Alexander |
| author_sort |
Delaplace Claire |
| title |
Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation? |
| title_short |
Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation? |
| title_full |
Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation? |
| title_fullStr |
Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation? |
| title_full_unstemmed |
Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation? |
| title_sort |
can we beat the square root bound for ecdlp over 𝔽p2 via representation? |
| publisher |
De Gruyter |
| series |
Journal of Mathematical Cryptology |
| issn |
1862-2976 1862-2984 |
| publishDate |
2020-08-01 |
| description |
We give a 4-list algorithm for solving the Elliptic Curve Discrete Logarithm (ECDLP) over some quadratic field 𝔽p2. Using the representation technique, we reduce ECDLP to a multivariate polynomial zero testing problem. Our solution of this problem using bivariate polynomial multi-evaluation yields a p1.314-algorithm for ECDLP. While this is inferior to Pollard’s Rho algorithm with square root (in the field size) complexity 𝓞(p), it still has the potential to open a path to an o(p)-algorithm for ECDLP, since all involved lists are of size as small as p34,$\begin{array}{}
p^{\frac 3 4},
\end{array}$ only their computation is yet too costly. |
| topic |
ecdlp representations multivariate polynomial zero-testing 11y99 11t06 11t71 |
| url |
https://doi.org/10.1515/jmc-2019-0025 |
| work_keys_str_mv |
AT delaplaceclaire canwebeatthesquarerootboundforecdlpoverfp2viarepresentation AT mayalexander canwebeatthesquarerootboundforecdlpoverfp2viarepresentation |
| _version_ |
1717767896319393792 |