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: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2020-08-01
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| Series: | Journal of Mathematical Cryptology |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/jmc-2019-0025 |
| Summary: | 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. |
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| ISSN: | 1862-2976 1862-2984 |