Although calculating the exact time/memory complexity of algorithms
based on the Grobner basis is not easy, the new approach is
interesting:
A new algorithms for computing discrete logarithms on elliptic
curves defined over finite fields is suggested. It is based on a new
method to find zeroes of summation polynomials. In binary elliptic
curves one is to solve a cubic system of Boolean equations. Under a
first fall degree assumption the regularity degree of the system is
at most $4$. Extensive experimental data which supports the assumption is
provided. An heuristic analysis suggests a new asymptotical
complexity bound $2^{c\sqrt{n\ln n}}, c\approx 1.69$ for computing discrete
logarithms on an elliptic curve over a field of size $2^n$. For
several binary elliptic curves recommended by FIPS the new method
performs better than Pollard's.
<http://eprint.iacr.org/2015/310.pdf> or
<http://arxiv.org/pdf/1504.01175v1>
--
Regards,
ASK
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