Paper 2016/704

High Saturation Complete Graph Approach for EC Point Decomposition and ECDL Problem

Nicolas T. Courtois


One of the key questions in contemporary applied cryptography is whether there exist an efficient algorithm for solving the discrete logarithm problem in elliptic curves. The primary approach for this problem is to try to solve a certain system of polynomial equations. Current attempts try to solve them directly with existing software tools which does not work well due to their very loosely connected topology and illusory reliance on degree falls. A deeper reflection on what makes systems of algebraic equations efficiently solvable is missing. In this paper we propose a new approach for solving this type of polynomial systems which is radically different than current approaches. We carefully engineer systems of equations with excessively dense topology obtained from a complete clique/biclique graphs and hypergraphs and unique special characteristics. We construct a sequence of systems of equations with a parameter K and argue that asymptotically when K grows the system of equations achieves a high level of saturation with lim_{K\to\infty} F/T = 1 which allows to reduce the "regularity degree" and makes that polynomial equations over finite fields may become efficiently solvable.

Note: more details on the software demonstrator

Available format(s)
Publication info
Preprint. MINOR revision.
cryptanalysisfinite fieldselliptic curvesECDL problemindex calculuserror correcting codeslinear codescodes on elliptic curvesSemaev polynomialsblock ciphersNP-hard problemsMQ problemphase transitionsXL algorithmGrobner basesblock ciphersElimLin
Contact author(s)
n courtois @ bettercrypto com
2016-09-11: last of 7 revisions
2016-07-18: received
See all versions
Short URL
Creative Commons Attribution


      author = {Nicolas T.  Courtois},
      title = {High Saturation Complete Graph Approach for EC Point Decomposition and ECDL Problem},
      howpublished = {Cryptology ePrint Archive, Paper 2016/704},
      year = {2016},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.