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

*Nicolas T. Courtois*

**Abstract: **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.

**Category / Keywords: **cryptanalysis, finite fields, elliptic curves, ECDL problem, index calculus, error correcting codes, linear codes, codes on elliptic curves, Semaev polynomials, block ciphers, NP-hard problems, MQ problem, phase transitions, XL algorithm, Grobner bases, block ciphers, ElimLin

**Date: **received 14 Jul 2016, last revised 11 Sep 2016

**Contact author: **n courtois at bettercrypto com

**Available format(s): **PDF | BibTeX Citation

**Note: **more details on the software demonstrator

**Version: **20160911:204337 (All versions of this report)

**Short URL: **ia.cr/2016/704

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