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Abstract

Models for cryptographic problems are often expressed as boolean polynomial systems, whose equivalent logical formulas can be treated using SAT solvers. Given the algebraic nature of the problem, the use of the logical XOR operator is common in SAT-based cryptanalysis. Recent works have focused on advanced techniques for handling parity (XOR) constraints, such as the Gaussian Elimination technique. First, we propose an original XOR-reasoning SAT solver, named WDSat, dedicated to a specific cryptographic problem. Secondly, we show that in some cases Gaussian Elimination on SAT instances does not work as well as Gaussian Elimination on algebraic systems. We demonstrate how this oversight is fixed in our solver, which is adapted to read instances in algebraic normal form (ANF). Finally, we propose a novel preprocessing technique based on the Minimal Vertex Cover Problem in graph theory. Our benchmarks use a model obtained from cryptographic instances for which a significant speedup is achieved using the findings in this paper. We further explain how our preprocessing technique can be used as an assessment of the security of a cryptographic system.

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BlueKrypt. Cryptographic key length recommendation. https://www.keylength.com, 2018. Accessed: 2019-11-15.
An extensible satsolver
  • Niklas Eén
  • Niklas Sörensson
Niklas Eén and Niklas Sörensson. An extensible satsolver. In Enrico Giunchiglia and Armando Tacchella, editors, Theory and Applications of Satisfiability Testing, pages 502-518, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg.
Improving the Complexity of Index Calculus Algorithms in Elliptic Curves over Binary Fields
  • Jean-Charles Faugère
  • Ludovic Perret
  • Christophe Petit
  • Guénaël Renault
Jean-Charles Faugère, Ludovic Perret, Christophe Petit, and Guénaël Renault. Improving the Complexity of Index Calculus Algorithms in Elliptic Curves over Binary Fields. In Advances in Cryptology -EUROCRYPT 2012 -31st Annual International Conference on the Theory and Applications of Cryptographic Techniques, Cambridge, UK, April 15-19, 2012. Proceedings, pages 27-44, 2012.
Summation polynomial algorithms for elliptic curves in characteristic two
  • D Steven
  • Shishay W Galbraith
  • Gebregiyorgis
Steven D. Galbraith and Shishay W. Gebregiyorgis. Summation polynomial algorithms for elliptic curves in characteristic two. In Willi Meier and Debdeep Mukhopadhyay, editors, Progress in Cryptology -IN-DOCRYPT 2014 -15th International Conference on Cryptology in India, volume 8885 of Lecture Notes in Computer Science, pages 409-427. Springer, 2014.
On Polynomial Systems Arising from a Weil Descent
  • Christophe Petit
  • Jean-Jacques Quisquater
Christophe Petit and Jean-Jacques Quisquater. On Polynomial Systems Arising from a Weil Descent. In Advances in Cryptology -ASIACRYPT 2012 -18th International Conference on the Theory and Application of Cryptology and Information Security, volume 7658 of Lecture Notes in Computer Science, pages 451-466. Springer, 2012.
Extending SAT Solvers to Cryptographic Problems
  • Mate Soos
  • Karsten Nohl
  • Claude Castelluccia
Mate Soos, Karsten Nohl, and Claude Castelluccia. Extending SAT Solvers to Cryptographic Problems. In SAT, volume 5584 of Lecture Notes in Computer Science, pages 244-257. Springer, 2009.