Designs, Codes and Cryptography

, Volume 75, Issue 1, pp 175–185 | Cite as

Speeding up deciphering by hypergraph ordering

  • Peter Horak
  • Zsolt Tuza


The “ Gluing Algorithm” of Semaev (Des. Codes Cryptogr. 49:47–60, 2008)—that finds all solutions of a sparse system of linear equations over the Galois field \(GF(q)\)—has average running time \(O(mq^{\max \left| \cup _{1}^{k}X_{j}\right| -k}),\) where \(m\) is the total number of equations, and \(\cup _{1}^{k}X_{j}\) is the set of all unknowns actively occurring in the first \(k\) equations. In order to make the implementation of the algorithm faster, our goal here is to minimize the exponent of \(q\) in the case where every equation contains at most three unknowns. The main result states that if the total number \(\left| \cup _{1}^{m}X_{j}\right| \) of unknowns is equal to \(m\), then the best achievable exponent is between \(c_{1}m\) and \(c_{2}m\) for some positive constants \(c_{1}\) and \(c_{2}.\)


Sparse systems of Boolean equations Hypergraph ordering 

Mathematics Subject Classification

94A60 (05C65, 68Q25) 



The authors are indebted to Noga Alon for discussions on expanders and on probabilistic methods, which lead to an improvement of the lower bound. This research of the first author was initiated thanks to the University of Washington—University of Bergen exchange program. Supported by a SPIRE Grant from University of Bergen, Norway, and by a research Grant from IAS, University of Washington. Supported in part by the Hungarian Scientific Research Fund, OTKA grant T-81493, and by the Hungarian State and the European Union under the Grant TAMOP-4.2.2.A-11/1/ KONV-2012-0072.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of Washington, TacomaTacomaUSA
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  3. 3.University of PannoniaVeszprémHungary

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