Synthesizing Shortest Linear Straight-Line Programs over GF(2) Using SAT

  • Carsten Fuhs
  • Peter Schneider-Kamp
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6175)


Non-trivial linear straight-line programs over the Galois field of two elements occur frequently in applications such as encryption or high-performance computing. Finding the shortest linear straight-line program for a given set of linear forms is known to be MaxSNP-complete, i.e., there is no ε-approximation for the problem unless P = NP.

This paper presents a non-approximative approach for finding the shortest linear straight-line program. In other words, we show how to search for a circuit of XOR gates with the minimal number of such gates. The approach is based on a reduction of the associated decision problem (“Is there a program of length k?”) to satisfiability of propositional logic. Using modern SAT solvers, optimal solutions to interesting problem instances can be obtained.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asín, R., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E.: Cardinality networks and their applications. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 167–180. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Boyar, J., Matthews, P., Peralta, R.: On the shortest linear straight-line program for computing linear forms. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 168–179. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Boyar, J., Peralta, R.: A new technique for combinational circuit optimization and a new circuit for the S-Box for AES. In: Patent Application Number 61089998 filed with the U.S. Patent and Trademark Office (2009)Google Scholar
  4. 4.
    Boyar, J., Peralta, R.: A new combinational logic minimization technique with applications to cryptology. In: Festa, P. (ed.) SEA 2010. LNCS, vol. 6049, pp. 178–189. Springer, Heidelberg (2010)Google Scholar
  5. 5.
    Codish, M., Lagoon, V., Stuckey, P.: Solving partial order constraints for LPO termination. Journal on Satisfiability, Boolean Modeling and Computation (JSAT) 5, 193–215 (2008)MathSciNetMATHGoogle Scholar
  6. 6.
    Eén, N., Sörensson, N.: Translating pseudo-boolean constraints into SAT. Journal on Satisfiability, Boolean Modelling and Computation (JSAT) 2(1-4), 1–26 (2006)MATHGoogle Scholar
  7. 7.
    Fuhs, C., Giesl, J., Middeldorp, A., Thiemann, R., Schneider-Kamp, P., Zankl, H.: SAT solving for termination analysis with polynomial interpretations. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 340–354. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: Automatic termination proofs in the dependency pair framework. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 281–286. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Grinchtein, O., Leucker, M., Piterman, N.: Inferring network invariants automatically. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 483–497. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Hong, H., Jakuš, D.: Testing positiveness of polynomials. Journal of Automated Reasoning (JAR) 21(1), 23–38 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kojevnikov, A., Kulikov, A.S., Yaroslavtsev, G.: Finding efficient circuits using SAT-solvers. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 32–44. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Le Berre, D., Parrain, A.: SAT4J,
  13. 13.
    Federal Information Processing Standard 197. The advanced encryption standard. Technical report, National Institute of Standards and Technology (2001)Google Scholar
  14. 14.
    Tseitin, G.: On the complexity of derivation in propositional calculus. Studies in Constructive Mathematics and Mathematical Logic, pp. 115–125 (1968); Reprinted in Automation of Reasoning 2, 466–483 (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Carsten Fuhs
    • 1
  • Peter Schneider-Kamp
    • 2
  1. 1.LuFG Informatik 2RWTH Aachen UniversityGermany
  2. 2.IMADAUniversity of Southern DenmarkDenmark

Personalised recommendations