Quadratic Time Algorithm for Inversion of Binary Permutation Polynomials

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10931)


In this paper, we propose a new version of the Lagrange interpolation applied to binary permutation polynomials and, more generally, permutation polynomials over prime power modular rings. We discuss its application to obfuscation and reverse engineering.


Permutation polynomial Lagrange interpolation Obfuscation 


  1. 1.
    Zhou, Y., Main, A., Gu, Y.X., Johnson, H.: Information hiding in software with mixed boolean-arithmetic transforms. In: Kim, S., Yung, M., Lee, H.-W. (eds.) WISA 2007. LNCS, vol. 4867, pp. 61–75. Springer, Heidelberg (2007). Scholar
  2. 2.
    Barthelemy, L., Eyrolles, N., Renault, G., Roblin, R.: Binary permutation polynomial inversion and application to obfuscation techniques. In: Proceedings of the 2nd International Workshop on Software Protection, Vienna, Austria. ACM, October 2016Google Scholar
  3. 3.
    Biondi, F., Josse, S., Legay, A., Sirvent, T.: Effectiveness of synthesis in concolic deobfuscation. Comput. Secur. 70, 500–515 (2017)CrossRefGoogle Scholar
  4. 4.
    Mullen, G., Stevens, H.: Polynomial functions (mod \(m\)). Acta Math. Hungar. 44(3–4), 237–241 (1984)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Rivest, R.L.: Permutation polynomials modulo \(2^w\). Finite Fields Appl. 7(2), 287–292 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 4th edn. Clarendon Press, Oxford (1960)zbMATHGoogle Scholar
  7. 7.
    Oruç, H., Phillips, G.M.: Explicit factorization of the Vandermonde matrix. Linear Algebra Appl. 315(1–3), 113–123 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Markovski, S., Šunić, Z., Gligoroski, D.: Polynomial functions on the units of \(Z_{2^n}\). Quasigr. Relat. Syst. 18(1), 59–82 (2010)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.QuarkslabParisFrance
  2. 2.Graduate CenterCUNYNew YorkUSA
  3. 3.Agence Nationale de la Sécurité des Systèmes d’InformationParis 07 SPFrance
  4. 4.Sorbonne Université, UPMC, LIP6Paris Cedex 5France
  5. 5.Department of MathematicsHofstra UniversityHempsteadUSA

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