A New Combinational Logic Minimization Technique with Applications to Cryptology

  • Joan Boyar
  • René Peralta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6049)


A new technique for combinational logic optimization is described. The technique is a two-step process. In the first step, the non-linearity of a circuit – as measured by the number of non-linear gates it contains – is reduced. The second step reduces the number of gates in the linear components of the already reduced circuit. The technique can be applied to arbitrary combinational logic problems, and often yields improvements even after optimization by standard methods has been performed. In this paper we show the results of our technique when applied to the S-box of the Advanced Encryption Standard (AES [6]). This is an experimental proof of concept, as opposed to a full-fledged circuit optimization effort. Nevertheless the result is, as far as we know, the circuit with the smallest gate count yet constructed for this function. We have also used the technique to improve the performance (in software) of several candidates to the Cryptographic Hash Algorithm Competition. Finally, we have experimentally verified that the second step of our technique yields significant improvements over conventional methods when applied to randomly chosen linear transformations.


Circuit complexity multiplicative complexity linear component minimization AES S-box 


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  1. 1.
    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
  2. 2.
    Boyar, J., Peralta, R.: Tight bounds for the multiplicative complexity of symmetric functions. Theoretical Computer Science 396(1-3), 223–246 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boyar, J., Peralta, R.: New logic minimization techniques with applications to cryptology. Cryptology ePrint Archive, Report 2009/191 (2009),
  4. 4.
    Boyar, J., Peralta, R.: Patent application number 61089998 filed with the U.S. Patent and Trademark Office. In: A new technique for combinational circuit optimization and a new circuit for the S-Box for AES (2009)Google Scholar
  5. 5.
    Canright, D.: A very compact Rijndael S-box. Technical Report NPS-MA-05-001, Naval Postgraduate School (2005)Google Scholar
  6. 6.
    FIPS. Advanced Encryption Standard (AES). National Institute of Standards and Technology (2001)Google Scholar
  7. 7.
    Itoh, T., Tsujii, S.: A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases. Inf. Comput. 78(3), 171–177 (1988)Google Scholar
  8. 8.
    Käsper, E., Schwabe, P.: Faster and timing-attack resistant AES-GCM. In: Clavier, C., Gaj, K. (eds.) CHES 2009. LNCS, vol. 5747, pp. 1–17. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Mastrovito, E.: VLSI architectures for computation in Galois fields. PhD thesis, Linköping University, Dept. Electr. Eng., Sweden (1991)Google Scholar
  10. 10.
    Paar, C.: Some remarks on efficient inversion in finite fields. In: 1995 IEEE International Symposium on Information Theory, Whistler, B.C. Canada, p. 58 (1995)Google Scholar
  11. 11.
    Paar, C.: Optimized arithmetic for Reed-Solomon encoders. In: IEEE International Symposium on Information Theory, p. 250 (1997)Google Scholar
  12. 12.
    Satoh, A., Morioka, S., Takano, K., Munetoh, S.: A compact rijndael hardware architecture with S-box optimization. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 239–254. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Joan Boyar
    • 1
  • René Peralta
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern Denmark 
  2. 2.Information Technology LaboratoryNIST 

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