Skip to main content
SpringerLink
Log in

An iterative method for linear decomposition of index generating functions

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Various methods for reducing hardware implementation cost of incompletely specified index generating functions have been proposed lately. Considering the methods based on linear decomposition, for the first time in this work, we provide necessary and sufficient conditions which describe the linear decomposition of these functions in general. These conditions are derived using the concept of functional degeneracy, and we show that the problem of linear decomposition can be translated into the problem of constructing suitable coordinate Boolean functions (which represent the generating functions) such that the linear decomposition is possible. In this context, we propose several design methods of such Boolean functions and furthermore we employ one particular design method to derive a new iterative semi-deterministic algorithm for linear decomposition. In addition, we provide a general result which describes all incompletely specified index generating functions for which the linear decomposition is (not) possible. Consequently, our results indicate that the functional degeneracy is a promising approach in derivation of new deterministic-like algorithms for linear decomposition of incompletely specified index generating functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. From the point of cryptographic applications, this term was introduced by Mitchell [10].

  2. Algorithm 1 was implemented in Wolfram Mathematica 9, on the machine with the following characteristics: Lenovo ThinkPad E540, OS Windows 7 (64-bit), Intel Core i5-4200M-2.5GHz, RAM 4Gb.

References

  1. Astola, J., Astola, P., Stanković, R., Tabus, I.: An algebraic approach to reducing the number of variables of incompletely defined discrete functions. In: 46th International Symposium on Multiple-Valued Logic, pp. 107–112 (2016)

  2. Astola, J., Astola, P., Stanković, R., Tabus, I.: Index generation functions based on linear and polynomial transformations. In: 46th International Symposium on Multiple-Valued Logic, pp. 102–106 (2016)

  3. Bhattacharyya, A., Indyk, P., Woodruff, D.P., Xie, N.: The complexity of linear dependence problems in vector spaces. Innovations Comput. Sci., 496–508. ISBN 978-7-302-24517-9 (2011)

  4. Crama, Y., Hammer, P.L.: Boolean Functions: Theory, Algorithms, and Applications. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  5. Een, N., Sorensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) Theory and Applications of Satisfiability Testing. Lecture Notes in Computer Science, SAT 2003, vol. 2919, pp 502–518. Springer, Berlin (2003)

  6. Karpovsky, M.G., Stankovic, R.S., Astola, J.T.: Spectral Logic and Its Applications for the Design of Digital Devices. Wiley, Hoboken (2007)

    Google Scholar 

  7. Kolomeec, N., Pavlov, A.: Bent functions on the minimal distance. IEEE Region 8, SIBIRCON, Irkutsk Listvyanka, Russia (2010)

  8. Lechner, R.J.: Harmonic analysis of switching functions. Recent Developments in Switching Theory, New York, Edited by: Amar Mukhopadhyay, pp. 121–228 (1971)

  9. Luba, T., Borowik, G., Jankowski, C.: Gate-based decomposition of index generation functions. Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments, vol. 10031 (2016)

  10. Mitchell, C.: Enumerating Boolean functions of cryptographic significance. J. Cryptol. 2(3), 155–170 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nagayama, S., Sasao, T., Butler, J.T.: An efficient heuristic for linear decomposition of index generation functions. In: 46nd International Symposium on Multiple-Valued Logic, pp. 96–101. Japan (2016)

  12. Nagayama, S., Sasao, T., Butler, J.T.: A balanced decision tree based heuristic for linear decomposition of index generation functions. IEICE Trans. Inf. Syst. E100.D (8), 1583–1591 (2017)

    Article  Google Scholar 

  13. Nečiporuk, E.I.: Network synthesis by using linear transformation of variables. Dokladi Akademii Nauk SSSR 123(4), 610–612 (1958)

    Google Scholar 

  14. Pagiamtzis, K., Sheikholeslami, A.: Content-addressable memory (CAM) circuits and architectures: A tutorial and survey. IEEE J. Solid State Circuits 41(3), 712–727 (2006)

    Article  Google Scholar 

  15. Sasao, T.: Index generation functions: tutorial. J. Multiple-Valued Logic Soft Comput. 23(3–4), 235–263 (2014)

    Google Scholar 

  16. Sasao, T.: Multiple-valued input index generation functions: optimization by linear transformation. In: 42nd International Symposium on Multiple-Valued Logic, pp. 185–190. Canada (2012)

  17. Sasao, T.: A reduction method for the number of variables to represent index generation functions: s-Min method. In: 45nd International Symposium on Multiple-Valued Logic, pp. 164–169. Canada (2015)

  18. Sasao, T.: Index generation functions: theory and applications. In: International Symposium on Communications and Information Technologies, pp. 585–590. Japan (2010)

  19. Sasao, T.: Linear transformations for variable reduction. ReedMuller Workshop (2011)

  20. Sasao, T.: On the numbers of variables to represent multi-valued incompletely specified functions. In: 13th Euromicro Conference on Digital System Design: Architectures, Methods and Tools. France (2010)

  21. Sasao, T.: On the numbers of variables to represent sparse logic functions. In: IEEE/ACM International Conference on Computer-Aided Design, pp. 45–51. San Jose, USA (2008)

  22. Sasao, T.: Index generation functions: recent developments. In: 41st IEEE International Symposium on Multiple-Valued Logic, pp. 235–263. Finland (2011)

  23. Sasao, T.: Memory-Based Logic Synthesis. Springer-Verlag, New York (2011)

    Book  Google Scholar 

  24. Sasao, T.: Linear decomposition of index generation functions. In: 17th Asia and South Pacific Design Automation Conference, pp. 781–788 (2012)

  25. Sasao, T.: Index generation functions: Minimization methods. In: 47th International Symposium on Multiple-Valued Logic, pp. 197–206 (2017)

  26. Sasao, T.: An application of autocorrelation functions to find linear decompositions for incompletely specified index generation functions. In: 43rd International Symposium on Multiple-Valued Logic, pp. 96–102 (2013)

  27. Sasao, T.: A linear decomposition of index generation functions: optimization using autocorrelation functions. J. Multiple-Valued Logic Soft Comput. 28(1), 105–127 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Sasao, T., Fumishi, I., Iguch, Y.: A method to minimize variables for incompletely specified index generation functions using a SAT solver. In: International Workshop on Logic and Synthesis, Mountain View, pp. 161–167 (2015)

  29. Sasao, T., Nakamura, T., Matsuura, M.: Representation of incompletely specified index generation functions using minimal number of compound variables. In: 12th Euromicro Conference on Digital System Design / Architectures, Methods and Tools, pp 765–772 (2009)

  30. Sasao, T., Matsuura, M., Nakahara, H.: A realization of Index generation functions using modules of uniform sizes. In: International Workshop on Logic and Synthesis, pp. 201–208. California (2010)

  31. Sasao, T., Urano, Y., Iguchi, Y.: A method to find linear decompositions for incompletely specified index generation functions using difference matrix. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E97.A(12), 2427–2433 (2014)

    Article  Google Scholar 

  32. Sasao, T., Urano, Y., Iguchi, Y.: A lower bound on the number of variables to represent incompletely specified index generation functions. In: 44th International Symposium on Multiple-Valued Logic, pp. 7–12. Germany (2014)

  33. Abboud, A., Lewi, K., Williams, R.: Losing weight by gaining edges. In: 22th Annual European Symposium on Algorithms, Lecture Notes in Computer Science, vol. 8737, pp. 1–12. Poland (2014)

  34. Simovici, D.A., Zimand, M., Pletea, D.: Several remarks on index generation functions. In: 42nd IEEE International Symposium on Multiple-Valued Logic, pp. 179–184. Canada (2012)

  35. Stanković, M., Stanković, R.: Variable reduction of index generating functions in Walsh-Hadamard domain. In: Proceedings Reed-Muller Workshop, pp. 80–85. Japan (2013)

  36. Troy Nagle, H., Carroll, B.D., David Irwin, J.: : An Introduction to Computer Logic (Prentice-Hall computer applications in electrical engineering series), 1st edn. Prentice-Hall, Englewood Cliffs (1975)

    Google Scholar 

  37. Varma, D., Trachtenberg, E.: Design automation tools for efficient implementation of logic functions by decomposition. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 8(8), 901–916 (1989)

    Article  Google Scholar 

  38. Wu, C.-K., Feng, D.: Boolean functions and their applications in cryptography. Advances in Computer Science and Technology (2016)

Download references

Acknowledgments

Samir Hodžić is supported in part by the Slovenian Research Agency (research program P3-0384 and Young Researchers Grant). Enes Pasalic is partly supported by the Slovenian Research Agency (research program P3-0384 and research project J1-9108). Also, the first two authors gratefully acknowledge the European Commission for funding the InnoRenew CoE project (Grant Agreement no. 739574) under the Horizon2020 Widespread-Teaming program and the Republic of Slovenia (Investment funding of the Republic of Slovenia and the European Union of the European regional Development Fund). Anupam Chattopadhyay will like to acknowledge Tsutomu Sasao for helpful discussions related to the complexity of index generation functions.

Author information

Authors and Affiliations

  1. FAMNIT, University of Primorska, Koper, Slovenia

    S. Hodžić

  2. FAMNIT & IAM, University of Primorska, Koper, Slovenia

    E. Pasalic

  3. School of Computer Science and Engineering, College of Engineering, NTU, Singapore, Singapore

    A. Chattopadhyay

Authors
  1. S. Hodžić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Pasalic
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Chattopadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Hodžić.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

The performance of Algorithm 1 shown in Tables 6, 7 and 8 uses Proposition 5.1 for finding a basis of \(\sigma ^{\perp }_{i}\) in Step 1.

Example A.1

Let us consider \(\sigma _{i}=(\alpha _{1},\ldots ,\alpha _{4})=(1,0,1,1)\in \mathbb {F}^{4}_{2}\) (n − (i − 1) = 4). We have that p = 1 is the minimal index in {1,…, 4} such that αp = 1. Note that one may consider other indices p as well (not necessary the minimal one). Using the vectors \(e_{1},\ldots ,e_{4}\in \mathbb {F}^{4}_{2}\), we have that \(\widehat {e}_{j}\) (for j ∈{2, 3, 4} = {1, 2, 3, 4}∖{p}), defined as in Proposition 5.1, are given as

$$\left( \begin{array}{c} \widehat{e}_{2} \\ \widehat{e}_{3} \\ \widehat{e}_{4} \end{array} \right)=\left( \begin{array}{c} e_{2} \\ e_{1}\oplus e_{3} \\ e_{1}\oplus e_{4} \end{array} \right)=\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{array} \right). $$

Thus the set \(\{\widehat {e}_{2},\widehat {e}_{3},\widehat {e}_{4}\}\) is basis of σ⊥.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hodžić, S., Pasalic, E. & Chattopadhyay, A. An iterative method for linear decomposition of index generating functions. Cryptogr. Commun. 11, 1079–1102 (2019). https://doi.org/10.1007/s12095-019-0351-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-019-0351-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation