104
R. S. Stankovi´ c et al.
The Gibbs dyadic derivative of a bent function is the set of eigenvalues of
this operator with the signs corresponding to that in the function values, and it
consists of first 2 n non-negative integers. In this way, the Gibbs dyadic derivative
associates with each bent function an easily determined Gibbs permutation matrix.
These matrices are not the same as matrices converting a bent function into another
in either Boolean or Walsh spectral domain, since they relate these functions in
the Gibbs dyadic derivative domain. They however still can be used to establish
relationships among bent functions and to generate them from the subsets of bent
functions.
We show that Gibbs permutation matrices have a particular block structure which
can be expressed in terms of four precisely defined submatrices.
References
1. Cusic, T.W., St˘ anic˘ a, P.: Cryptographic Boolean Functions and Applications. Academic/Elsevier, Cambridge (2009)
2. Gibbs, J.E.: Walsh spectrometry, a form of spectral analysis well suited to binary digital
computation, p. 24. National Physical Laboratory, Teddington (1967)
3. Gibbs, J.E.: Sine waves and Walsh waves in physics. In: Proc. Sympos. Applic. Walsh
Functions, pp. 260–274, Washington, DC, 1970 March 31–April 3 (1970)
4. Gibbs, J.E., Gebbie, H.A.: Application of Walsh functions to transform spectroscopy. Nature
224(5223), 1012–1013 (1969).
5. Gibbs, J.E., Ireland, B.: Walsh Functions and differentiation. In: Schreiber, H., Sandy, G.F.
(eds.) Applications of Walsh Functions and Sequency Theory, pp. 147–176. IEEE, New York
(1974)
6. Karpovsky, M.G., Stankovi´ c, R.S., Astola, J.T.: Spectral Logic and Its Application in the
Design of Digital Devices. Wiley, Hoboken (2008)
7. Pichler, F.: Walsh Functions and Linear System Theory. Tech. Res. Rept, Dept. of Electrical
Engineering, University of Maryland, Report T-70-05 (1970)
8. Rothaus, O.S.: On ‘bent’ functions. J. Combin. Theory, Ser. A 20, 300–305 (1976)
9. Stankovi´ c, R.S., Moraga, C., Astola, J.T.: Fourier Analysis on Finite Non-Abelian Groups with
Applications in Signal Processing and System Design. Wiley/IEEE Press, Hoboken (2005)
10. Stankovi´ c, R.S., Astola, J.T., Moraga, C., Stankovi´ c, M., Gaji´ c, D.: Remarks on characterization of bent functions in terms of Gibbs dyadic derivatives, EUROCAST 2015. In:
Moreno-Diaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) Computer Aided Systems Theory
– EUROCAST 2015, 15th International Conference, Las Palmas de Gran Canaria, February
8–13, 2015. Revised Selected Papers LNCS, vol. 9520, pp. 632–639. Springer, Berlin (2015)
11. Stankovi´ c, R.S., Stankovi´ c, M., Astola, J.T., Moraga, C.: Gibbs characterization of binary
and ternary bent functions. In: Proceedings 46th International Symposium on Multiple-Valued
Logic, pp. 205–210, Sapporo, May 18–20 (2016)
12. Stankovi´ c, R.S., Stankovi´ c, M., Astola, J.T., Moraga, C.: Towards the Gibbs characterization
of a class of quaternary bent functions. In: Proceedings 47th International Symposium on
Multiple-Valued Logic, pp. 73–78, Novi Sad, May 22–24 (2017)
13. Stankovi´ c, R.S., Stankovi´ c, M., Astola, J.T., Moraga, C.: Quaternary generalized Boolean bent
functions obtained through permutation of binary Boolean bent functions. In: Proceedings 48th
International Symposium on Multiple-Valued Logic, Linz, May 16–18 (2018)
R. S. Stankovi´ c et al.
The Gibbs dyadic derivative of a bent function is the set of eigenvalues of
this operator with the signs corresponding to that in the function values, and it
consists of first 2 n non-negative integers. In this way, the Gibbs dyadic derivative
associates with each bent function an easily determined Gibbs permutation matrix.
These matrices are not the same as matrices converting a bent function into another
in either Boolean or Walsh spectral domain, since they relate these functions in
the Gibbs dyadic derivative domain. They however still can be used to establish
relationships among bent functions and to generate them from the subsets of bent
functions.
We show that Gibbs permutation matrices have a particular block structure which
can be expressed in terms of four precisely defined submatrices.
References
1. Cusic, T.W., St˘ anic˘ a, P.: Cryptographic Boolean Functions and Applications. Academic/Elsevier, Cambridge (2009)
2. Gibbs, J.E.: Walsh spectrometry, a form of spectral analysis well suited to binary digital
computation, p. 24. National Physical Laboratory, Teddington (1967)
3. Gibbs, J.E.: Sine waves and Walsh waves in physics. In: Proc. Sympos. Applic. Walsh
Functions, pp. 260–274, Washington, DC, 1970 March 31–April 3 (1970)
4. Gibbs, J.E., Gebbie, H.A.: Application of Walsh functions to transform spectroscopy. Nature
224(5223), 1012–1013 (1969).
5. Gibbs, J.E., Ireland, B.: Walsh Functions and differentiation. In: Schreiber, H., Sandy, G.F.
(eds.) Applications of Walsh Functions and Sequency Theory, pp. 147–176. IEEE, New York
(1974)
6. Karpovsky, M.G., Stankovi´ c, R.S., Astola, J.T.: Spectral Logic and Its Application in the
Design of Digital Devices. Wiley, Hoboken (2008)
7. Pichler, F.: Walsh Functions and Linear System Theory. Tech. Res. Rept, Dept. of Electrical
Engineering, University of Maryland, Report T-70-05 (1970)
8. Rothaus, O.S.: On ‘bent’ functions. J. Combin. Theory, Ser. A 20, 300–305 (1976)
9. Stankovi´ c, R.S., Moraga, C., Astola, J.T.: Fourier Analysis on Finite Non-Abelian Groups with
Applications in Signal Processing and System Design. Wiley/IEEE Press, Hoboken (2005)
10. Stankovi´ c, R.S., Astola, J.T., Moraga, C., Stankovi´ c, M., Gaji´ c, D.: Remarks on characterization of bent functions in terms of Gibbs dyadic derivatives, EUROCAST 2015. In:
Moreno-Diaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) Computer Aided Systems Theory
– EUROCAST 2015, 15th International Conference, Las Palmas de Gran Canaria, February
8–13, 2015. Revised Selected Papers LNCS, vol. 9520, pp. 632–639. Springer, Berlin (2015)
11. Stankovi´ c, R.S., Stankovi´ c, M., Astola, J.T., Moraga, C.: Gibbs characterization of binary
and ternary bent functions. In: Proceedings 46th International Symposium on Multiple-Valued
Logic, pp. 205–210, Sapporo, May 18–20 (2016)
12. Stankovi´ c, R.S., Stankovi´ c, M., Astola, J.T., Moraga, C.: Towards the Gibbs characterization
of a class of quaternary bent functions. In: Proceedings 47th International Symposium on
Multiple-Valued Logic, pp. 73–78, Novi Sad, May 22–24 (2017)
13. Stankovi´ c, R.S., Stankovi´ c, M., Astola, J.T., Moraga, C.: Quaternary generalized Boolean bent
functions obtained through permutation of binary Boolean bent functions. In: Proceedings 48th
International Symposium on Multiple-Valued Logic, Linz, May 16–18 (2018)