SeMA Journal

, Volume 72, Issue 1, pp 13–36

# On the construction of new bent functions from the max-weight and min-weight functions of old bent functions

• Joan-Josep Climent
• Francisco J. García
• Verónica Requena
Article

## Abstract

Given a bent function $$f(\varvec{x})$$ of n variables, its max-weight and min-weight functions are introduced as the Boolean functions $${f}^{+}(\varvec{x})$$ and $${f}^{-}(\varvec{x})$$ whose supports are the sets $$\{\varvec{a} \in {\mathbb {F}}_{2}^{n} \ | \ w(f \oplus l_{\varvec{a}}) = 2^{n-1}+2^{\frac{n}{2}-1}\}$$ and $$\{\varvec{a} \in {\mathbb {F}}_{2}^{n} \ | \ w(f \oplus l_{\varvec{a}}) = 2^{n-1}-2^{\frac{n}{2}-1}\}$$ respectively, where $$w(f \oplus l_{\varvec{a}})$$ denotes the Hamming weight of the Boolean function $$f(\varvec{x}) \oplus l_{\varvec{a}}(\varvec{x})$$ and $$l_{\varvec{a}}(\varvec{x})$$ is the linear function defined by $$\varvec{a} \in {\mathbb {F}}_{2}^{n}$$. $${f}^{+}(\varvec{x})$$ and $${f}^{-}(\varvec{x})$$ are proved to be bent functions. Furthermore, combining the 4 minterms of 2 variables with the max-weight or min-weight functions of a 4-tuple $$(f_{0}(\varvec{x}), f_{1}(\varvec{x}), f_{2}(\varvec{x}), f_{3}(\varvec{x}))$$ of bent functions of n variables such that $$f_{0}(\varvec{x}) \oplus f_{1}(\varvec{x}) \oplus f_{2}(\varvec{x}) \oplus f_{3}(\varvec{x}) = 1$$, a bent function of $$n+2$$ variables is obtained. A family of 4-tuples of bent functions satisfying the above condition is introduced, and finally, the number of bent functions we can construct using the method introduced in this paper are obtained. Also, our construction is compared with other constructions of bent functions.

## Keywords

Boolean function Linear function Bent function  Support Minterm Max-weight function

06E30 94A60

## References

1. 1.
Borissov, Y., Braeken, A., Nikova, S., Preneel, B.: On the covering radii of binary Reed-Muller codes in the set of resilient Boolean functions. IEEE Trans. Inform. Theory 51(3), 1182–1189 (2005)
2. 2.
Braeken, A., Borissov, Y., Nikova, S., Preneel, B.: Classification of Boolean functions of $$6$$ variables or less with respect to some cryptographic properties. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) Automata. Languages and Programming, vol. 3580 of Lecture Notes in Computer Science, pp. 324–334. Springer-Verlag, Berlin (2005)Google Scholar
3. 3.
Braeken, A., Nikov, V., Nikova, S., Prenee, B.: On Boolean functions with generalized cryptographic properties. In: Canteaut, A., Viswanathan, K. (eds.) Progress in Cryptology INDOCRYPT 2004. Lecture Notes in Computer Science, vol. 3348, pp. 120–135. Springer-Verlag, Berlin (2004)
4. 4.
Canteaut, A., Daum, M., Dobbertin, H., Leander, G.: Finding nonnormal bent functions. Discrete Appl. Math. 154, 202–218 (2006)
5. 5.
Carlet, C., Tarannikov, Y.: Covering sequences of Boolean functions and their cryptographic significance. Des. Codes Crypt. 25, 263–279 (2002)
6. 6.
Carlet, C.: Two new classes of bent functions. In: Helleseth, T. (ed.) Advances in Cryptology EUROCRYPT’93. Lecture Notes in Computer Science, vol. 765, pp. 77–101. Springer-Verlag, Berlin (1994)Google Scholar
7. 7.
Carlet, C.: On the secondary constructions of resilient and bent functions. Prog. Comput. Sci. Appl. Logic 23, 3–28 (2004)
8. 8.
Carlet, C.: On bent and highly nonlinear balanced/resilient functions and their algebraic immunities. In: Fossorier, M., Imai, H., Lin, S., Poli, A. (eds.) Applied Algebra. Algebraic Algorithms and Error-Correcting Codes (AAECC-16), vol. 3857 of Lecture Notes in Computer Science, pp. 1–28. Springer-Verlag, Berlin (2006)Google Scholar
9. 9.
Carlet, C., Yucas, J.L.: Piecewise constructions of bent and almost optimal Boolean functions. Des. Codes Crypt. 37, 449–464 (2005)
10. 10.
Chang, D.K.: Binary bent sequences of order $$64$$. Utilitas Math. 52, 141–151 (1997)
11. 11.
Climent, J.J., García, F.J., Requena, V.: On the construction of bent functions of $$n+2$$ variables from bent functions of $$n$$ variables. Adv. Math. Commun. 2(4), 421–431 (2008)
12. 12.
Daum, M., Dobbertin, H., Leander, G.: An algorithm for checking normality of Boolean functions. In: Proceedings of the 2003 International Workshop on Coding and Cryptography (WCC 2003), pp. 133–142 (2003)Google Scholar
13. 13.
John, F.D.: Elementary Hadamard Difference Sets. Ph.D thesis, University of Maryland (1974)Google Scholar
14. 14.
Joanne, F., Ed, D., William, M.: Evolutionary generation of bent functions for cryptography. In: Proceedings of the 2003 Congress on Evolutionary Computation, vol. 2, pp. 1655–1661. IEEE (2003)Google Scholar
15. 15.
Hou, X.D., Langevin, P.: Results on bent functions. J. Comb. Theory Ser. A 80, 232–246 (1997)
16. 16.
Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized bent functions and their properties. J. Comb. Theory Ser. A 40, 90–107 (1985)
17. 17.
Kurosawa, K., Iwata, T., Yoshiwara, T.: New covering radius of Reed-Muller codes for $$t$$-resilient functions. IEEE Trans. Inform. Theory 50(3), 468–475 (2004)
18. 18.
Langevin, P., Leander, G.: Counting all bent functions in dimension eight $$99270589265934370305785861242880$$. Des. Codes Crypt. 59, 193–201 (2011)
19. 19.
Losev, V.V.: Decoding of sequences of bent functions by means of a fast Hadamard transform. Sov. J. Commun. Technol. Electron. 32(10), 155–157 (1987)
20. 20.
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes, 6th edn. North-Holland, Amsterdam (1988)Google Scholar
21. 21.
McFarland, R.L.: A family of difference sets in non-cyclic groups. J. Comb. Theory Ser. A 15, 1–10 (1973)
22. 22.
Meier, W., Staffelbach, O.: Nonlinearity criteria for cryptographic functions. In: Quisquater, J.J., Vandewalle, J. (eds.) Advances in Cryptology EUROCRYPT’89. Lecture Notes in Computer Science, vol. 434, pp. 549–562. Springer-Verlag, Berlin (1990)Google Scholar
23. 23.
Olejár, D., Stanek, M.: On cryptographic properties of random Boolean functions. J. Univ. Comput. Sci. 4(8), 705–717 (1998)
24. 24.
Pasalic, E., Johansson, T.: Further results on the relation between nonlinearity and resiliency for Boolean functions. In: Walker, M. (ed.) Crytography and Coding. Lecture Notes in Computer Science, vol. 1746, pp. 35–44. Springer-Verlag, Berlin (1999)Google Scholar
25. 25.
Pieprzyk, J., Finkelstein, G.: Towards effective nonlinear cryptosystem design. IEEE Proc. 135(6), 325–335 (1988)Google Scholar
26. 26.
Preneel, B.: Analysis and Design of Cryptographic Hash Functions. PhD thesis, Katholieke University Leuven (1993)Google Scholar
27. 27.
Rothaus, O.S.: On “bent” functions. J. Comb. Theory Ser. A 20, 300–305 (1976)
28. 28.
Sarkar, P., Maitra, S.: Construction of nonlinear Boolean functions with important cryptographic properties. In: Preneel, B. (ed.) Advances in Cryptology EUROCRYPT 2000. Lecture Notes in Computer Science, vol. 1807, pp. 485–506. Springer-Verlag, Berlin (2000)
29. 29.
Seberry, J., Zhang, X.M.: Constructions of bent functions from two known bent functions. Australas. J. Comb. 9, 21–35 (1994)
30. 30.
Seberry, J., Zhang, X.M., Zheng, Y.: Nonlinearity and propagation characteristics of balanced Boolean functions. Inform. Comput. 119, 1–13 (1995)
31. 31.
Scott, A.V., van Paul, C.O.: An Introduction to Error Correcting Codes with Applications. Kluwer Academic Publishers, Boston (1989)
32. 32.
Yarlagadda, R., Hershey, J.E.: Analysis and synthesis of bent sequences. IEE Proc. 136(2), 112–123 (1989)Google Scholar
33. 33.
Yu, N.Y., Gong, G.: Constructions of quadratic bent functions in polynomial forms. IEEE Trans. Inform. Theory 52(7), 3291–3299 (2006)

## Authors and Affiliations

• Joan-Josep Climent
• 1
Email author
• Francisco J. García
• 2
• Verónica Requena
• 3
1. 1.Departament de MatemàtiquesUniversitat d’AlacantAlacantSpain
2. 2.Departament de Mètodes Quantitatius i Teoria EconòmicaUniversitat d’AlacantAlacantSpain