Abstract
We establish a new connection between invariant theory and the theory of bent functions. This enables us to construct Boolean functions with a prescribed symmetry group action. Besides the quadratic bent functions the only other previously known homogeneous bent functions are the six variable degree three functions constructed in [16]. We show that these bent functions arise as invariants under an action of the symmetric group on four letters and determine the stabilizer which turns out to be a matrix group of order 10752. We apply the machinery of invariant theory in order to construct homogeneous bent functions of degree three in 8, 10, and 12 variables. This approach gives a great computational advantage over the unstructured search problem and yields Boolean functions which have a concise description in terms of certain designs and graphs. We consider the question of linear equivalence of the constructed bent functions and study the properties of the associated elementary abelian difference sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Th. Beth, D. Jungnickel and H. Lenz, Design Theory, Vol. I, Cambridge University Press, 2nd ed. (1999).
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp., Vol. 24 (1997) pp. 235–266.
C. Carlet, Two new classes of bent functions, In: Advances in Cryptology-Eurocrypt '93, Lecture Notes in Computer Science, Vol. 765, Springer (1993) pp. 77–101.
C. Charnes, M. Rötteler and Th. Beth, On homogeneous bent functions, In Proceedings Applied Algebra, Algebraic Algorithms and Error Correcting Codes (AAECC-14), Lecture Notes in Computer Science, Vol. 2227 (2001) pp. 249–259.
C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, CRC Press (1996).
J. H. Conway, R. T. Curtis, S. P. Norton and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford (1985).
J. Dillon, A Survey of Bent Functions, NSA Technical Journal Special Issue (1972) pp. 191–215.
J. Dixon and B. Mortimer, Permutation Groups, Graduate Texts in Mathematics, Vol. 163, Springer (1996).
H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity, In (B. Preneel, ed.), Fast Software Encryption, Lecture Notes in Computer Science, Vol. 1008 (1995) pp. 61–74.
GAP—Groups, Algorithms, and Programming, V4.2. The GAP Group, Aachen, St Andrews (2000) (http://www-gap.dcs.st-and.ac.uk/007E;gap).
C. Godsil, Algebraic Combinatorics. Chapman and Hall, New York, London (1993).
X. Hou, Cubic Bent Functions, Discrete Mathematics Vol. 189 (1998) pp. 149–161.
I. G. MacDonald, Symmetric Functions and Hall Polynomials. Oxford University Press (1979).
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977).
Z. Nagy, A constructive estimation of the Ramsey number (in Hungarian), Mat. Lapok, Vol. 23 (1972) pp. 301–302.
C. Qu, J. Seberry and J. Pieprzyk, On the symmetric property of homogeneous Boolean functions, In J. Pieprzyk, R. Safavi-Naini and T. Sebury, eds., Proceedings of the Australian Conference on Information Security and Privacy (ACISP), Lecture Notes in Computer Science, Vol. 1587, Springer (1999) pp. 26–35.
E. M. Rains and N. J. A. Sloane, Handbook of Coding Theory, Vol. I, Chapt. Self-dual codes, Elsevier (1998) pp. 177–294.
O. S. Rothaus, On “bent” functions, Journal of Combinatorial Theory, Series A, Vol. 20 (1976) pp. 300–305.
P. Savický, On the bent Boolean functions that are symmetric, Europ. J. Combinatorics, Vol. 15 (1994) pp. 407–410.
L. Smith, Polynomial Invariants of Finite Groups, A. K. Peters (1995).
B. Sturmfels, Algorithms in Invariant Theory, Springer (1993).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Charnes, C., Rötteler, M. & Beth, T. Homogeneous Bent Functions, Invariants, and Designs. Designs, Codes and Cryptography 26, 139–154 (2002). https://doi.org/10.1023/A:1016509410000
Issue Date:
DOI: https://doi.org/10.1023/A:1016509410000