Twin bent functions, strongly regular Cayley graphs, and Hurwitz-Radon theory

Abstract

The real monomial representations of Clifford algebras give rise to two sequences of bent functions. For each of these sequences, the corresponding Cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. Even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. The proof of this non-isomorphism is a simple consequence of a theorem of Radon.

Keywords

• Bent functions
• Strongly regular graphs
• Clifford algebras
• Hurwitz-Radon

References

1. [1] A. Bernasconi, B. Codenotti, Spectral analysis of Boolean functions as a graph eigenvalue problem, IEEE Trans. Comput. 48(3) (1999) 345–351.
2. [2] A. Canteaut, C. Carlet, P. Charpin, C. Fontaine, On cryptographic properties of the cosets of $R(1,m)$, IEEE Trans. Inform. Theory 47(4) (2001) 1494–1513.
3. [3] A. Canteaut, P. Charpin, Decomposing bent functions, IEEE Trans. Inform. Theory 49(8) (2003) 2004–2019.
4. [4] R. Craigen, Signed groups, sequences, and the asymptotic existence of Hadamard matrices, J. Combin. Theory Ser. A 71(2) (1995) 241–254.
5. [5] J. F. Dillon, Elementary Hadamard Difference Sets, PhD thesis, University of Maryland College Park, Ann Arbor, USA, 1974.
6. [6] A. V. Geramita, N. J. Pullman, A theorem of Hurwitz and Radon and orthogonal projective modules, Proc. Amer. Math. Soc. 42(1) (1974) 51–56.
7. [7] A. Hurwitz, Über die Komposition der quadratischen Formen, Math. Ann. 88(1–2) (1922) 1–25.
8. [8] P. Leopardi, Classifying bent functions by their Cayley graphs, arXiv:1705.04507 [math.CO].

9. [9] P. Leopardi, A generalized FFT for Clifford algebras, Bull. Belg. Math. Soc. Simon Stevin 11(5) (2005) 663–688.
10. [10] P. Leopardi, Constructions for Hadamard matrices using Clifford algebras, and their relation to amicability / anti–amicability graphs, Australas. J. Combin. 58(2) (2014) 214–248.
11. [11] P. Leopardi, Twin Bent Functions and Clifford Algebras. In: Colbourn C. (eds) Algebraic Design Theory and HadamardMatrices. Springer Proceedings in Mathematics & Statistics, vol 133. Springer, Cham.
12. [12] P. Leopardi, Boolean–Cayley–graphs, (2016). http://tinyurl.com/Boolean-Cayley-graphs Sage- MathCloud public folder. Last accessed 16 April 2017.
13. [13] J. Radon, Lineare Scharen orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1(1) (1922) 1–14.
14. [14] SageMath, Inc., SageMathCloud Online Computational Mathematics, (2016).
15. [15] N. Tokareva, On the number of bent functions from iterative constructions: lower bounds and hypotheses, Adv. Math. Commun. 5(4) (2011) 609–621.
16. [16] J. Williamson, Hadamard’s determinant theorem and the sum of four squares, Duke Math. J. 11(1) (1944) 65–81.
17. [17] P. Y. Yiu, Strongly regular graphs and Hurwitz–Radon numbers, Graphs and Combin. 6(1) (1990) 61–69.