A004491 Number of bent functions of 2n variables.

2, 8, 896, 5425430528, 99270589265934370305785861242880

Offset

0,1

Comments

The old entry with this sequence number was a duplicate of A004483.

References

Carlet, C. & Mesnager, S., Four decades of research on bent functions, Designs, Codes and Cryptography, January 2016, Volume 78, Issue 1, pp. 5-50.

J. F. Dillon, Elementary Hadamard Difference Sets, Ph. D. Thesis, Univ. Maryland, 1974.

J. F. Dillon, Elementary Hadamard Difference Sets, in Proc. 6th South-Eastern Conf. Combin. Graph Theory Computing (Utilitas Math., Winnipeg, 1975), pp. 237-249.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977. [Section 5 of Chap. 14 deals with bent functions. For a(2) see page 418.]

B. Preneel, Analysis and design of cryptographic hash functions, Ph. D. thesis, Katholieke Universiteit Leuven, Belgium, 1993. [Confirms a(3).]

Links

Table of n, a(n) for n=0..4.

Elwyn R. Berlekamp and Lloyd R.Welch, Weight distributions of the cosets of the (32,6) Reed-Muller code, IEEE Trans. Information Theory IT-18 (1972), 203-207. [Not strictly relevant because it deals with the case of five variables. Included for completeness.]

L. Budaghyan and P. Stanica, What is a cryptographic Boolean function?, Notices Amer. Math. Soc., 66 (Jan 2019), 60-63.

Philippe Langevin, Classification of Boolean Quartics Forms in Eight Variables [Broken link]

James A. Maiorana, A classification of the cosets of the Reed-Muller code R(1,6), Math. Comp. 57 (1991), no. 195, 403-414. [Gives a(3).]

Meng Qing-shu, Yang Zhang and Cui Jing-song, A novel algorithm enumerating bent functions, IACR, Report 2004/274, 2004. [Also confirms a(3).]

O. S. Rothaus, On "bent" functions, J. Combinat. Theory, 20A (1976), 300-305.

N. J. A. Sloane and R. J. Dick, On the Enumeration of Cosets of First-Order Reed-Muller Codes, Proc. IEEE International Conf. Commun., Montreal 1971, IEEE Press, NY, 7 (1971), pp. 36-2 to 36-6.

Crossrefs

See A099090 for a normalized version.

Sequence in context: A120802 A120838 A282890 * A132573 A322142 A061591

Adjacent sequences: A004488 A004489 A004490 * A004492 A004493 A004494

Keyword

nonn,hard,nice,more

Author

N. J. A. Sloane, Sep 23 2008, based on emails from Philippe Langevin, Gregor Leander and Pante Stanica.

Extensions

a(4) found in 2008 by Philippe Langevin and Gregor Leander.

Status

approved