Piecewise constructions of bent and almost optimal boolean functions

Designs, Codes, and Cryptography

Volumn 37, Issue 3, 2005, Pages 449-464

  Carlet, Claude ^a,b   Yucas, Joseph L ^c  

^a INRIA   (France)

^b UNIVERSITÉ PARIS 8   (France)

^c Southern Illinois University at Carbondale   (United States)

Author keywords

Bent functions; Codes; Finite fields; Quadratic forms

Indexed keywords

BENT FUNCTIONS; FINITE FIELDS; QUADRATIC FORMS;

BOOLEAN FUNCTIONS; CODES (STANDARDS); OPTIMAL CONTROL SYSTEMS; QUADRATIC PROGRAMMING;

CRYPTOGRAPHY;

EID: 27244445150     PISSN: 09251022     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10623-005-4036-2     Document Type: Article

References (39)

1. E. F. Assmus and J. D. Key, Designs and their Codes, Cambridge University Press, Cambridge.
2. E. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York (1968).
3. S. Boztas and P. V. Kumar, Binary sequences with Gold-like correlation but large linear span, IEEE Trans. Inform. Theory, Vol. 40, No. 2 (1994) pp. 532-537.
4. A. Canteaut, C. Carlet, P. Charpin and C. Fontaine, On cryptographic properties of the cosets of R(1, m). IEEE Trans. Inform. Theory, Vol. 47, No. 4 (2001) pp. 1494-1513.
5. A. Canteaut and P. Charpin, Decomposing bent functions. In Proc. 2002 IEEE International Symposium on Information Theory, Lausanne, 2002.
6. And IEEE Transactions on Information Theory Vol. 49 (2003) pp. 2004-2019.
7. C. Carlet, Partially bent functions, Designs Codes Cryptography, Vol. 3, (1993) pp. 135-145
8. and proceedings of CRYPTO' 92, Advances in Cryptology, Lecture Notes in Computer Science 740, Springer Verlag, Berlin (1993) pp. 280-291.
9. C. Carlet, Two new classes of bent functions, EUROCRYPT' 93, Advances in Cryptology, Lecture Notes in Computer Science 765, Springer-Verlag, Berlin (1994) pp. 77-101.
10. C. Carlet, Generalized partial spreads, IEEE Trans. Inform. Theory, Vol. 41, No. 5 (1995) pp. 1482-1487.
11. C. Carlet, A construction of bent functions. Finite Fields and Applications, London Mathematical Society, Lecture Series 233, Cambridge University Press, Cambridge (1996) pp. 47-58.
12. C. Carlet, More correlation-immune and resilient functions over Galois fields and Galois rings. Advances in Cryptology, EUROCRYPT'97, Lecture Notes in Computer Science 1233, Springer-Verlag, (1997) pp. 422-433.
13. C. Carlet, Recent results on binary bent functions. International Conference on Combinatorics, Information Theory and Statistics; Journal of Combinatorics, Information and System Sciences, Vol. 24, No. 3-4, (1999) pp. 275-291.
14. C. Carlet, On Kerdock codes, American Mathematical Society (Proceedings of the conference Finite Fields and Applications Fq4) Contemporary Mathematics 225, (1999) pp. 155-163.
15. C. Carlet. On the confusion and diffusion properties of Maiorana-McFarland's and extended Maiorana-McFarland's functions. Journal of Complexity, Vol. 20 (2004) pp. 182-204, dedicated to Prof. Harald Niederreiter on the occasion of his 60th birthday.
16. C. Carlet. On the secondary constructions of resilient and bent functions. Proceedings of "Coding, Cryptography and Combinatorics", Progress in Computer Science and Applied Logic, Vol. 23, Birkhäuser Verlag, Basel, (2004) pp. 3-28.
17. C. Carlet, H. Dobbertin et and G. Leander, Normal extensions of bent functions. IEEE Trans. Inform. Theory, Vol. 50 (2004) pp. 2880-2885.
18. C. Carlet and E. Prouff, On plateaued Boolean functions and their constructions, In Proc. of "Fast Software Encryption 2003", Lecture Notes in Computer Science, Vol. 2887 (2003) pp. 54-73.
19. P. Delsarte and J. Goethals, Irreducible binary codes of even dimension, In 1970 Proc. Second Chapel Hill Conference on Combinatorial Mathematics and its Applications, Univ. North Carolina, Chapel Hill, NC (1970) pp. 100-113.
20. J. Detombe and S. Tavares, Constructing large cryptographically strong S-boxes, Advances in Cryptology-AUSCRYPT'92, Springer-Verlag, Berlin, Heidelberg, New York (1993).
21. J. F. Dillon, Elementary Hadamard Difference sets. Ph. D. Thesis, Univ. of Maryland (1974).
22. J. F. Dillon, Elementary Hadamard difference sets, In Proc. Sixth S-E Conf. Comb. Graph Theory and Comp., F. Hoffman et al. (Eds), Winnipeg Utilitas Math, (1975) pp. 237-249.
23. H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity. Fast Software Encryption, Second International Workshop, Lecture Notes in Computer Science 1008, (1995) pp. 61-74.
24. X. -D. Hou, New constructions of bent functions, International Conference on Combinatorics, Information Theory and Statistics; Journal of Combinatorics, Information and System Sciences, Vol. 24, No. 3-4 pp. 275-291
25. R. Fitzgerald and J. Yucas, Pencils of quadratic forms over finite fields, Discrete Mathematics, Vol. 283 (2004) pp. 71-79.
26. R. Gold, Maximal recursive sequences with 3-valued cross correlation functions, IEEE Trans. Inform. Theory, Vol. 14, (1968) pp. 154-156.
27. G. van der Geer and M. van der Vlugt, Quadratic forms, generalized Hamming weights of codes and curves with many points, J. Number Theory, Vol. 59 (1996) pp. 20-36.
28. T. Helleseth and P. V. Kumar, Sequences with low correlation, In eds. (V. Pless and C. Huffman), Handbook of Coding Theory, Elsevier Science Publishers, Amsterdam (1998).
29. X. -D. Hou, New constructions of bent functions, International Conference on Combinatorics, Information Theory and Statistics; Journal of Combinatorics, Information and System Sciences, Vol. 24, No. 3-4 (1999) pp. 275-291
30. X. -D. Hou, and P. Langevin, Results on bent functions, J. Combin. Theory, Series A, Vol. 80 (1997) pp. 232-246.
31. K. Khoo, G. Gong and D. Stinson, New families of Gold-like sequences, IEEE Int. Symp. Inform. Theory 02, Vol. 12 (2002) p. 281.
32. A. Lempel and M. Cohn, Maximal families of bent sequences, IEEE Trans. Infor. Theory, IT-28 No. 6 (1982) pp. 865-868.
33. R. Lidl and H. Niederreiter, Finite fields, Cambridge University Press, Cambridge (1997).
34. F. MacWilliams and N. Sloane, The Theory of Error Correcting Codes, North Holland, Amsterdam/New York/Oxford (1977).
35. J. D. Olsen, R. A. Scholtz and L. R. Welch, Bent function sequences, IEEE Trans. Inform. Theory, IT-28 No. 6, (1982) pp. 858-864.
36. O. S. Rothaus, On bent functions, J. Comb. Theory, Ser A 20 (1976) pp. 300-305.
37. M. Simon, J. Omura, R. Scholtz and B. Levitt, Spread-Spectrum Communications, Vol. 1, Computer Science Press, Rock Ville, MD (1985).
38. J. Wolfman, Formes quadratiques et codes a deux poids, C. R. Acad. Sci. Paris Ser. A-B, Vol. 281 (1975) pp. 533-535.
39. Y. Zheng and X. M. Zhang, Plateaued functions. ICICS'99, Lecture Notes in Computer Science, Vol. 1726. Heidelberg, Ed., Springer-Verlag, (1999) pp. 284-300.