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Four decades of research on bent functions

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Abstract

In this survey, we revisit the Rothaus paper and the chapter of Dillon’s thesis dedicated to bent functions, and we describe the main results obtained on these functions during these last 40 years. We also cover more briefly super-classes of Boolean functions, vectorial bent functions and bent functions in odd characteristic.

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Notes

  1. The book [186] contains a complete and detailed mathematical study of bent functions as well as a systematic overview of their generalizations and their applications.

  2. We respect the usual notation: \(\widetilde{f}\) in the case of binary functions and \(f^\star \) in the case of p-ary functions.

  3. In multivariate representation, with the usual inner product, the matrices of \(L^{-1}\) and \(L'\) are transposed of each other.

  4. A hyperoval of \(PG_{2}(2^m)\) is a set of \(2^m+2\) points no three of them are collinear; it can then be represented as \(D(f)=\{(1,t,G(t)), t\in {\mathbb {F}}_{2^{n}}\}\cup \{(0,1,0), (0,0,1)\}\) where G is an o-polynomial on \({\mathbb {F}}_{2^{m}}\).

  5. The study of bent functions embedded into the recursive framework of \(\mathbb {Z}\)-bent functions has been initiated by Dobbertin in 2005.

  6. A hyperplane in \({\mathbb {F}}_{2^{n}}\) is an \((n-1)\)-dimensional linear subspace of \({\mathbb {F}}_{2^{n}}\), viewing \({\mathbb {F}}_{2^{n}}\) as an n-dimensional vector space over \({\mathbb {F}}_{2^{}}\), and has an equation of the form \(Tr_{1}^{n} (ax)=0\), \(a\in {\mathbb {F}}_{2^{n}}^*\).

  7. A variant of nonlinearity exists for stream ciphers [51] and parameters of vectorial functions more dedicated to block cipher environment which have been introduced for quantifying the balancedness of the derivatives [33, 38] can also be considered as nonlinearity parameters. But here we are only interested in the nonlinearity parameter which quantifies the bentness of vectorial functions.

References

  1. Alon N., Goldreich O., Hastad J., Peralta R.: Simple constructions of almost k-wise independent random variables. Random Struct. Algorithms 3(3), 289–304 (1992).

  2. Assmus E.F., Key J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992).

  3. Bending T.D.: Bent functions, SDP designs and their automorphism groups. Ph.D. Thesis, Queen Mary and Westfield College (1993).

  4. Beth T., Jungnickel D., Lenz H.: Design Theory, vol. 1. Cambridge University Press, Cambridge (1999).

  5. Biham E., Shamir A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991).

  6. Braeken A., Borisov Y., Nikova S., Preneel B.: Classification of Boolean functions of 6 variables or less with respect to cryptographic properties. In: Proceedings of ICALP 2005. Lecture Notes in Computer Science, vol. 3580, pp. 324–334 (2005).

  7. Budaghyan L., Carlet C.: CCZ-equivalence of single and multi output Boolean functions. In: AMS Contemporary Mathematics, vol. 518. Post-proceedings of the conference Fq9, pp. 43–54 (2010).

  8. Budaghyan L., Carlet C.: CCZ-equivalence of bent vectorial functions and related constructions. In: Post-proceedings of WCC 2009. Des. Codes Cryptogr. 59(1–3), 69–87 (2011).

  9. Budaghyan L., Helleseth T.: New commutative semifields by new PN multinomials. Cryptogr. Commun. 3(1), 1–16 (2011).

  10. Budaghyan L., Carlet C., Pott A.: New classes of almost bent and almost perfect nonlinear functions. IEEE Trans. Inf. Theory 52(3), 1141–1152 (2006).

  11. Budaghyan L., Carlet C., Helleseth T.: On bent functions associated to AB functions. In: Proceedings of IEEE Information Theory Workshop (ITW’11), Paraty (2011).

  12. Budaghyan L., Carlet C., Helleseth T., Kholosha A.: Generalized bent functions and their relation to Maiorana–Mcfarland class. In: Proceedings of the 2012 IEEE International Symposium on Information Theory (ISIT), pp. 1217–1220 (2012).

  13. Budaghyan L., Carlet C., Helleseth T., Kholosha A., Mesnager S.: Further results on Niho bent functions. IEEE Trans. Inf. Theory 58(11), 6979–6985 (2012).

  14. Budaghyan L., Carlet C., Helleseth T., Kholosha A.: On o-equivalence of niho bent functions. In: Proceedings of International Workshop on the Arithmetic of Finite Fields, pp. 155–168 (2014).

  15. Budaghyan L., Kholosha A., Carlet C., Helleseth T.: Niho bent functions from quadratic o-monomials. In: Proceedings of ISIT 2014, pp 1827–1831 (2014).

  16. Canteaut C., Charpin P.: Decomposing bent functions. IEEE Trans. Inf. Theory 49(8), 2004–2019 (2003).

  17. Canteaut A., Carlet C., Charpin P., Fontaine C.: On cryptographic properties of the cosets of R(1, m). IEEE Trans. Inf. Theory 47, 1494–1513 (2001).

  18. Canteaut A., Daum M., Dobbertin H., Leander G.: Normal and non-normal bent functions. In: Proceedings of the Workshop on Coding and Cryptography 2003, pp. 91–100 (2003).

  19. Canteaut A., Charpin P., Kyureghyan G.: A new class of monomial bent functions. Finite Fields Appl. 14(1), 221–241 (2008).

  20. Cao X., Chen H., Mesnager S.: Further results on semi-bent functions in polynomial form. J. Adv. Math. Commun. (to appear).

  21. Carlet C.: Partially-bent functions. Des. Codes Cryptogr. 3, 135–145 (1993), and Proceedings of CRYPTO’ 92. Lecture Notes in Computer Science, vol. 740, pp. 280–291 (1993).

  22. Carlet C.: Two new classes of bent functions. In: Proceedings of EUROCRYPT’93. Lecture Notes in Computer Science, vol. 765, pp. 77–101 (1994).

  23. Carlet C.: Generalized partial spreads. IEEE Trans. Inf. Theory 41(5), 1482–1487 (1995).

  24. Carlet C.: A construction of bent functions. In: Finite Fields and Applications. London Mathematical Society, Lecture Series, vol. 233, pp. 47–58. Cambridge University Press, Cambridge (1996).

  25. Carlet C.: On cryptographic propagation criteria for Boolean functions. In: Information and Computation, vol. 151, pp. 32–56. Academic Press, New York (1999).

  26. Carlet C.: Recent results on binary bent functions. Proceedings of the International Conference on Combinatorics, Information Theory and Statistics. J. Comb. Inf. Syst. Sci. 25(1–4), 133–149 (2000).

  27. Carlet C.: On the secondary constructions of resilient and bent functions. In: Proceedings of the Workshop on Coding, Cryptography and Combinatorics 2003, published by Birkhäuser Verlag, pp. 3–28 (2004).

  28. Carlet C.: On the degree, nonlinearity, algebraic thickness and non-normality of Boolean functions, with developments on symmetric functions. IEEE Trans. Inf. Theory 50, 2178–2185 (2004).

  29. Carlet C.: On the confusion and diffusion properties of Maiorana–McFarland’s and extended Maiorana–McFarland’s functions. Special Issue “Complexity Issues in Coding and Cryptography”, dedicated to Prof. Harald Niederreiter on the occasion of his 60th birthday. J. Complex. 20, 182–204 (2004).

  30. Carlet C.: On bent and highly nonlinear balanced/resilient functions and their algebraic immunities. In: Proceedings of AAECC 16. Lecture Notes in Computer Science, vol. 3857, pp. 1–28 (2006).

  31. Carlet C.: Boolean Functions for Cryptography and Error Correcting Codes. In: Crama Y., Hammer P. (eds.) Chapter of the monography Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press, Cambridge (2010). Preliminary version available at http://www.math.univ-paris13.fr/~carlet/pubs.html.

  32. Carlet C.: Vectorial Boolean Functions for Cryptography. Chapter of the monography. In: Crama Y., Hammer P. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 398–469. Cambridge University Press, Cambridge (2010). Preliminary version available at http://www.math.univ-paris13.fr/~carlet/pubs.html.

  33. Carlet C.: Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions. Des. Codes Cryptogr. 59(1–3), 89–109 (2011).

  34. Carlet C.: Open problems on binary bent functions. Proceeding of the conference. In: Open Problems in Mathematical and Computational Sciences, 18–20 Sept, 2013, Istanbul, pp. 203–241. Springer, Berlin (2014).

  35. Carlet C.: Boolean and vectorial plateaued functions, and APN functions. IEEE Trans. Inf. Theory 61(11), 6272–6289 (2015). A shorter version has appeared in the Proceedings of C2SI 2015 with the title “On the Properties of Vectorial Functions with Plateaued Components and Their Consequences on APN Functions”, pp. 63–73 (2015)

  36. Carlet C.: More \({\cal {P}}{\cal {S}}\)-like bent functions. In: IACR Cryptology ePrint Archive, Report 2015/168, 2015. Presented in the International conference Finite field and their Applications Fq12, New York, July (2015).

  37. Carlet C., Ding C.: Highly Nonlinear Mappings. Special Issue “Complexity Issues in Coding and Cryptography”, dedicated to Prof. Harald Niederreiter on the occasion of his 60th birthday. J. Complex. 20, 205–244 (2004).

  38. Carlet C., Ding C.: Nonlinearities of S-boxes. J. Finite Fields Appl. 13(1), 121–135 (2007).

  39. Carlet C., Dubuc S.: On generalized bent and q-ary perfect nonlinear functions. In: Proceedings of the Fifth International Conference “Finite Fields and Applications”, pp. 81–94 (2001).

  40. Carlet C., Gaborit P.: Hyper-bent functions and cyclic codes. J. Comb. Theory Ser. A 113(3), 466–482 (2006).

  41. Carlet C., Gao G.: A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions. J. Comb. Theory Ser. A 127(1), 161–175 (2014).

  42. Carlet C., Guillot P.: A characterization of binary bent functions. J. Comb. Theory Ser. A 76, 328–335 (1996).

  43. Carlet C., Guillot P.: An alternate characterization of the bentness of binary functions, with uniqueness. Des. Codes Cryptogr. 14, 133–140 (1998).

  44. Carlet C., Guillot P.: Bent, resilient functions and the Numerical Normal Form. DIMACS Ser. Discret. Math. Theor. Comput. Sci. 56, 87–96 (2001).

  45. Carlet C., Helleseth T.: Sequences, Boolean functions, and cryptography. In: Boztas S. (ed.) Handbook of Codes, Sequences and Their Applications. CRC Press (to appear).

  46. Carlet C., Klapper A.: Upper bounds on the numbers of resilient functions and of bent functions. This paper was meant to appear in an issue of Lecture Notes in Computer Sciences dedicated to Philippe Delsarte, Editor Jean-Jacques Quisquater. But this issue finally never appeared. A shorter version has appeared in the Proceedings of the 23rd Symposium on Information Theory in the Benelux, Louvain-La-Neuve (2002). The results are given in [30].

  47. Carlet C., Mesnager S.: On the construction of bent vectorial functions. J. Inf. Coding Theory Algebr. Comb. Coding Theory 1(2), 133–148 (2010).

  48. Carlet C., Mesnager S.: On Dillon’s class \(H\) of bent functions, Niho bent functions and o-polynomials. J. Comb. Theory Ser. A 118, 2392–2410 (2011).

  49. Carlet C., Mesnager S.: On semi-bent Boolean functions. IEEE Trans. Inf. Theory 58, 3287–3292 (2012).

  50. Carlet C., Prouff E.: On plateaued functions and their constructions. In: Proceedings of Fast Software Encryption (FSE), Lecture Notes in Computer Science, vol. 2887, pp. 54–73 (2003).

  51. Carlet C., Prouff E.: On a new notion of nonlinearity relevant to multi-output pseudo-random generators. In: Matsui M., Zuccherato R.J. (eds.) Proceedings of Selected Areas in Cryptography 2003. Lecture Notes in Computer Science, vol. 3006, pp. 291–305 (2004).

  52. Carlet C., Charpin P., Zinoviev V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998).

  53. Carlet C., Dobbertin H., Leander G.: Normal extensions of bent functions. IEEE Trans. Inf. Theory 50(11), 2880–2885 (2004).

  54. Carlet C., Danielsen L.E., Parker M.G., Solé P.: Self dual bent functions. Special Issue of the Int. J. Inf. Coding Theory (IJICoT) (dedicated to Vera Pless) 1(4), 384–399 (2010).

  55. Carlet C., Helleseth T., Kholosha A., Mesnager S.: On the duals of bent functions with \(2^r\) Niho exponents. IEEE Int. Symp. Inf. Theory 2011, 703–707 (2011).

  56. Carlet C., Zhang F., Hu Y.: Secondary constructions of bent functions and their enforcement. Adv. Math. Commun. 6(3), 305–314 (2012)

  57. Carlet C., Gao G., Liu W.: Results on constructions of rotation symmetric bent and semi-bent functions. In: Proceedings of Sequences and Their Applications (SETA 2014). Lecture Notes in Computer Science, vol. 8865, pp. 21–33 (2014).

  58. Ceşmelioğlu A., Meidl W.: A construction of bent functions from plateaued functions. Des. Codes Cryptogr. 66(1–3), 231–242 (2013).

  59. Ceşmelioğlu A., Meidl W., Pott A.: Generalized Maiorana-McFarland Class and normality of p-ary bent functions. Finite Fields Appl. 24, 105–117 (2013).

  60. Ceşmelioğlu A., Meidl W., Topuzoglu A.: Partially bent functions and their properties. Appl. Algebra Number Theory, p. 22 (2014).

  61. Ceşmelioğlu A., Meidl W., Pott A.: Bent functions, spreads, and o-polynomials. SIAM J. Discret. Math. 29(2), 854–867 (2015).

  62. Chabaud F., Vaudenay S.: Links between differential and linear cryptanalysis. In: Proceedings of EUROCRYPT’94. Lecture Notes in Computer Science, vol. 950, pp. 356–365 (1995).

  63. Charnes C., Rotteler M., Beth T.: On homogeneous bent functions. In: Proceedings Applied Algebra, Algebraic Algorithms and Error Correcting Codes (AAECC-14). Lecture Notes in Computer Science, vol. 2227, pp. 249–259 (2001).

  64. Charnes C., Rotteler M., Beth T.: Homogeneous bent functions, invariants, and designs. Des. Codes Cryptogr. 26(1–3), 139–154 (2002).

  65. Charpin P.: Normal Boolean functions.: Special Issue “Complexity Issues in Coding and Cryptography”, dedicated to Prof. Harald Niederreiter on the occasion of his 60th birthday. J. Complex. 20, 245–265 (2004).

  66. Charpin P., Gong G.: Hyperbent functions, Kloosterman sums and Dickson polynomials. IEEE Trans. Inf. Theory 54(9), 4230–4238 (2008).

  67. Charpin P., Kyureghyan G.: Cubic monomial bent functions: a subclass of M. SIAM J. Discret. Math. 22(2), 650–665 (2008).

  68. Charpin P., Pasalic E., Tavernier C.: On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inf. Theory 51(12), 4286–4298 (2005).

  69. Chase P.J., Dillon J.F., Lerche K.D.: Bent functions and difference sets. R41 Technical Paper. April (1971).

  70. Chee S., Lee S., Kim K.: Semi-bent Functions. Advances in Cryptology-ASIACRYPT94. In: Pieprzyk J., Safavi-Naini R. (eds.) Proceedings of 4th International Conference on the Theory and Applications of Cryptology, Wollongong. Lecture Notes in Computer Science, vol. 917, pp. 107–118 (1994).

  71. Cherowitzo W.E.: Hyperovals in Desarguesian planes of even order. Ann. Discret. Math. 37, 87–94 (1988).

  72. Cusick T., Cheon Y.: Affine equivalence of quartic homogeneous rotation symmetric Boolean functions. Inf. Sci. 259, 192–211 (2014).

  73. Cohen G., Mesnager S.: On constructions of semi-bent functions from bent functions. J. Contemp. Math. Discret. Geom. Algebr. Comb. 625, 141–154 (2014).

  74. Coulter R.S., Matthews, R.W.: Planar functions and planes of Lenz-Barlotti class II. Des. Codes Cryptogr. 10(2), 167–184 (1997)

  75. Dalai D.K., Maitra S., Sarkar S.: Results on rotation symmetric bent functions. Discret. Math. 309, 2398–2409 (2009).

  76. Dembowski P.: Finite Geometries. Springer, Berlin (1968).

  77. Dempwolff U.: Automorphisms and equivalence of bent functions and of difference sets in elementary abelian 2-groups. Commun. Algebra 34(3), 1077–1131 (2006).

  78. Dillon J.F.: A survey of bent functions. NSA Technical Journal Special Issue, pp. 191–215 (1972).

  79. Dillon J.F.: Elementary Hadamard difference sets. PhD Dissertation, University of Maryland, College Park (1974).

  80. Dillon J.F.: Elementary Hadamard Difference sets. In: Hoffman F. et al. (eds.) Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 237–249. Congressus Numerantium XIV, Winnipeg Utilitas Math (1975).

  81. Dillon J.F.: Multiplicative difference sets via additive characters. Des. Codes Cryptogr. 17, 225–235 (1999).

  82. Dillon J.F., Dobbertin H.: New cyclic difference sets with Singer parameters. Finite Fields Appl. 10(3), 342–389 (2004).

  83. Dillon J.F., McGuire G.: Near bent functions on a hyperplane. Finite Fields Appl. 14(3), 715–720 (2008).

  84. Dillon J.F., Schatz J.R.: Block designs with the symmetric difference property. In: Ward, R.L. (ed.) Proceedings of NSA Mathematical Sciences Meetings, pp. 159–164. U.S. Govt. Printing Office, Washington, DC (1987). http://www.openmathtexts.org/papers/dillon-shatz-designs.

  85. Dobbertin H.: Construction of bent functions and balanced Boolean functions with high nonlinearity. In: Proceedings of Fast Software Encryption 1994, Second International Workshop. Lecture Notes in Computer Science, vol. 1008, pp. 61–74 (1995).

  86. Dobbertin H.: Another proof of Kasami’s theorem. Des. Codes Cryptogr. 17, 177–180 (1999).

  87. Dobbertin H.: Bent functions embedded into the recursive framework of \({\mathbb{Z}}\)-bent functions. Preprint (January 2005).

  88. Dobbertin H., Leander G.: A survey of some recent results on bent functions. In: Proceeding of SETA 2004. Lecture Notes in Computer Science, vol. 3486, pp. 1–29 (2005).

  89. Dobbertin H., Leander G.: Cryptographer’s Toolkit for Construction of 8-Bit Bent Functions. IACR Cryptol. ePrint Archive, Report 2005/089. http://eprint.iacr.org/.

  90. Dobbertin H., Leander G.: Bent functions embedded into the recursive framework of \(\mathbb{Z}\)-bent functions. Des. Codes Cryptogr. 49(1–3), 3–22 (2008).

  91. Dobbertin H., Leander G., Canteaut A., Carlet C., Felke P., Gaborit P.: Construction of bent functions via Niho power functions. J. Comb. Theory Ser. A 113, 779–798 (2006).

  92. Dong D., Zhang X., Qu L., Fu S.: A note on vectorial bent functions. Inf. Process. Lett. 113(22–24), 866–870 (2013).

  93. Feng K., Luo J.: Value distributions of exponential sums from perfect nonlinear functions and their applications. IEEE Trans. Inf. Theory 53(9), 3035–3041 (2007).

  94. Feulner T., Sok L., Solé P., Wassermann A.: Towards the classification of self-dual bent functions in eight variables. Des. Codes Cryptogr. 68(1–3), 395–406 (2013).

  95. Filiol E., Fontaine C.: Highly nonlinear balanced Boolean functions with a good correlation-immunity. In: Proceedings of EUROCRYPT’98. Lecture Notes in Computer Science, vol. 1403, pp. 475–488 (1998).

  96. Fontaine C.: On some cosets of the first-order Reed–Muller code with high minimum weight. IEEE Trans. Inf. Theory 45, 1237–1243 (1999).

  97. Flori J.P., Mesnager S.: Dickson polynomials, hyperelliptic curves and hyper-bent functions. In: Proceedings of the 7th International Conference Sequences and Their Applications, SETA 2012, Waterloo. Lecture Notes in Computer Science, vol. 7780, pp. 40–52. Springer, Berlin (2012).

  98. Flori J.P., Mesnager S.: An efficient characterization of a family of hyper-bent functions with multiple trace terms. J. Math. Cryptol. 7(1), 43–68 (2013).

  99. Flori J.P., Mesnager S., Cohen G.: The Value 4 of Binary Kloosterman Sums. In: Proceedings of the Thirteenth International Conference on Cryptography and Coding, Oxford, United Kingdom, IMACC 2011. Lecture Notes in Computer Science, vol. 7089, pp. 61–78. Springer, Berlin (2011).

  100. Fu S., Qu L., Li C., Sun B.: Balanced \(2p\)-variable rotation symmetric Boolean functions with maximum algebraic immunity. Appl. Math. Lett. 24, 2093–2096 (2011).

  101. Gao G., Zhang X., Liu W., Carlet C.: Constructions of quadratic and cubic rotation symmetric bent functions. IEEE Trans. Inf. Theory 58, 4908–4913 (2012).

  102. Gangopadhyay S., Joshi A., Leander G., Sharma R.K.: A new construction of bent functions based on \({\mathbb{Z}}\)-bent functions. Des. Codes Cryptogr. 66(I.1–3), 243–256 (2013).

  103. Gold R.: Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inf. Theory 14(1), 154–156 (1968).

  104. Göloglŭ F.: Almost Bent and Almost Perfect Nonlinear Functions, Exponential Sums, Geometries and Sequences. PhD Dissertation, University of Magdeburg, Magdeburg (2009).

  105. Gong G., Golomb S.: Transform domain analysis of DES. IEEE Trans. Inf. Theory 45(6), 2065–2073 (1999).

  106. Gong G., Helleseth T., Hu H., Kholosha A.: On the dual of certain ternary weakly regular bent functions. IEEE Trans. Inf. Theory 58(4), 2237–2243 (2012).

  107. Guillot P.: Partial bent functions. In: Proceedings of the World Multiconference on Systemics, Cybernetics and Informatics, SCI 2000 (2000).

  108. Guillot P.: Completed GPS covers all bent functions. J. Comb. Theory Ser. A 93, 242–260 (2001).

  109. Helleseth T.: Some results about the cross-correlation function between two maximal linear sequences. Discret. Math. 16, 209–232 (1976).

  110. Helleseth T.: Correlation of m-sequences and related topics. In: Ding C., Helleseth T., Niederreiter H. (eds.) Proceedings of SETA’98. Discrete Mathematics and Theoretical Computer Science, pp. 49–66. Springer, London (1999).

  111. Helleseth T., Kholosha A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(5), 2018–2032 (2006).

  112. Helleseth T., Kholosha A.: New binomial bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 56(9), 4646–4652 (2010).

  113. Helleseth T., Kholosha A.: On generalized bent functions. Information Theory and Applications Workshop (ITA), pp. 1–6 (2010).

  114. Helleseth T., Kholosha A.: Bent functions and their connections to combinatorics. In: Surveys in Combinatorics 2013, pp. 91–126. Cambridge University Press, Cambridge (2013).

  115. Helleseth T., Kumar P.V.: Sequences with low correlation. Chapter 21. In: Pless V.S., Huffman W.C., Brualdi R.A. (eds.) Handbook of Coding Theory, Part 3: Applications, pp. 1765–1853. Elsevier, Amsterdam (1998).

  116. Helleseth T., Hollmann H.D.L., Kholosha A., Wang Z., Xiang Q.: Proofs of two conjectures on ternary weakly regular bent functions. IEEE Trans. Inf. Theory 55(11), 5272–5283 (2009).

  117. Helleseth T., Kholosha A., Mesnager S.: Niho bent functions and subiaco hyperovals. In: Proceedings of the 10th International Conference on Finite Fields and Their Applications (Fq’10). Contemporary Mathematics, vol. 579, pp. 91–101. AMS, Providence (2012).

  118. Hou X.D.: Cubic bent functions. Discret. Math. 189, 149–161 (1998).

  119. Hou X.D.: q-ary bent functions constructed from chain rings. Finite Fields Appl. 4, 55–61 (1998).

  120. Hou X.D.: On the coefficients of binary bent functions. Proc. Am. Math. Soc. 128(4), 987–996 (2000).

  121. Hou X.D.: New constructions of bent functions. Proceedings of the International Conference on Combinatorics, Information Theory and Statistics; J. Comb. Inf. Syst. Sci. 25(1-4), 173–189 (2000).

  122. Hou X.D.: Classification of self dual quadratic bent functions. Designs Codes Cryptogr.63(2), 183–198 (2012).

  123. Hou X.D., Langevin P.: Results on bent functions. J. Comb. Theory Ser. A 80, 232–246 (1997).

  124. Hu H., Feng D.: On quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 53(7), 2610–2615 (2007).

  125. Hubrechts H.: Point counting in families of hyperelliptic curves in characteristic 2. LMS J. Comput. Math. 10, 207–234 (2007).

  126. Hyun J.Y., Lee H., Lee Y.: Nonexistence of certain types of plateaued functions. Discret. Appl. Math. 161(16–17), 2745–2748 (2013).

  127. Jiang Y., Deng Y.: New results on nonexistence of generalized bent functions. Des. Codes Cryptogr. 75, 375–385 (2015).

  128. Johnson N., Jha V., Biliotti M.: Handbook of finite translation planes. In: Pure and Applied Mathematics, vol. 289. Chapman & Hall/CRC, London (2007)

  129. Kantor W.M.: Symplectic groups, symmetric designs, and line ovals. J. Algebra 33, 43–58 (1975).

  130. Kantor W.M.: Spreads, translation planes and Kerdock sets. II. SIAM J. Algebr. Discret. Methods 3, 308–318 (1982).

  131. Kantor W.M.: Exponential numbers of two-weight codes, difference sets and symmetric designs. Discret. Math. 46(1), 95–98 (1983).

  132. Kantor W.M.: Commutative semifields and symplectic spreads. J. Algebra 270, 96–114 (2003).

  133. Kantor W.M.: Bent functions generalizing Dillon’s partial spread functions. arXiv:1211.2600 (2012).

  134. Karpovsky, M.G., Kulikowski, K.J., Wang, Z.: On-line self error detection with equal protection against all errors. Int. J. High. Reliab. Electron. Syst. Des. (2008)

  135. Kasami T.: The weight enumerators for several classes of subcodes of the 2nd order Reed–Muller codes. Inf. Control. 18, 369–394 (1971).

  136. Kavut S., Maitra S., Yücel M.D.: Search for Boolean functions with excellent profiles in the rotation symmetric class. IEEE Trans. Inf. Theory 53(5), 1743–1751 (2007).

  137. Kholosha A., Pott A.: Bent and related functions. Handbook of Finite Fields, Subsection 9.3, pp. 262–273 (2013).

  138. Khoo K., Gong G., Stinson D.R.: A new family of Gold-like sequences. In: Proceedings of IEEE International Symposium on Information Theory, Lausanne (2002).

  139. Khoo K., Gong G., Stinson D.: A new characterization of semi-bent and bent functions on finite fields. Des. Codes Cryptogr. 38, 279–295 (2006).

  140. Khoo, K., Gong, G., Stinson, D.R.: A new characterization of semi-bent and bent functions on finite fields. Des. Codes Cryptogr. 38(2), 279–295 (2006).

  141. Koçak N., Mesnager S., Özbudak F.: Bent and semi-bent functions via linear translators. Preprint (2015).

  142. Kolokotronis N., Limniotis K.: A greedy algorithm for checking normality of cryptographic Boolean functions. In: Proceedings of ISITA 2012, Honolulu, pp. 28–31 (2012).

  143. Kumar P.V., Scholtz R.A., Welch L.R.: Generalized bent functions and their properties. J. Comb. Theory Ser. A 40, 90–107 (1985).

  144. Lachaud G., Wolfmann J.: The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inf. Theory 36(3), 686–692 (1990).

  145. Lahtonen J., McGuire G., Ward H.N.: Gold and Kasami–Welch functions, quadratic forms and bent functions. Adv. Math. Commun. 1, 243–250 (2007).

  146. Langevin P.: On generalized bent functions. CISM Courses and Lectures, vol. 339 (Eurocode), pp. 147–157 (1992).

  147. Langevin P., Hou X.D.: Counting partial spread functions in eight variables. IEEE Trans. Inf. Theory 57(4), 2263–2269 (2011).

  148. Langevin P., Leander G.: Counting all bent functions in dimension eight 99270589265934370305785861242880. Des. Codes Cryptogr. 59(1–3), 193–205 (2011).

  149. Langevin P., Rabizzoni P., Veron P., Zanotti J.-P.: On the number of bent functions with 8 variables. In: Proceedings of the conference BFCA 2006, Publications des universités de Rouen et du Havre, pp. 125–136 (2007).

  150. Leander G.: Monomial bent functions. In: Proceedings of WCC 2006, pp. 462–470, (2005). And IEEE Trans. Inf. Theory, 52(2), 738–743 (2006).

  151. Leander G., Kholosha A.: Bent functions with \(2^r\) Niho exponents. IEEE Trans. Inf. Theory 52, 5529–5532 (2006).

  152. Leander G., McGuire G.: Spectra of functions, subspaces of matrices, and going up versus going down. In: Proceedings of AAECC. Lecture Notes in Computer Science, vol. 4851, pp. 51–66. Springer, Berlin (2007).

  153. Leander G., McGuire G.: Construction of bent functions from near-bent functions. J. Comb. Theory Ser. A 116, 960–970 (2009).

  154. Lin B., Lin D.: New constructions of quaternary bent functions. arXiv:1309.0199 (2013).

  155. Li N., Tang X., Helleseth T.: New classes of generalized Boolean bent functions over \(Z_4\). In: IEEE International Symposium on Information Theory Proceedings (ISIT 2012) (2012).

  156. Li N., Helleseth T., Tang X., Kholosha A.: Several new classes of bent functions from Dillon exponents. IEEE Trans. Inf. Theory 59(3), 1818–1831 (2013).

  157. Li N., Tang X., Helleseth T.: New constructions of quadratic bent functions in polynomial form. IEEE Trans. Inf. Theory 60(9), 5760–5767 (2014).

  158. Li Y., Wang M.: On EA-equivalence of certain permutations to power mappings. Des. Codes Cryptogr. 58, 259–269 (2011).

  159. Lisoněk P.: An efficient characterization of a family of hyperbent functions. IEEE Trans. Inf. Theory 57(9), 6010–6014 (2011).

  160. Lisoněk P., Lu H.Y.: Bent functions on partial spreads. Des. Codes Cryptogr. 73(1), 209–216 (2014).

  161. Lisoněk P., Marko M.: On zeros of Kloosterman sums. Des. Codes Cryptogr. 59, 223–230 (2011).

  162. Ma W., Lee M., Zhang F.: A new class of bent functions. EICE Trans. Fundam. E88-A(7), 2039-2040 (2005).

  163. MacWilliams F.J., Sloane N.J.: The theory of error-correcting codes. North Holland, Amsterdam (1977).

  164. McFarland R.L.: A family of noncyclic difference sets. J. Comb. Theory Ser. A 15, 1–10 (1973).

  165. Mann H.B.: Addition Theorems. Inderscience, New York (1965).

  166. Matsui M.: Linear cryptanalysis method for DES cipher. In: Proceedings of EUROCRYPT’93. Lecture Notes in Computer Science, vol. 765, pp. 386–397 (1994).

  167. Meier W., Staffelbach O.: Fast correlation attacks on stream ciphers. In: Proceedings of EUROCRYPT’88. Lecture Notes in Computer Science, vol. 330, pp. 301–314 (1988).

  168. Meng Q., Zhang H., Yang M., Cui J.: On the degree of homogeneous bent functions. Discret. Appl. Math. 155(5), 665–669 (2007).

  169. Meng Q., Chena L., Fub F.-W.: On homogeneous rotation symmetric bent functions. Discret. Appl. Math. 158(10), 111–1117 (2010).

  170. Mesnager S.: A new family of hyper-bent Boolean functions in polynomial form. In: Proceedings of Twelfth International Conference on Cryptography and Coding (IMACC 2009). Lecture Notes in Computer Science, vol. 5921, pp. 402–417. Springer, Heidelberg (2009).

  171. Mesnager S.: Hyper-bent Boolean functions with multiple trace terms. In: Proceedings of International Workshop on the Arithmetic of Finite Fields (WAIFI 2010). Lecture Notes in Computer Science, vol. 6087, pp. 97–113 (2010).

  172. Mesnager S.: A new class of bent and Hyper-Bent Boolean functions in polynomial forms. Des. Codes Cryptogr. 59(1–3), 265–279 (2011) (see also proceedings of WCC 2009).

  173. Mesnager S.: Bent and hyper-bent functions in polynomial form and their link with some exponential sums and dickson polynomials. IEEE Trans. Inf. Theory 57(9), 5996–6009 (2011).

  174. Mesnager S.: Semi-bent functions from Dillon and Niho exponents, Kloosterman sums, and Dickson polynomials. IEEE Trans. Inf. Theory 57, 7443–7458 (2011).

  175. Mesnager S.: Semi-bent functions with multiple trace terms and hyperelliptic curves. In: Proceeding of International Conference on Cryptology and Information Security in Latin America (IACR) (Latincrypt 2012). Lecture Notes in Computer Science, vol. 7533, pp. 18–36. Springer, Berlin (2012)

  176. Mesnager S.: Semi-bent functions from oval polynomials. In: Proceedings of the Fourteenth International Conference on Cryptography and Coding, Oxford, United Kingdom (IMACC 2013). Lecture Notes in Computer Science, vol. 8308, pp. 1–15. Springer, Heidelberg (2013).

  177. Mesnager S.: Characterizations of plateaued and bent functions in characteristic \(p\). In: Proceedings of the 8th International Conference on Sequences and Their Applications (SETA 2014). Lecture Notes in Computer Science, pp. 72–82. Springer, Berlin (2014).

  178. Mesnager S.: On semi-bent functions and related plateaued functions over the Galois field \(F_{2^n}\). In: Proceedings of Open Problems in Mathematics and Computational Science. Lecture Notes in Computer Science, pp. 243–273. Springer, Berlin (2014).

  179. Mesnager S.: Several new infinite families of bent functions and their duals. IEEE Trans. Inf. Theory 60(7), 4397–4407 (2014).

  180. Mesnager S.: Bent functions and spreads. Proceedings of the 11th International conference on Finite Fields and their Applications (Fq’11), J. Am. Math. Soc., Contemporary Mathematics, 632, 295–316 (2015).

  181. Mesnager S.: Bent vectorial functions and linear codes from o-polynomials. Des. Codes Cryptogr. 77(1), 99–116 (2015).

  182. Mesnager S.: On p-ary bent functions from (maximal) partial spreads. In: Presented in the International conference Finite field and their Applications Fq12. New York, July (2015).

  183. Mesnager S.: A note on constructions of bent functions from involutions. Cryptology ePrint Archive: Report 2015/982 (2015).

  184. Mesnager S.: Linear codes with few weights from weakly regular bent functions based on a generic construction. Cryptology ePrint Archive: Report 2015/1103 (2015).

  185. Mesnager S.: Further constructions of infinite families of bent functions from new permutations and their duals. In: Cryptography and Communications. Springer, Berlin (to appear).

  186. Mesnager S.: Bent Functions: Fundamentals and Results. Springer, New York (to appear).

  187. Mesnager S., Cohen G.: On the link of some semi-bent functions with Kloosterman sums. In: Proceedings of International Workshop on Coding and Cryptology (IWCC 2011). Lecture Notes in Computer Science 6639, pp. 263–272. Springer, Berlin (2011).

  188. Mesnager S., Cohen G., Madore D.: On existence (based on an arithmetical problem) and constructions of bent functions. In: Proceedings of International Conference on Cryptography and Coding (IMACC 2015), Oxford, UK, LNCS, pp. 3–19. Springer, Heidelberg (2015).

  189. Mesnager S., Flori J.P.: Hyper-bent functions via Dillon-like exponents. IEEE Trans. Inf. Theory 59(5), 3215–3232 (2013).

  190. Mesnager S., Özbudak F., Sınak A.: Characterizations of plateaued functions in arbitrary characteristic. In: Proceedings of the International Conference on Coding theory and Cryptography (ICCC 2015), Algers (2015).

  191. Mesnager S., Özbudak F., Sınak A.: Results on characterizations of plateaued functions in arbitrary characteristic. In: Proceedings of BalkanCryptSec 2015. Lecture Notes in Computer Science (to appear).

  192. Mullen G.L., Panario D.: Handbook of Finite Fields. Series: Discrete Mathematics and Its Applications. CRC Press, Boca Raton (2013).

  193. Muratović-Ribić A., Pasalic E., Bajrić S.: Vectorial bent functions from multiple terms trace Functions. IEEE Trans. Inf. Theory 60(2), 1337–1347 (2014).

  194. Muratović-Ribić A., Pasalic E., Bajrić S.: Vectorial hyperbent trace functions from the \(PS_{ap}\) class: their exact number and specification. IEEE Trans. Inf. Theory 60(7), 4408–4413 (2014).

  195. Niho Y.: Multi-valued cross-correlation functions between two maximal linear recursive sequences. Ph.D. Dissertation, University of Sothern California, Los Angeles (1972).

  196. Nyberg K.: Perfect nonlinear S-boxes. In: Proceedings of EUROCRYPT’ 91. Lecture Notes in Computer Science, vol. 547, pp. 378–386 (1992).

  197. Nyberg K.: On the construction of highly nonlinear permutations. In: Proceedings of EUROCRYPT’92. Lecture Notes in Computer Science, vol. 658, pp. 92–98 (1993).

  198. Nyberg K.: New bent mappings suitable for fast implementation. In: International Workshop on Fast Software Encryption (FSE). Lecture Notes in Computer Science, vol. 809, pp. 179–184 (1994).

  199. Olsen J.D., Scholtz R.A., Welch L.R.: Bent function sequences. IEEE Trans. Inf. Theory 28(6), 858–864 (1982).

  200. Parker M.G., Pott A.: On Boolean functions which are bent and negabent. In: Sequences, Subsequences, and Consequences, International Workshop (SSC 2007), Los Angeles. Lecture Notes in Computer Science, vol. 4893 (2007).

  201. Pasalic E.: A note on nonexistence of vectorial bent functions with binomial trace representation in the PS- class. Inf. Process. Lett. 115(2), 139–140 (2015).

  202. Pasalic E.: Corrigendum to “A note on nonexistence of vectorial bent functions with binomial trace representation in the PS- class” [Inf. Process. Lett. 115(2) (2015) 139–140]. Inf. Process. Lett. 115(4), 520 (2015).

  203. Pasalic E., Charpin P.: Some results concerning cryptographically significant mapping over \(GF(2^n)\). Des. Codes Cryptogr. 57, 257–269 (2010).

  204. Penttila T., Budaghyan L., Carlet C., Helleseth T. Kholosha A.: Projective equivalence of ovals and EA-equivalence of Niho bent functions. Invited talk at the Finite geometries fourth Irsee conference (2014).

  205. Perlis S.: Normal bases of cyclic fields of prime-power degree. Duke Math J. 9, 507–517 (1942).

  206. Pieprzyk J., Qu C.: Fast Hashing and rotation symmetric functions. J. Univ. Comput. Sci. 5, 20–31 (1999).

  207. Piret G., Roche T., Carlet C.: PICARO—a block cipher allowing efficient higher-order side-channel resistance. In: Proceedings of ACNS 2012. Lecture Notes in Computer Science, vol. 7341, pp. 311–328 (2012).

  208. Preneel B.: Analysis and design of cryptographic hash functions. Ph.D. Thesis, Katholieke Universiteit Leuven, Leuven, U.D.C. 621.391.7 (1993).

  209. Preneel B., Govaerts R., Vandevalle J.: Boolean functions satisfying higher order propagation criteria. In: Proceedings of EUROCRYPT’91. Lecture Notes in Computer Sciences, vol. 547, pp. 141–152 (1991).

  210. Qu C., Seberry J., Pieprzyk J.: On the symmetric property of homogeneous Boolean functions. In: Proceedings of the Australian Conference on Information Security and Privacy (ACISP). Lecture Notes in Computer Science, vol. 1587, pp. 26–35. Springer, Berlin (1999).

  211. Qu C., Seberry J., Pieprzyk J.: Homogeneous bent functions. Discret. Appl. Math. 102(1–2), 133–139 (2000).

  212. Quisquater M., Preneel B., Vandewalle J.: A new inequality in discrete Fourier theory. IEEE Trans. Inf. Theory 49, 2038–2040 (2003).

  213. Rothaus O.S.: On bent functions. J. Comb. Theory Ser. A 20, 300–305 (1976).

  214. Riera C., Parker M.G.: Generalised bent criteria for Boolean functions (I). IEEE Trans. Inf. Theory 52(9), 4142–4159 (2006).

  215. Satoh T., Iwata T., Kurosawa K.: On cryptographically secure vectorial Boolean functions. In: Proceedings of Asiacrypt’99. Lecture Notes in Computer Science, pp. 20–28. Springer, Berlin (1999).

  216. Schmidt K.U.: Quaternary constant-amplitude codes for multicode CDMA. In: Proceedings of the IEEE International Symposium on Information Theory ISIT’2007, Nice, France, 24–29 June, 2007. Proceedings of 2007. pp. 278–2785 and IEEE Trans. Inf. Theory 55(4), 1824–1832 (2009).

  217. Schmidt K.U.: Z4-valued quadratic forms and quaternary sequence families. IEEE Trans. Inf. Theory 55(12), 5803–5810 (2009).

  218. Schmidt K.-U., Parker M.G., Pott A.: Negabent functions in the Maiorana–McFarland class. In: Proceedings Sequences and Their Applications (SETA 2008). Lecture Notes in Computer Science, vol. 5203, pp. 14–18 (2008).

  219. Shannon C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948).

  220. Singh R., Sarma B., Saikia A.: Public key cryptography using permutation P-polynomials over finite fields. IACR Cryptol. ePrint Archive 2009/208 (2009).

  221. Solé P., Tokareva N.: Connections between quaternary and binary bent functions. IACR Cryptol. ePrint Archive. http://www.eprint.iacr.org/2009/544 (2009).

  222. Stǎnicǎ P., Maitra S.: Rotation symmetric Boolean functions-count and cryptographic properties. Discret. Appl. Math. 156, 1567–1580 (2008).

  223. Su S., Tang X.: On the systematic constructions of rotation symmetric bent functions with any possible algebraic degrees. IACR Cryptol. ePrint Archive: Report 2015/451 (2015).

  224. Tang C., Qi Y.: Constructing hyper-bent functions from boolean functions with the Walsh spectrum taking the same value twice. In: Proceedings of SETA 2014. Lecture Notes in Computer Science, pp. 60–71 (2014).

  225. Tang C., Li N., Qi Y., Zhou Z., Helleseth T.: Linear codes with two or three weights from weakly regular bent functions. arXiv:1507.06148v3 (2015).

  226. Tokareva N.: On the number of bent functions from iterative constructions: lower bounds and hypotheses. Adv. Math. Commun. 5(4), 609–621 (2011).

  227. Tokareva N.: Bent functions, results and applications to cryptography. Elsevier, Amsterdam (2015).

  228. Vercauteren, F.: Advances in point counting. Advances in elliptic curve cryptography. London Mathematical Society Lecture Note Series, vol. 317, pp. 103–132. Cambridge University Press, Cambridge (2005).

  229. Wang X., Zhou J.: Generalized Partially Bent Functions. In Proceedings of Future Generation Communication and Networking (FGCN 2007) 1, 16–21 (2007).

  230. Wu B.: \({\cal {P}}{\cal {S}}\) bent functions constructed from finite pre-quasifield spreads. ArXiv e-prints (2013).

  231. Wu G., Parker M.G.: A complementary construction using mutually unbiased bases. Cryptogr. Commun. 6(1), 3–25 (2014).

  232. Wolfmann J.: Bent functions and coding theory. In: Pott A., Kumar P.V., Helleseth T., Jungnickel D. (eds.) Difference Sets, Sequences and Their Correlation Properties, pp. 393–417. Kluwer, Amsterdam (1999).

  233. Wolfmann J.: Cyclic code aspects of bent functions. In: Finite Fields Theory and Applications. Contemporary Mathematics Series of the AMS, vol. 518, pp. 363–384. American Mathematical Society, Providence (2010).

  234. Wolfmann J.: Bent and near-bent functions. arXiv:1308.6373 (2013).

  235. Wolfmann J.: Special bent and near-bent functions. Adv. Math. Commun. 8(1), 21–33 (2014).

  236. Xia T., Seberry J., Pieprzyk J., Charnes C.: Homogeneous bent functions of degree \(n\). Discret. Appl. Math. 142(1–3), 127–132 (2004).

  237. Youssef A.M.: Generalized hyper-bent functions over \(GF(p)\). Discret. Appl. Math. 155(8), 1066–1070 (2007).

  238. Youssef A.M., Gong G.: Hyper-bent functions. In: Proceedings of EUROCRYPT 2001. Lecture Notes in Computer Science, vol. 2045, pp. 406–419. Springer, Berlin (2001).

  239. Yu N.Y., Gong G.: Construction of quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 7(52), 3291–3299 (2006).

  240. Yuan J., Carlet C., Ding C.: The weight distribution of a class of linear codes from perfect nonlinear functions. IEEE Trans. Inf. Theory 52(2), 712–717 (2006).

  241. Zhang F., Carlet C., Hu Y., Zhang W.: New secondary constructions of bent functions. Preprint (2015).

  242. Zheng Y., Zhang X.M.: Plateaued functions. In: Advances in Cryptology ICICS 1999. Lecture Notes in Computer Science, vol. 1726, pp. 284–300. Springer, Berlin (1999).

  243. Zheng Y., Zhang X.M.: Relationships between bent functions and complementary plateaued functions. Lecture Notes in Computer Science, vol. 1787, pp. 60–75 (1999).

  244. Zheng Y., Zhang X.M.: On plateaued functions. IEEE Trans. Inf. Theory 47(3), 1215–1223 (2001).

  245. Zheng D., Zeng X., Hu L.: A family of \(p\)-ary binomial bent functions. EICE Trans. Fundam. 94A(9), 1868–1872 (2011).

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Acknowledgments

We thank John Dillon for his kind recollections and detailed information on the early days of bent functions and Bill Kantor for useful information on designs.

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  1. LAGA, Department of Mathematics, UMR 7539, CNRS, Universities of Paris 8 and Paris 13, 2 rue de la liberté, 93526, Saint-Denis Cedex 02, France

    Claude Carlet & Sihem Mesnager

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This is one of several papers published in Designs, Codes and Cryptography comprising the 25th Anniversary Issue.

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Carlet, C., Mesnager, S. Four decades of research on bent functions. Des. Codes Cryptogr. 78, 5–50 (2016). https://doi.org/10.1007/s10623-015-0145-8

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