Abstract
This chapter gives a survey of the recent results on Boolean bent functions and lists some open problems in this domain. It includes also new results. We recall the definitions and basic results, including known and new characterizations of bent functions; we describe the constructions (primary and secondary; known and new) and give the known infinite classes, in multivariate representation and in trace representation (univariate and bivariate). We also focus on the particular class of rotation symmetric (RS) bent functions and on the related notion of bent idempotent: we give the known infinite classes and secondary constructions of such functions, and we describe the properties of a recently introduced transformation of RS functions into idempotents.
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Notes
- 1.
In public-key cryptography, the only key which must be kept secret is the decryption key, but as far as we know, bent functions play no big role in such ciphers.
References
L. Budaghyan, C. Carlet, CCZ-equivalence of single and multi output Boolean functions. AMS Contemp. Math. 518, 43–54 (2010). Post-proceedings of the Conference Fq9
L. Budaghyan, C. Carlet, CCZ-equivalence of bent vectorial functions and related constructions. Des. Codes Cryptogr. 59(1–3), 69–87 (2011). Post-proceedings of WCC 2009
L. Budaghyan, C. Carlet, A. Pott, New classes of almost bent and almost perfect nonlinear functions. IEEE Trans. Inf. Theory 52(3), 1141–1152 (2006)
L. Budaghyan, C. Carlet, T. Helleseth, On bent functions associated to AB functions, in Proceedings of IEEE Information Theory Workshop, ITW’11, Paraty, October 2011
L. Budaghyan, C. Carlet, T. Helleseth, A. Kholosha, S. Mesnager, Further results on Niho bent functions. IEEE Trans. Inf. Theory 58(11), 6979–6985 (2012)
A. Canteaut, P. Charpin, G. Kyureghyan, A new class of monomial bent functions. Finite Fields Appl. 14(1), 221–241 (2008)
C. Carlet, Two new classes of bent functions, in Proceedings of EUROCRYPT’93. Lecture Notes in Computer Science, vol. 765 (1994), pp. 77–101
C. Carlet, A construction of bent functions, in Finite Fields and Applications. London Mathematical Society. Lecture Series, vol. 233 (Cambridge University Press, Cambridge, 1996), pp. 47–58
C. Carlet, On the secondary constructions of resilient and bent functions, in Proceedings of the Workshop on Coding, Cryptography and Combinatorics 2003 (Birkhäuser, Basel, 2004), pp. 3–28
C. Carlet, On bent and highly nonlinear balanced/resilient functions and their algebraic immunities, in Proceedings of AAECC 16. Lecture Notes in Computer Science, vol. 3857 (2006), pp. 1–28
C. Carlet, Boolean functions for cryptography and error correcting codes. Chapter of the monography, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering, ed. by Y. Crama, P. Hammer (Cambridge University Press, Cambridge, 2010), pp. 257–397. Preliminary version available at http://www-rocq.inria.fr/codes/Claude.Carlet/pubs.html
C. Carlet, Vectorial Boolean functions for cryptography. Chapter of the monography, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering, ed. by Y. Crama, P. Hammer (Cambridge University Press, Cambridge, 2010), pp. 398–469. Preliminary version available at http://www-rocq.inria.fr/codes/Claude.Carlet/pubs.html
C. Carlet, C. Ding, Highly nonlinear mappings. J. Complex. 20, 205–244 (2004). Special Issue “Complexity Issues in Coding and Cryptography”, dedicated to Prof. Harald Niederreiter on the occasion of his 60th birthday
C. Carlet, P. Gaborit, Hyper-bent functions and cyclic codes. J. Combin. Theory Ser. A 113(3), 466–482 (2006)
C. Carlet, G. Gao and W. Liu. A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions. Journal of Combinatorial Theory, Series A, vol. 127(1), pp. 161–175 (2014).
C. Carlet, T. Helleseth, Sequences, Boolean functions, and cryptography, in Handbook of Codes, Sequences and Their Applications, ed. by S. Boztas (CRC Press, to appear)
C. Carlet, A. Klapper, 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, Belgium, 2002. The results are given in [11]
C. Carlet, S. Mesnager, On Dillon’s class H of bent functions, Niho bent functions and o-polynomials. J. Combin. Theory Ser. A 118, 2392–2410 (2011)
C. Carlet, S. Mesnager, On semi-bent Boolean functions. IEEE Trans. Inform. Theory 58, 3287–3292 (2012)
C. Carlet, P. Charpin, V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998)
C. Carlet, L.E. Danielsen, M.G. Parker, P. Solé, Self dual bent functions. Int. J. Inf. Coding Theory 1(4), 384–399 (2010). Special Issue, dedicated to Vera Pless
C. Carlet, F. Zhang, Y. Hu, Secondary constructions of bent functions and their enforcement. Adv. Math. Commun. 6(3), 305–314 (2012)
P. Charpin, G. Gong, Hyperbent functions, Kloosterman sums and Dickson polynomials. IEEE Trans. Inf. Theory 54(9), 4230–4238 (2008)
P. Charpin, G.M. Kyureghyan, Cubic monomial bent functions: a subclass of M. SIAM J. Discrete Math. 22(2), 650–665 (2008)
P. Charpin, E. Pasalic, C. Tavernier, On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inform. Theory 51, 4286–4298 (2005)
T. Cusick, Y. Cheon, Affine equivalence of quartic homogeneous rotation symmetric Boolean functions. Inf. Sci. 259, 192–211 (2014)
D.K. Dalai, S. Maitra, S. Sarkar, Results on rotation symmetric bent functions. Discrete Math. 309, 2398–2409 (2009)
J. Dillon, Elementary Hadamard difference sets. Ph.D. dissertation, University of Maryland, 1974
J.F. Dillon, H. Dobbertin, New cyclic difference sets with Singer parameters. Finite Fields Appl. 10(3), 342–389 (2004)
J.F. Dillon, G. McGuire, Near bent functions on a hyperplane. Finite Fields Appl. 14(3), 715–720 (2008)
H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke, P. Gaborit, Construction of bent functions via Niho power functions. J. Combin. Theory Ser. A 113, 779–798 (2006)
E. Filiol, C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, in Proceedings of EUROCRYPT’98. Lecture Notes in Computer Science, vol. 1403 (1998), pp. 475–488
C. Fontaine, On some cosets of the first-order Reed-Muller code with high minimum weight. IEEE Trans. Inform. Theory 45, 1237–1243 (1999)
S. Fu, L. Qu, C. Li, B. Sun, Balanced 2p-variable rotation symmetric Boolean functions with maximum algebraic immunity. Appl. Math. Lett. 24, 2093–2096 (2011)
G. Gao, X. Zhang, W. Liu, C. Carlet, Constructions of quadratic and cubic rotation symmetric bent functions. IEEE Trans. Inform. Theory 58, 4908–4913 (2012)
C. Carlet, G. Gao and W. Liu. Results on Constructions of Rotation Symmetric Bent and Semi-bent Functions. To appear in the proceedings of Sequences and Their Applications - Seta 2014: 8th International Conference, Melbourne, Vic, Australia, November 24–28, LNCS 8865, 2014.
F. Gologlu, Almost bent and almost perfect nonlinear functions, exponential sums, geometries and sequences. Ph.D. dissertation, University of Magdeburg, 2009
P. Guillot, Completed GPS covers all bent functions. J. Combin. Theory Ser. A 93, 242–260 (2001)
T. Helleseth, A. Kholosha, Private communication (2013)
T. Helleseth, A. Kholosha, S. Mesnager, Niho bent functions and Subiaco hyperovals. Contemp. Math. AMS 579, 91–101 (2012). Proceedings of the 10-th International Conference on Finite Fields and Their Applications (Fq’10)
S. Kavut, S. Maitra, M.D. Yücel, Search for Boolean functions with excellent profiles in the rotation symmetric class. IEEE Trans. Inf. Theory 53(5), 1743–1751 (2007)
K. Khoo, G. Gong, D. Stinson, A new characterization of semi-bent and bent functions on finite fields. Des. Codes Cryptogr. 38, 279–295 (2006)
P. Langevin, G. Leander, Counting all bent functions in dimension eight 99270589265934370305785861242880. Des. Codes Cryptogr. 59(1–3), 193–205 (2011)
G. Leander, Monomial bent functions, in Proceedings of WCC 2006 (2005), pp. 462–470; and IEEE Trans. Inf. Theory 52(2), 738–743 (2006)
G. Leander, A. Kholosha, Bent functions with 2r Niho exponents. IEEE Trans. Inform. Theory 52, 5529–5532 (2006)
N. Li, T. Helleseth, X. Tang, A. Kholosha, Several new classes of bent functions From Dillon exponents. IEEE Trans. Inf. Theory 59(3), 1818–1831 (2013)
P. Lisoněk, M. Marko, On zeros of Kloosterman sums. Des. Codes Cryptogr. 59, 223–230 (2011)
W. Ma, M. Lee, F. Zhang, A new class of bent functions. EICE Trans. Fundam. E88-A(7), 2039–2040 (2005)
F.J. MacWilliams, N.J. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977)
S. Mesnager, 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 (Springer, Heidelberg, 2009), pp. 402–417
S. Mesnager, 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 (2010), pp. 97–113
S. Mesnager, 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)
S. Mesnager, 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)
S. Mesnager, Semi-bent functions from Dillon and Niho exponents, Kloosterman sums, and Dickson polynomials. IEEE Trans. Inform. Theory 57, 7443–7458 (2011)
S. Mesnager, J.P. Flori, Hyper-bent functions via Dillon-like exponents. IEEE Trans. Inf. Theory 59(5), 3215–3232 (2013)
G.L. Mullen, D. Panario, Handbook of Finite Fields. Series: Discrete Mathematics and Its Applications (CRC Press, West Palm Beach, 2013)
S. Perlis, Normal bases of cyclic fields of prime-power degree. Duke Math J. 9, 507–517 (1942)
J. Pieprzyk, C. Qu, Fast Hashing and rotation symmetric functions. J. Univ. Comput. Sci. 5, 20–31 (1999)
O.S. Rothaus, On bent functions. J. Combin. Theory Ser. A 20, 300–305 (1976)
R. Singh, B. Sarma, A. Saikia, Public key cryptography using permutation p-polynomials over finite fields. IACR Cryptology ePrint Archive 2009: 208 (2009)
P. Stǎnicǎ, S. Maitra, Rotation symmetric Boolean functions-count and cryptographic properties. Discrete Appl. Math. 156, 1567–1580 (2008)
P. Stǎnicǎ, S. Maitra, J. Clark, Results on rotation symmetric bent and correlation immune Boolean functions, in Proceedings of Fast Software Encryption 2004. Lecture Notes in Computer Science, vol. 3017 (2004), pp. 161–177
N. Tokareva, On the number of bent functions from iterative constructions: lower bounds and hypotheses. Adv. Math. Commun. 5(4), 609–621 (2011)
A.M. Youssef, G. Gong, Hyper-bent functions, in Proceedings of EUROCRYPT 2001. Lecture Notes in Computer Science, vol. 2045 (2001), pp. 406–419
N.Y. Yu, G. Gong, Construction of quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 7(52), 3291–3299 (2006)
F. Zhang, C. Carlet, Y. Hu, W. Zhang, New secondary constructions of bent functions. Preprint, 2013
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We thank Thomas Cusick, Guangpu Gao, and Sihem Mesnager for useful information.
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Carlet, C. (2014). Open Problems on Binary Bent Functions. In: Koç, Ç. (eds) Open Problems in Mathematics and Computational Science. Springer, Cham. https://doi.org/10.1007/978-3-319-10683-0_10
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