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Almost perfect and planar functions

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In this article I survey some recent results on planar functions, almost planar functions and modified planar functions from the perspective of difference sets.

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References

  1. Albert A.A.: Finite division algebras and finite planes. In: Proceedings of Symposia in Applied Mathematics, vol. 10, pp. 53–70. American Mathematical Society, Providence (1960).

  2. Albert A.A.: Generalized twisted fields. Pac. J. Math. 11, 1–8 (1961).

  3. Aubry Y., Rodier F.: Differentially 4-uniform functions. In: Arithmetic, Geometry, Cryptography and Coding Theory 2009. Contemporary Mathematics, vol. 521, pp. 1–8. American Mathematical Society, Providence (2010).

  4. Aubry Y., McGuire G., Rodier F.: A few more functions that are not APN infinitely often. In: Finite Fields: Theory and Applications. Contemporary Mathematics, vol. 518, pp. 23–31. American Mathematical Society, Providence (2010).

  5. Bending, T.D., Fon-Der-Flaass D.: Crooked functions, bent functions, and distance regular graphs. Electron. J. Comb. 5, Research Paper 34, 14 pp (electronic) (1998).

  6. Berger T.P., Canteaut A., Charpin P., Laigle-Chapuy Y.: On almost perfect nonlinear functions over \(F^n_2\). IEEE Trans. Inf. Theory 52, 4160–4170 (2006).

  7. Beth T., Ding C.: On almost perfect nonlinear permutations. In: Advances in Cryptology—EUROCRYPT ’93 (Lofthus, 1993). Lecture Notes in Computer Science, vol. 765, pp. 65–76. Springer, Berlin (1993).

  8. Beth T., Jungnickel D., Lenz H.: Design Theory, vol. 1, 2nd edn. Cambridge University Press, Cambridge (1999).

  9. Beth T., Jungnickel D., Lenz H.: Design Theory, vol. 2, 2nd edn. Cambridge University Press, Cambridge (1999).

  10. Bierbrauer J.: New commutative semifields and their nuclei. In: Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes. Lecture Notes in Computer Science, vol. 5527, pp. 179–185. Springer, Berlin (2009).

  11. Bierbrauer J.: New semifields, PN and APN functions. Des. Codes Cryptogr. 54, 189–200 (2010).

  12. Bierbrauer J.: Commutative semifields from projection mappings. Des. Codes Cryptogr. 61, 187–196 (2011).

  13. Bierbrauer J.: Projective polynomials, a projection construction and a family of semifields. Des. Codes Cryptogr. (2015). doi:10.1007/s10623-015-0044-z.

  14. Bierbrauer J., Kyureghyan G.M.: Crooked binomials. Des. Codes Cryptogr. 46, 269–301 (2008).

  15. Blokhuis A., Jungnickel D., Schmidt B.: Proof of the prime power conjecture for projective planes of order \(n\) with abelian collineation groups of order \(n^2\). In: Proceedings of the American Mathematical Society, vol. 130, pp. 1473–1476 (2002).

  16. Blondeau C., Nyberg K.: New links between differential and linear cryptanalysis. In: Advances in Cryptology—EUROCRYPT 2013. Lecture Notes in Computer Science, vol. 7881, pp. 388–404. Springer, Heidelberg (2013).

  17. Blondeau, C., Nyberg, K.: Perfect nonlinear functions and cryptography. Finite Fields Appl. 32, 120–147 (2015)

  18. Blondeau C., Canteaut A., Charpin P.: Differential properties of power functions. Int. J. Inf. Coding Theory 1, 149–170 (2010).

  19. Blondeau C., Canteaut A., Charpin P.: Differential properties of \(x\mapsto x^{2^t-1}\). IEEE Trans. Inf. Theory 57, 8127–8137 (2011).

  20. Bluher A.W.: On \(x^{q+1}+ax+b\). Finite Fields Appl. 10, 285–305 (2004).

  21. Bluher A.W.: On existence of Budaghyan–Carlet APN hexanomials. Finite Fields Appl. 24, 118–123 (2013).

  22. Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997).

  23. Bracken C., Zha Z.: On the Fourier spectra of the infinite families of quadratic APN functions. Adv. Math. Commun. 3, 219–226 (2009).

  24. Bracken C., Byrne E., Markin N., McGuire G.: On the Walsh spectrum of a new APN function. In: Cryptography and Coding. Lecture Notes in Computer Science, vol. 4887, pp. 92–98. Springer, Berlin (2007).

  25. Bracken C., Byrne E., Markin N., McGuire G.: New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields Appl. 14, 703–714 (2008).

  26. Bracken C., Byrne E., Markin N., McGuire G.: Fourier spectra of binomial APN functions. SIAM J. Discret. Math. 23, 596–608 (2009).

  27. Bracken C., Byrne E., Markin N., McGuire G.: A few more quadratic APN functions. Cryptogr. Commun. 3, 43–53 (2011).

  28. Bracken C., Tan C.H., Tan Y.: On a class of quadratic polynomials with no zeros and its application to APN functions. Finite Fields Appl. 25, 26–36 (2014).

  29. Brinkmann M., Leander G.: On the classification of APN functions up to dimension five. Des. Codes Cryptogr. 49, 273–288 (2008).

  30. Brouwer A.E., Tolhuizen L.M.G.M.: A sharpening of the Johnson bound for binary linear codes and the nonexistence of linear codes with Preparata parameters. Des. Codes Cryptogr. 3, 95–98 (1993).

  31. Browning K.A., Dillon J.F., McQuistan M.T., Wolfe A.J.: An APN permutation in dimension six. In: Finite Fields: Theory and Applications. Contemporary Mathematics, vol. 518, pp. 33–42. American Mathematical Society, Providence (2010).

  32. Budaghyan L.: Construction and Analysis of Cryptographic Functions. Springer, Cham (2014).

  33. Budaghyan L., Carlet C.: Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Trans. Inf. Theory 54, 2354–2357 (2008).

  34. Budaghyan L., Helleseth T.: New perfect nonlinear multinomials over \({\bf F}_{p^{2k}}\) for any odd prime p. In: Sequences and Their Applications—SETA 2008. Lecture Notes in Computer Science, vol. 5203, pp. 403–414. Springer, Berlin (2008).

  35. Budaghyan L., Helleseth T.: On isotopisms of commutative presemifields and CCZ-equivalence of functions. Int. J. Found. Comput. Sci. 22, 1243–1258 (2011).

  36. Budaghyan L., Carlet C., Felke P., Leander G.: An infinite class of quadratic APN functions which are not equivalent to power mappings. In: IEEE International Symposium on Information Theory, pp. 2637–2641. IEEE, Piscataway (2006).

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

  38. Budaghyan L., Carlet C., Leander G.: Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inf. Theory 54, 4218–4229 (2008).

  39. Budaghyan L., Carlet C., Leander G.: Constructing new APN functions from known ones. Finite Fields Appl. 15, 150–159 (2009).

  40. Canteaut A., Naya-Plasencia M.: Structural weaknesses of permutations with a low differential uniformity and generalized crooked functions. In: Finite Fields: Theory and Applications. Contemporary Mathematics, vol. 518, pp. 55–71. American Mathematical Society, Providence (2010).

  41. Canteaut A., Charpin P., Dobbertin H.: Binary \(m\)-sequences with three-valued crosscorrelation: a proof of Welch’s conjecture. IEEE Trans. Inf. Theory 46, 4–8 (2000).

  42. Canteaut A., Charpin P., Dobbertin H.: Weight divisibility of cyclic codes, highly nonlinear functions on \({F}_{2^m}\), and crosscorrelation of maximum-length sequences. SIAM J. Discret. Math. 13, 105–138 (2000) (electronic).

  43. Carlet C.: Boolean functions for cryptography and error correcting codes. In: Crama Y., Hammer P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering. Encyclopedia of Mathematics and Its Applications, vol. 134, Chap. 8, pp. 257–397. Cambridge University Press, Cambridge (2010).

  44. Carlet C.: Vectorial boolean functions for cryptography. In: Crama Y., Hammer P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering. Encyclopedia of Mathematics and Its Applications, vol. 134, Chap. 9, pp. 398–469. Cambridge University Press, Cambridge (2010).

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

  46. Carlet C.: Boolean functions. In: Mullen G.L., Panario D. (eds.) Handbook of Finite Fields, Chap. 9.1, pp. 241–252. CRC Press, Boca Raton (2013).

  47. Carlet C.: On the Properties of Vectorial Functions with Plateaued Components and Their Consequences on APN Functions. In: Said El Hajji C.C., Nitaj A., Souidi E.M. (eds.) Lecture Notes in Computer Science, vol. 9084, pp. 63–73, Springer International Publishing, Cham (2015).

  48. Carlet C., Ding C.: Highly nonlinear mappings. J. Complex. 20, 205–244 (2004).

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

  50. Carlet C., Ding C., Yuan J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51, 2089–2102 (2005).

  51. Carlet C., Gong G., Tan Y.: Quadratic zero-difference balanced functions, APN functions and strongly regular graphs. CoRR. arXiv:1410.2903 (2014).

  52. Caullery F.: A new large class of functions not APN infinitely often. Des. Codes Cryptogr. 73, 601–614 (2014).

  53. Caullery F., Schmidt K.-U.: On the classification of hyperovals. Adv. Math. 283, 195–203 (2015).

  54. Çeşmelioğlu A., McGuire G., Meidl W.: A construction of weakly and non-weakly regular bent functions. J. Comb. Theory Ser. A 119, 420–429 (2012).

  55. Chabaud F., Vaudenay S.: Links between differential and linear cryptanalysis. In: Advances in Cryptology-EUROCRYPT ’94 (Perugia). Lecture Notes in Computer Science, vol. 950, pp. 356–365. Springer, Berlin (1995).

  56. Charpin P.: Open problems on cyclic codes. In: Handbook of Coding Theory, vol. I, II. North-Holland, Amsterdam (1998).

  57. Charpin P.: PN and APN functions. In: Mullen G.L., Panario D. (eds.) Handbook of Finite Fields, Chap. 9.2, pp. 253–261. CRC Press, Boca Raton (2013).

  58. Clark D., Jungnickel D., Tonchev V.D.: Affine geometry designs, polarities, and Hamada’s conjecture. J. Comb. Theory Ser. A 118, 231–239 (2011).

  59. Cohen S.D., Ganley M.J.: Commutative semifields, two-dimensional over their middle nuclei. J. Algebra 75, 373–385 (1982).

  60. Colbourn C.J., Dinitz J.H. (eds.): Handbook of Combinatorial Designs, 2nd edn. Discrete Mathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton (2007).

  61. Coulter R.S.: The classification of planar monomials over fields of prime square order. In: Proceedings of American Mathematical Society, vol. 134, pp. 3373–3378 (2006) (electronic).

  62. Coulter R.S.: \(\kappa \)-Polynomials and related algebraic objects. In: Mullen G.L., Panario D. (eds.) Handbook of Finite Fields, Chap. 9.3, pp. 273–278. CRC Press, Boca Raton (2013).

  63. Coulter R.S.: Planar functions and commutative semifields. In: Mullen G.L., Panario D. (eds.) Handbook of Finite Fields, Chap. 9.3, pp. 278–282. CRC Press, Boca Raton (2013).

  64. Coulter R.S., Henderson M.: Commutative presemifields and semifields. Adv. Math. 217, 282–304 (2008).

  65. Coulter R.S., Kosick P.: Commutative semifields of order 243 and 3125. In: Finite Fields: Theory and Applications. Contemporary Mathematics, vol. 518, pp. 129–136. American Mathematical Society, Providence (2010).

  66. Coulter R.S., Lazebnik F.: On the classification of planar monomials over fields of square order. Finite Fields Appl. 18, 316–336 (2012).

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

  68. Coulter R.S., Senger S.: On the number of distinct values of a class of functions with finite domain. Ann. Comb. 18, 233–243 (2014).

  69. Daemen J., Rijmen V.: The design of Rijndael, Information Security and Cryptography. Springer-Verlag, Berlin. AES-the advanced encryption standard (2002).

  70. Coulter R.S., Henderson M., Kosick P.: Planar polynomials for commutative semifields with specified nuclei. Des. Codes Cryptogr. 44, 275–286 (2007).

  71. Davis J.A., Jedwab J.: A unifying construction for difference sets. J. Comb. Theory Ser. A 80, 13–78 (1997).

  72. Delgado M., Janwa H.: On the Conjecture on APN Functions. ArXiv e-prints (2012).

  73. Dembowski P., Ostrom T.: Planes of order n with collineation groups of order \(n^{2}\). Math. Z. 103, 239–258 (1968).

  74. Dembowski P., Piper F.: Quasiregular collineation groups of finite projective planes. Math. Z. 99, 53–75 (1967).

  75. Dempwolff U.: Semifield planes of order 81. J. Geom. 89, 1–16 (2008).

  76. Dempwolff U.: More translation planes and semifields from Dembowski–Ostrom polynomials. Des. Codes Cryptogr. 68, 81–103 (2013).

  77. Dempwolff U., Reifart A.: The classification of the translation planes of order \(16\). I. Geom. Dedicata 15, 137–153 (1983).

  78. Dempwolff U., Röder M.: On finite projective planes defined by planar monomials. Innov. Incid. Geom. 4, 103–108 (2006).

  79. Dickson L.E.: Linear algebras with associativity not assumed. Duke Math. J. 1, 113–125 (1935).

  80. Dillon J.: Elementary Hadamard Difference-Sets. ProQuest LLC, Ann Arbor (1974). Ph.D. Thesis, University of Maryland, College Park.

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

  82. Dillon J.: Slides from talk given at “Polynomials over Finite Fields and Appliocations”, held at Banff International Research Station (2006).

  83. Dillon J.: Slides from talk given at conference “Finite Fields and Their Applications” in Dublin (2009).

  84. Dillon J.: On the dimension of an APN code. Cryptogr. Commun. 3, 275–279 (2011).

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

  86. Ding C.: Codes from difference sets. World Scientific Publishing Co., Pte. Ltd., Hackensack (2015).

  87. Ding C., Yuan J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006).

  88. Ding C., Wang Z., Xiang Q.: Skew Hadamard difference sets from the Ree–Tits slice symplectic spreads in \({{\rm PG}}(3,3^{2h+1})\). J. Comb. Theory Ser. A 114, 867–887 (2007).

  89. Dobbertin H.: Almost perfect nonlinear power functions on \({{\rm GF}}(2^n)\): the Niho case. Inf. Comput. 151, 57–72 (1999).

  90. Dobbertin H.: Almost perfect nonlinear power functions on \({{\rm GF}}(2^n)\): the Welch case. IEEE Trans. Inf. Theory 45, 1271–1275 (1999).

  91. Dobbertin H.: Almost perfect nonlinear power functions on \({{\rm GF}}(2^n)\) : a new case for n divisible by 5. In: Finite Fields and Applications (Augsburg, 1999), pp. 113–121. Springer, Berlin (2001).

  92. Edel Y.: On quadratic APN functions and dimensional dual hyperovals. Des. Codes Cryptogr. 57, 35–44 (2010).

  93. Edel Y., Kyureghyan G., Pott A.: A new APN function which is not equivalent to a power mapping. IEEE Trans. Inf. Theory 52, 744–747 (2006).

  94. Edel Y., Pott A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3, 59–81 (2009).

  95. Edel Y., Pott A.: On the equivalence of nonlinear functions. In: Enhancing cryptographic primitives with techniques from error correcting codes. NATO Science for Peace and Security Series D: Information and Communication Security, vol. 23, pp. 87–103. IOS, Amsterdam (2009).

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

  97. Férard E., Oyono R., Rodier F.: Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents. In: Arithmetic, Geometry, Cryptography and Coding Theory, Contemporary Mathematics, vol. 574, pp. 27–36. American Mathematical Society, Providence (2012).

  98. Fitzgerald R.W.: Highly degenerate quadratic forms over finite fields of characteristic 2. Finite Fields Appl. 11, 165–181 (2005).

  99. Ganley M.J.: On a paper of P. Dembowski and T. G. Ostrom: “Planes of order n with collineation groups of order \(n^{2}\)” (Math. Z. 103 (1968), 239–258), Arch. Math. 27, 93–98 (1976).

  100. Ghinelli D., Jungnickel D.: Finite projective planes with a large abelian group. In: Surveys in Combinatorics, 2003 (Bangor). London Mathematical Society Lecture Note Series, vol. 307, pp. 175–237. Cambridge University Press, Cambridge (2003).

  101. Gluck D.: Affine planes and permutation polynomials. In: Coding Theory and Design Theory, Part II, pp. 99–100. Springer, New York (1990).

  102. Godsil C., Roy A.: Two characterizations of crooked functions. IEEE Trans. Inf. Theory 54, 864–866 (2008).

  103. Gold R.: Maximal recursive sequences with \(3\)-valued recursive cross-correlation function. IEEE Trans. Inf. Theory 14, 154–156 (1968).

  104. Göloğlu F.: Slides from talk given at conference “Finite Fields and Their Applications” in Saratoga Springs (2015).

  105. Göloğlu F.: Almost perfect nonlinear trinomials and hexanomials. Finite Fields Appl. 33, 258–282 (2015).

  106. Göloğlu F., Pott A.: Almost perfect nonlinear functions: A possible geometric approach. In: Nikova S., Preneel B., Storme L., Thas J. (eds.) Coding Theory and Cryptography II. Koninklijke Vlaamse Academie van België voor Wetenschappen en Kunsten, pp. 75–100 (2007).

  107. Golomb S.W., Gong G.: Signal Design for Good Correlation. Cambridge University Press, Cambridge (2005). For wireless communication, cryptography, and radar.

  108. Hamada N.: On the \(p\)-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes. Hiroshima Math. J. 3, 153–226 (1973).

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

  110. Helleseth T., Kumar P.V.: Sequences with low correlation. In: Handbook of Coding Theory, vol. I, II, pp. 1765–1853. North-Holland, Amsterdam (1998).

  111. Hernando F., McGuire G.: Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions. J. Algebra 343, 78–92 (2011).

  112. 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, 5272–5283 (2009).

  113. Hernando F., McGuire, G.: Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes. Des. Codes Cryptogr. 65, 275–289 (2012).

  114. Hernando F., McGuire G.: On The Classification of Perfect Nonlinear and Almost Perfect Nonlinear (APN) Monomial Functions. In: Charpin P., Pott A., Winterhof A. (eds.) Finite Fields and Their Applications. Radon Series on Computational and Applied Mathematics, vol. 11, pp. 145–168. De Gruyter, Berlin (2013).

  115. Hernando F., McGuire G., Monserrat F.: On the classification of exceptional planar functions over \({\mathbb{F}}_p\). Geom. Dedicata 173, 1–35 (2014).

  116. Hertel D., Pott A.: Two results on maximum nonlinear functions. Des. Codes Cryptogr. 47, 225–235 (2008).

  117. Hiramine Y.: On planar functions. J. Algebra 133, 103–110 (1990).

  118. Hollmann H.D.L., Xiang Q.: A proof of the Welch and Niho conjectures on cross-correlations of binary \(m\)-sequences. Finite Fields Appl. 7, 253–286 (2001).

  119. Horadam K.J.: Hadamard matrices and their applications. Princeton University Press, Princeton (2007).

  120. Hou X.-D.: A note on the proof of Niho’s conjecture. SIAM J. Discret. Math. 18, 313–319 (2004).

  121. Hou X.-D.: Affinity of permutations of \({\mathbb{F}}_2^n\). Discret. Appl. Math. 154, 313–325 (2006).

  122. Hou X.-D.: On the dual of a Coulter-Matthews bent function. Finite Fields Appl. 14, 505–514 (2008).

  123. Janwa H., Wilson R.M.: Hyperplane sections of Fermat varieties in \(\mathbf{P}^3\) in char. 2 and some applications to cyclic codes. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (San Juan, PR, 1993). Lecture Notes in Computer Science, vol. 673, pp. 180–194. Springer, Berlin (1993).

  124. Janwa H., McGuire G.M., Wilson R.M.: Double-error-correcting cyclic codes and absolutely irreducible polynomials over \({{\rm GF}}(2)\). J. Algebra 178, 665–676 (1995).

  125. Jedlicka D.: APN monomials over \({{\rm GF}}(2^n)\) for infinitely many \(n\). Finite Fields Appl. 13, 1006–1028 (2007).

  126. Johnson N.L., Jha V., Biliotti M.: Handbook of Finite Translation Planes. Pure and Applied Mathematics (Boca Raton), vol. 289. Chapman & Hall/CRC, Boca Raton (2007).

  127. Jungnickel D.: On automorphism groups of divisible designs. Can. J. Math. 34, 257–297 (1982).

  128. Jungnickel D., Tonchev V.D.: Polarities, quasi-symmetric designs, and Hamada’s conjecture. Des. Codes Cryptogr. 51, 131–140 (2009).

  129. Jungnickel D., Pott A., Smith K.W.: Difference sets. In: Colbourn C.J., Dinitz J.H. (eds.) The CRC Handbook of Combinatorial Designs, pp. 419–434. CRC Press, Boca Raton (2006).

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

  131. Kantor W.M., Williams M.E.: Symplectic semifield planes and \({\mathbb{Z}}_4\)-linear codes. Trans. Am. Math. Soc. 356, 895–938 (2004).

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

  133. Kholosha A., Pott A.: Bent and related functions. In: Mullen G.L., Panario D. (eds.) Handbook of Finite Fields, Chap. 9.3, pp. 262–273. CRC Press, Boca Raton (2013).

  134. Knarr N., Stroppel M.: Polarities and unitals in the Coulter–Matthews planes. Des. Codes Cryptogr. 55, 9–18 (2010).

  135. Knuth D.E.: Finite semifields and projective planes. J. Algebra 2, 182–217 (1965)

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

  137. Kyureghyan G.M.: Crooked maps in \({\mathbb{F}}_{2^n}\). Finite Fields Appl. 13, 713–726 (2007).

  138. Kyureghyan G.M.: The only crooked power functions are \(x^{2^k}+2^l\). Eur. J. Comb. 28, 1345–1350 (2007).

  139. Kyureghyan G.M.: Special Mappings of Finite Fields. In: Charpin P., Pott A., Winterhof A. (eds.) Finite Fields and Their Applications. Radon Series on Computational and Applied Mathematics, vol. 11, pp. 117–144. De Gruyter, Berlin (2013).

  140. Kyureghyan G.M., Pott A.: Some theorems on planar mappings. In: Arithmetic of Finite Fields. Lecture Notes in Computer Science, vol. 5130, pp. 117–122. Springer, Berlin (2008).

  141. Kyureghyan G.M., Suder V.: On inversion in \({\mathbb{Z}}_{2^n-1}\). Finite Fields Appl. 25, 234–254 (2014).

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

  143. Lander E.S.: Symmetric Designs: An Algebraic Approach. London Mathematical Society Lecture Note Series, vol. 74. Cambridge University Press, Cambridge (1983).

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

  145. Lavrauw M., Polverino O.: Finite Semifields. In: Current Research Topics in Galois Geometry, pp. 131–159. Nova Science Publishers, New York (2012).

  146. Leducq E.: Functions which are PN on infinitely many extensions of \({\mathbb{F}}_p\), p odd. Des. Codes Cryptogr. 75, 281–299 (2015).

  147. Li C.H.: On isomorphisms of finite Cayley graphs—a survey. Discret. Math. 256, 301–334 (2002).

  148. Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, 2nd edn, vol. 20. Cambridge University Press, Cambridge (1997).

  149. Lidl R., Mullen G.L., Turnwald G.: Dickson polynomials. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65. Longman Scientific & Technical, Harlow; copublished in the United States with Wiley, New York (1993).

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

  151. Lunardon, G., Marino, G., Polverino, O., Trombetti, R.: Symplectic semifield spreads of \({{\rm PG}}(5, q)\) and the Veronese surface. Ric. Mat. 60, 125–142 (2011)

  152. Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4, 221–261 (1994).

  153. Marino G., Polverino O.: On isotopisms and strong isotopisms of commutative presemifields. J. Algebr. Comb. 36, 247–261 (2012).

  154. Marino G., Polverino O., Trombetti R.: Towards the classification of rank 2 semifields 6-dimensional over their center. Des. Codes Cryptogr. 61, 11–29 (2011).

  155. Menichetti G.: On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field. J. Algebra 47, 400–410 (1977).

  156. Menichetti G.: \(n\)-Dimensional algebras over a field with a cyclic extension of degree \(n.\) Geom. Dedicata 63, 69–94 (1996).

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

  158. Müller P., Zieve M.E.: Low-degree planar monomials in characteristic two. J. Algebr. Comb. 42, 695–699 (2015). doi:10.1007/s10801-015-0597-y.

  159. Muzychuk M.: On Skew Hadamard difference sets. ArXiv e-prints (2010).

  160. No J.-S., Chung H., Yun M.-S.: Binary pseudorandom sequences of period \({2^m}-1\). IEEE Trans. Inf. Theory 44, 1278–1282 (1998).

  161. Nyberg K.: Perfect nonlinear S-boxes. In: Advances in Cryptology—EUROCRYPT ’91 (Brighton, 1991), Lecture Notes in Computer Science, vol. 547, pp. 378–386. Springer, Berlin (1991).

  162. Nyberg K.: Differentially uniform mappings for cryptography. In: Advances in Cryptology—EUROCRYPT ’93 (Lofthus 1993). Lecture Notes in Computer Science, vol. 765, pp. 55–64. Springer, Berlin (1994).

  163. Nyberg K., Knudsen L.R.: Provable security against differential cryptanalysis. In: Advances in Cryptology—CRYPTO ’92 (Santa Barbara, 1992). Lecture Notes in Computer Science, vol. 740, pp. 566–574. Springer, Berlin (1993).

  164. Ott U.: Endliche zyklische Ebenen. Math. Z. 144, 195–215 (1975).

  165. Penttila T.: Configurations of ovals. J. Geom. 76, 233–255. Combinatorics 2002 (Maratea) (2003).

  166. Penttila T., Williams B.: Ovoids of parabolic spaces. Geom. Dedicata 82, 1–19 (2000).

  167. Pott A.: Finite Geometry and Character Theory. Lecture Notes in Mathematics, vol. 1601. Springer, Berlin (1995).

  168. Pott A.: A survey on relative difference sets. In: Arasu K.T., Dillon J., Harada K., Sehgal S., Solomon R. (eds.), Groups, Difference Sets, and the Monster. Proceedings of a Special Research Quarter at theOhio State University, Spring 1993. Walter de Gruyter, Berlin, pp. 195–232 (1996).

  169. Pott A., Zhou Y.: Switching construction of planar functions on finite fields. In: Arithmetic of Finite Fields. Lecture Notes in Computer Science, vol. 6087, pp. 135–150. Springer, Berlin (2010).

  170. Pott A., Zhou Y.: A character theoretic approach to planar functions. Cryptogr. Commun. 3, 293–300 (2011).

  171. Pott A., Zhou Y.: CCZ and EA equivalence between mappings over finite Abelian groups. Des. Codes Cryptogr. 66, 99–109 (2013).

  172. Rodier F.: Functions of degree \(4e\) that are not APN infinitely often. Cryptogr. Commun. 3, 227–240 (2011).

  173. Pott A., Schmidt K.-U., Zhou Y.: Semifields, relative difference sets, and bent functions. In: Niederreiter H., Ostafe A., Panario D., Winterhof A. (eds.) Algebraic Curves and Finite Fields. Radon Series on Computational and Applied Mathematics, vol. 16, pp. 161–178. De Gruyter, Berlin (2014).

  174. Rónyai L., Szőnyi T.: Planar functions over finite fields. Combinatorica 9, 315–320 (1989).

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

  176. Rúa I.F., Combarro E.F., Ranilla J.: Classification of semifields of order 64. J. Algebra 322, 4011–4029 (2009).

  177. Scherr Z., Zieve M.E.: Some planar monomials in characteristic 2. Ann. Comb. 18, 723–729 (2014).

  178. Schmidt B.: Characters and Cyclotomic Fields in Finite Geometry. Lecture Notes in Mathematics, vol. 1797. Springer, Berlin (2002).

  179. Schmidt K.-U., Zhou Y.: Planar functions over fields of characteristic two. J. Algebr. Comb. 40, 503–526 (2014).

  180. Segre B., Bartocci U.: Ovali ed altre curve nei piani di Galois di caratteristica due. Acta Arith. 18, 423–449 (1971).

  181. Storme L.: Finite geometry. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs. Discrete Mathematics and Its Applications (Boca Raton), 2nd edn., pp. 702–729. Chapman & Hall/CRC, Boca Raton (2007).

  182. Walker R.J.: Determination of division algebras with \(32\) elements. In: Proceeding of Symposia in Applied Mathematics, vol. XV, pp. 83–85. American Mathematical Society, Providence (1963).

  183. Weng G., Zeng X.: Further results on planar DO functions and commutative semifields. Des. Codes Cryptogr. 63, 413–423 (2012).

  184. Weng G., Qiu W., Wang Z., Xiang Q.: Pseudo-Paley graphs and skew Hadamard difference sets from presemifields. Des. Codes Cryptogr. 44, 49–62 (2007).

  185. Weng G., Feng R., Qiu W., Zheng Z.: The ranks of Maiorana-McFarland bent functions. Sci. China Ser. A 51, 1726–1731 (2008).

  186. Xiang Q.: Maximally nonlinear functions and bent functions. Des. Codes Cryptogr. 17, 211–218 (1999).

  187. Yoshiara S.: Dimensional dual arcs-a survey. In: Finite Geometries, Groups, and Computation, pp. 247–266. De Gruyter, Berlin (2006).

  188. Yoshiara S.: Equivalences of quadratic APN functions. J. Algebr. Comb. 35, 461–475 (2012).

  189. Yu Y., Wang M., Li Y.: A matrix approach for constructing quadratic apn functions. Cryptology ePrint Archive, Report 2013/007 (2013). http://eprint.iacr.org/.

  190. Yu Y., Wang M., Li Y.: A matrix approach for constructing quadratic APN functions. Des. Codes Cryptogr. 73, 587–600 (2014).

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

  192. Zha Z., Kyureghyan G.M., Wang X.: Perfect nonlinear binomials and their semifields. Finite Fields Appl. 15, 125–133 (2009).

  193. Zhou Y.: \((2^n,2^n,2^n,1)\)-relative difference sets and their representations. J. Comb. Des. 21, 563–584 (2013).

  194. Zhou Y., Pott A.: A new family of semifields with 2 parameters. Adv. Math. 234, 43–60 (2013).

  195. Zieve M.E.: Planar functions and perfect nonlinear monomials over finite fields. Des. Codes Cryptogr. 75, 71–80 (2015).

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Acknowledgments

The author thanks Razi Arshad, Jürgen Bierbrauer, Ayça Çeşmelioğlu, Dieter Jungnickel, Gohar Kyureghyan, Wilfried Meidl, Kai-Uwe Schmidt and Yue Zhou for their careful reading of (parts of) the manuscript. The author is of course responsible for all remaining mistakes.

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    Alexander Pott

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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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Pott, A. Almost perfect and planar functions. Des. Codes Cryptogr. 78, 141–195 (2016). https://doi.org/10.1007/s10623-015-0151-x

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