Almost perfect and planar functions

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

[1]  Alexander Pott,et al.  Finite Geometry and Character Theory , 1995 .

[2]  Tor Helleseth,et al.  Proofs of Two Conjectures on Ternary Weakly Regular Bent Functions , 2009, IEEE Transactions on Information Theory.

[3]  Daniel Panario,et al.  Handbook of Finite Fields , 2013, Discrete mathematics and its applications.

[4]  Michael E. Zieve.  Planar functions and perfect nonlinear monomials over finite fields , 2013, Des. Codes Cryptogr..

[5]  Lajos Rónyai,et al.  Planar functions over finite fields , 1989, Comb..

[6]  Faruk Göloglu.  Almost perfect nonlinear trinomials and hexanomials , 2015, Finite Fields Their Appl..

[7]  Yin Tan,et al.  On a class of quadratic polynomials with no zeros and its application to APN functions , 2014, Finite Fields Their Appl..

[8]  A. Pott,et al.  A new family of semifields with 2 parameters , 2011, 1103.4555.

[9]  Lilya Budaghyan,et al.  Construction and Analysis of Cryptographic Functions , 2015, Springer International Publishing.

[10]  Andries E. Brouwer,et al.  A sharpening of the Johnson bound for binary linear codes and the nonexistence of linear codes with preparata parameters , 1993, Des. Codes Cryptogr..

[11]  Kai-Uwe Schmidt,et al.  On the classification of hyperovals , 2014, 1403.2880.

[12]  David Jedlicka,et al.  APN monomials over GF(2n) for infinitely many n , 2007, Finite Fields Their Appl..

[13]  K. Horadam.  Hadamard Matrices and Their Applications , 2006 .

[14]  P. Dembowski,et al.  Planes of ordern with collineation groups of ordern2 , 1968 .

[15]  Robert S. Coulter.  The classification of planar monomials over fields of prime square order , 2006 .

[16]  Cunsheng Ding,et al.  Linear codes from perfect nonlinear mappings and their secret sharing schemes , 2005, IEEE Transactions on Information Theory.

[17]  Alexander Pott,et al.  A new APN function which is not equivalent to a power mapping , 2005, IEEE Transactions on Information Theory.

[18]  Robert W. Fitzgerald,et al.  Highly degenerate quadratic forms over finite fields of characteristic 2 , 2005, Finite Fields Their Appl..

[19]  François Rodier.  Functions of degree 4e that are not APN infinitely often , 2011, Cryptography and Communications.

[20]  H. Hollmann,et al.  A Proof of the Welch and Niho Conjectures on Cross-Correlations of Binary m-Sequences , 2001 .

[21]  Robert S. Coulter,et al.  On the Number of Distinct Values of a Class of Functions with Finite Domain , 2012, 1205.4801.

[22]  Gohar M. M. Kyureghyan,et al.  On inversion in ℤ2n-1 , 2014, Finite Fields Their Appl..

[23]  Rongquan Feng,et al.  The ranks of Maiorana-McFarland bent functions , 2008 .

[24]  Anne Canteaut,et al.  Binary m-sequences with three-valued crosscorrelation: A proof of Welch's conjecture , 2000, IEEE Trans. Inf. Theory.

[25]  G. Marino,et al.  On isotopisms and strong isotopisms of commutative presemifields , 2011, 1105.5940.

[26]  H. Dobbertin.  Almost Perfect Nonlinear Power Functions on GF(2n): A New Case for n Divisible by 5 , 2001 .

[27]  Alexander Pott,et al.  Two results on maximum nonlinear functions , 2008, Des. Codes Cryptogr..

[28]  Udo Ott.  Endliche zyklische Ebenen , 1975 .

[29]  Gohar M. M. Kyureghyan.  The only crooked power functions are x2k+2l , 2007, Eur. J. Comb..

[30]  Yutaka Hiramine.  On planar functions , 1990 .

[31]  Xiangyong Zeng,et al.  Further results on planar DO functions and commutative semifields , 2012, Des. Codes Cryptogr..

[32]  William M. Kantor,et al.  Commutative semifields and symplectic spreads , 2003 .

[33]  Claude Carlet.  Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions , 2011, Des. Codes Cryptogr..

[34]  P. Vijay Kumar,et al.  Generalized Bent Functions and Their Properties , 1985, J. Comb. Theory, Ser. A.

[35]  Carl Bracken,et al.  On the Fourier spectra of the infinite families of quadratic APN functions , 2009, Adv. Math. Commun..

[36]  Lars R. Knudsen,et al.  Provable Security Against Differential Cryptanalysis , 1992, CRYPTO.

[37]  Gohar M. Kyureghyan.  Special Mappings of Finite Fields , 2013, Finite Fields and Their Applications.

[38]  Claude Carlet,et al.  Boolean Functions for Cryptography and Error-Correcting Codes , 2010, Boolean Models and Methods.

[39]  W. Kantor,et al.  Symplectic semifield planes and ℤ₄–linear codes , 2003 .

[40]  Cunsheng Ding,et al.  A family of skew Hadamard difference sets , 2006, J. Comb. Theory, Ser. A.

[41]  Gary McGuire,et al.  Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions , 2009, ArXiv.

[42]  Tor Helleseth,et al.  Monomial and quadratic bent functions over the finite fields of odd characteristic , 2006, IEEE Transactions on Information Theory.

[43]  Robert S. Coulter,et al.  Commutative presemifields and semifields , 2008 .

[44]  Jürgen Bierbrauer.  Commutative semifields from projection mappings , 2011, Des. Codes Cryptogr..

[45]  Michael E. Zieve,et al.  Low-degree planar monomials in characteristic two , 2013, 1305.6597.

[46]  Francisco Monserrat,et al.  On the classification of exceptional planar functions over $$\mathbb {F}_{p}$$Fp , 2013, ArXiv.

[47]  C. Ding.  Codes From Difference Sets , 2014 .

[48]  Claude Carlet,et al.  Two Classes of Quadratic APN Binomials Inequivalent to Power Functions , 2008, IEEE Transactions on Information Theory.

[49]  Eimear Byrne,et al.  On the Walsh Spectrum of a New APN Function , 2007, IMACC.

[50]  Wilfried Meidl,et al.  A construction of weakly and non-weakly regular bent functions , 2010, J. Comb. Theory, Ser. A.

[51]  Vincent Rijmen,et al.  The Design of Rijndael , 2002, Information Security and Cryptography.

[52]  Gohar M. M. Kyureghyan.  Crooked maps in F22 , 2007, Finite Fields Their Appl..

[53]  Jürgen Bierbrauer.  New Commutative Semifields and Their Nuclei , 2009, AAECC.

[54]  Tor Helleseth,et al.  New Perfect Nonlinear Multinomials over Ffor Any Odd Prime p , 2008, SETA.

[55]  Cai Heng Li,et al.  On isomorphisms of finite Cayley graphs--a survey , 2002, Discret. Math..

[56]  Gregor Leander,et al.  On the classification of APN functions up to dimension five , 2008, Des. Codes Cryptogr..

[57]  Some Planar Monomials in Characteristic 2 , 2013, 1302.1244.

[58]  Hans Dobbertin,et al.  Almost Perfect Nonlinear Power Functions on GF(2n): The Niho Case , 1999, Inf. Comput..

[59]  Claude Carlet.  On the Properties of Vectorial Functions with Plateaued Components and Their Consequences on APN Functions , 2015, C2SI.

[60]  A. Blokhuis,et al.  Proof of the prime power conjecture for projective planes of order $n$ with Abelian collineation groups of order $n^2$ , 2001 .

[61]  Hans Dobbertin,et al.  New cyclic difference sets with Singer parameters , 2004, Finite Fields Their Appl..

[62]  Cunsheng Ding,et al.  On Almost Perfect Nonlinear Permutations , 1994, EUROCRYPT.

[63]  Qing Xiang.  Maximally Nonlinear Functions and Bent Functions , 1999, Des. Codes Cryptogr..

[64]  Claude Carlet,et al.  Constructing new APN functions from known ones , 2009, Finite Fields Their Appl..

[65]  Dieter Jungnickel,et al.  Finite projective planes with a large abelian group , 2003 .

[66]  Heeralal Janwa,et al.  Double-Error-Correcting Cyclic Codes and Absolutely Irreducible Polynomials over GF(2) , 1995 .

[67]  Jong-Seon No,et al.  Binary Pseudorandom Sequences of Period 2m-1 with Ideal Autocorrelation Generated by the Polynomial zd + (z+1)d , 1998, IEEE Trans. Inf. Theory.

[68]  Tor Helleseth,et al.  Planar functions and commutative semifields , 2010 .

[69]  Alexander Pott,et al.  Bent and related functions , 2013, Handbook of Finite Fields.

[70]  Mikhail Muzychuk.  Skew Hadamard Difference Sets , 2010 .

[71]  Eimear Byrne,et al.  A few more quadratic APN functions , 2008, Cryptography and Communications.

[72]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[73]  Serge Vaudenay,et al.  Links Between Differential and Linear Cryptanalysis , 1994, EUROCRYPT.

[74]  Cunsheng Ding,et al.  Skew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3, 32h+1) , 2007, J. Comb. Theory, Ser. A.

[75]  Tim Penttila,et al.  Configurations of ovals , 2003 .

[76]  Vladimir D. Tonchev,et al.  Polarities, quasi-symmetric designs, and Hamada’s conjecture , 2009, Des. Codes Cryptogr..

[77]  Yue Zhou,et al.  A character theoretic approach to planar functions , 2011, Cryptography and Communications.

[78]  Claude Carlet,et al.  Vectorial Boolean Functions for Cryptography , 2006 .

[79]  Xiang-dong Hou,et al.  Affinity of permutations of P2n , 2006, Discret. Appl. Math..

[80]  Peter L. Hammer,et al.  Boolean Functions , 2013, Discrete Applied Mathematics.

[81]  Eimear Byrne,et al.  New families of quadratic almost perfect nonlinear trinomials and multinomials , 2008, Finite Fields Their Appl..

[82]  Robert Gold,et al.  Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.) , 1968, IEEE Trans. Inf. Theory.

[83]  José Ranilla,et al.  Classification of semifields of order 64 , 2009 .

[84]  D. Jungnickel.  On Automorphism Groups of Divisible Designs , 1982, Canadian Journal of Mathematics.

[85]  Petr Lisonek,et al.  Bent functions on partial spreads , 2014, Des. Codes Cryptogr..

[86]  A. Pott.  A survey on relative difference sets , 1996 .

[87]  D. Knuth.  Finite semifields and projective planes , 1965 .

[88]  Gary McGuire,et al.  Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes , 2010, Des. Codes Cryptogr..

[89]  Yongqiang Li,et al.  A matrix approach for constructing quadratic APN functions , 2014, Des. Codes Cryptogr..

[90]  Robert S. Coulter,et al.  Planar Functions and Planes of Lenz-Barlotti Class II , 1997, Des. Codes Cryptogr..

[91]  G. Lachaud,et al.  The weights of the orthogonals of the extended quadratic binary Goppa codes , 1990, IEEE Trans. Inf. Theory.

[92]  Antonia W. Bluher,et al.  On xq+1+ax+b , 2004, Finite Fields Their Appl..

[93]  Cunsheng Ding,et al.  The weight distribution of a class of linear codes from perfect nonlinear functions , 2006, IEEE Transactions on Information Theory.

[94]  Kaisa Nyberg,et al.  New Links Between Differential and Linear Cryptanalysis , 2015, IACR Cryptol. ePrint Arch..

[95]  A. Adrian Albert,et al.  Generalized twisted fields. , 1961 .

[96]  Dmitry Fon-Der-Flaass,et al.  Crooked Functions, Bent Functions, and Distance Regular Graphs , 1998, Electron. J. Comb..

[97]  G. Lunardon,et al.  Symplectic semifield spreads of PG(5, q) and the veronese surface , 2011 .

[98]  Yves Edel.  On quadratic APN functions and dimensional dual hyperovals , 2010, Des. Codes Cryptogr..

[99]  Claude Carlet,et al.  Classes of Quadratic APN Trinomials and Hexanomials and Related Structures , 2008, IEEE Transactions on Information Theory.

[100]  Norman L. Johnson,et al.  Handbook of Finite Translation Planes , 2007 .

[101]  Hans Dobbertin,et al.  Almost Perfect Nonlinear Power Functions on GF(2n): The Welch Case , 1999, IEEE Trans. Inf. Theory.

[102]  Elodie Leducq,et al.  Functions which are PN on infinitely many extensions of $$\mathbb {F}_p,\,p$$Fp,p odd , 2010, Des. Codes Cryptogr..

[103]  Tadao Kasami,et al.  The Weight Enumerators for Several Clauses of Subcodes of the 2nd Order Binary Reed-Muller Codes , 1971, Inf. Control..

[104]  Claude Carlet,et al.  New classes of almost bent and almost perfect nonlinear polynomials , 2006, IEEE Transactions on Information Theory.

[105]  Richard M. Wilson,et al.  Hyperplane Sections of Fermat Varieties in P3 in Char.2 and Some Applications to Cyclic Codes , 1993, AAECC.

[106]  Jürgen Bierbrauer.  Projective polynomials, a projection construction and a family of semifields , 2016, Des. Codes Cryptogr..

[107]  Giuseppe Marino,et al.  Towards the classification of rank 2 semifields 6-dimensional over their center , 2011, Des. Codes Cryptogr..

[108]  Marie Henderson,et al.  Planar polynomials for commutative semifields with specified nuclei , 2007, Des. Codes Cryptogr..

[109]  François Rodier.  Some More Functions That Are Not APN Infinitely Often. The Case of Kasami exponents , 2011, ArXiv.

[110]  Cunsheng Ding,et al.  Highly nonlinear mappings , 2004, J. Complex..

[111]  Stephen D. Cohen,et al.  Commutative semifields, two dimensional over their middle nuclei , 1982 .

[112]  Anne Canteaut,et al.  Weight Divisibility of Cyclic Codes, Highly Nonlinear Functions on F2m, and Crosscorrelation of Maximum-Length Sequences , 2000, SIAM J. Discret. Math..

[113]  Kaisa Nyberg,et al.  Differentially Uniform Mappings for Cryptography , 1994, EUROCRYPT.

[114]  Kaisa Nyberg,et al.  Perfect nonlinear functions and cryptography , 2015, Finite Fields Their Appl..

[115]  Giampaolo Menichetti.  On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field , 1977 .

[116]  Florian Caullery.  A new large class of functions not APN infinitely often , 2014, Des. Codes Cryptogr..

[117]  Michael J. Ganley.  On a paper of dembowski and ostrom , 1976 .

[118]  Ulrich Dempwolff.  More translation planes and semifields from Dembowski–Ostrom polynomials , 2013, Des. Codes Cryptogr..

[119]  Claude Carlet,et al.  An infinite class of quadratic APN functions which are not equivalent to power mappings , 2006, 2006 IEEE International Symposium on Information Theory.

[120]  Vladimir D. Tonchev,et al.  Affine geometry designs, polarities, and Hamada's conjecture , 2011, J. Comb. Theory, Ser. A.

[121]  O. S. Rothaus,et al.  On "Bent" Functions , 1976, J. Comb. Theory, Ser. A.

[122]  Alexander Pott,et al.  A new almost perfect nonlinear function which is not quadratic , 2008, Adv. Math. Commun..

[123]  Semifield Planes of Order 81 , 2008 .

[124]  Beniamino Segre,et al.  Ovali ed altre curve nei piani di Galois di caratteristica due , 1971 .

[125]  J. Dillon.  Elementary Hadamard Difference Sets , 1974 .

[126]  Anne Canteaut,et al.  Differential properties of power functions , 2010, 2010 IEEE International Symposium on Information Theory.

[127]  Xueli Wang,et al.  Perfect nonlinear binomials and their semifields , 2009, Finite Fields Their Appl..

[128]  N. Hamada,et al.  On the $p$-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes , 1973 .

[129]  E. Lander.  Symmetric Designs: An Algebraic Approach , 1983 .

[130]  Leonard Eugene Dickson,et al.  Linear algebras with associativity not assumed , 1935 .

[131]  Guang Gong,et al.  Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar , 2005 .

[132]  Claude Carlet,et al.  Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems , 1998, Des. Codes Cryptogr..

[133]  John F. Dillon,et al.  Multiplicative Difference Sets via Additive Characters , 1999, Des. Codes Cryptogr..

[134]  Tim Penttila,et al.  Dimensional dual arcs – a survey , 2006 .

[135]  Tim Penttila,et al.  Ovoids of Parabolic Spaces , 2000 .

[136]  Keqin Feng,et al.  Value Distributions of Exponential Sums From Perfect Nonlinear Functions and Their Applications , 2007, IEEE Transactions on Information Theory.

[137]  Thierry P. Berger,et al.  On Almost Perfect Nonlinear Functions Over$mmb F_2^n$ , 2006, IEEE Transactions on Information Theory.

[138]  Kaisa Nyberg,et al.  Perfect Nonlinear S-Boxes , 1991, EUROCRYPT.

[139]  n-Dimensional algebras over a field with a cyclic extension of degree n , 1996 .

[140]  Jürgen Bierbrauer.  New semifields, PN and APN functions , 2010, Des. Codes Cryptogr..

[141]  Anne Canteaut,et al.  Differential Properties of ${x\mapsto x^{2^{t}-1}}$ , 2011, IEEE Transactions on Information Theory.

[142]  James A. Davis,et al.  A Unifying Construction for Difference Sets , 1997, J. Comb. Theory, Ser. A.

[143]  Fred Piper,et al.  Quasiregular collineation groups of finite projective planes , 1967 .

[144]  Philippe Langevin,et al.  Counting all bent functions in dimension eight 99270589265934370305785861242880 , 2011, Des. Codes Cryptogr..

[145]  Xiang-dong Hou.  A Note on the Proof of Niho's Conjecture , 2004, SIAM J. Discret. Math..

[146]  Vincent Rijmen,et al.  The Design of Rijndael: AES - The Advanced Encryption Standard , 2002 .

[147]  A. Pott,et al.  Semifields, relative difference sets, and bent functions , 2014, 1401.3294.

[148]  Xiang-Dong Hou.  On the dual of a Coulter-Matthews bent function , 2008, Finite Fields Their Appl..

[149]  Yves Aubry,et al.  Differentially 4-uniform functions , 2009, 0907.1734.

[150]  Tor Helleseth,et al.  On Isotopisms of Commutative presemifields and CCZ-Equivalence of Functions , 2011, Int. J. Found. Comput. Sci..

[151]  Siu Lun Ma,et al.  A survey of partial difference sets , 1994, Des. Codes Cryptogr..

[152]  Claude Carlet,et al.  Quadratic zero-difference balanced functions, APN functions and strongly regular graphs , 2014, Des. Codes Cryptogr..

[153]  Aidan Roy,et al.  Two Characterizations of Crooked Functions , 2008, IEEE Transactions on Information Theory.

[154]  Yue Zhou,et al.  CCZ and EA equivalence between mappings over finite Abelian groups , 2013, Des. Codes Cryptogr..

[155]  Bernhard Schmidt,et al.  Characters and Cyclotomic Fields in Finite Geometry , 2002 .

[156]  Arthur Reifart.  The classification of the translation planes of order 16, II , 1983 .

[157]  Jürgen Bierbrauer,et al.  Crooked binomials , 2008, Des. Codes Cryptogr..

[158]  Alexander Pott,et al.  Some Theorems on Planar Mappings , 2008, WAIFI.

[159]  Qing Xiang,et al.  Pseudo-Paley graphs and skew Hadamard difference sets from presemifields , 2007, Des. Codes Cryptogr..

[160]  Eimear Byrne,et al.  Fourier Spectra of Binomial APN Functions , 2009, SIAM J. Discret. Math..

[161]  Satoshi Yoshiara.  Equivalences of quadratic APN functions , 2012 .

[162]  Moises Delgado,et al.  On the Conjecture on APN Functions , 2012, ArXiv.

[163]  Rudolf Lide,et al.  Finite fields , 1983 .

[164]  Antonia W. Bluher.  On existence of Budaghyan-Carlet APN hexanomials , 2013, Finite Fields Their Appl..

[165]  Yue Zhou,et al.  Switching Construction of Planar Functions on Finite Fields , 2010, WAIFI.

[166]  Felix Lazebnik,et al.  On the classification of planar monomials over fields of square order , 2012, Finite Fields Their Appl..

[167]  Yue Zhou.  (2n,2n,2n,1)‐Relative Difference Sets and Their Representations , 2013 .

[168]  John F. Dillon.  On the dimension of an APN code , 2011, Cryptography and Communications.

[169]  Yue Zhou,et al.  Planar functions over fields of characteristic two , 2013, ArXiv.

[170]  Markus J. Stroppel,et al.  Polarities and unitals in the Coulter–Matthews planes , 2010, Des. Codes Cryptogr..