Abstract
In this chapter we present all necessary definitions and preliminary results on cryptographic functions which are used throughout this book.
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References
A. A. Albert. On nonassociative division algebras. Trans. Amer. Math. Soc. 72, pp. 296–309, 1952.
A. A. Albert. Generalized twisted fields. Pacific J. Math. 11, pp. 1–8, 1961.
T. Bending, D. Fon-Der-Flaass. Crooked functions, bent functions and distance-regular graphs. Electron. J. Comb., 5 (R34), 1–4, 1998.
T. Beth and C. Ding. On almost perfect nonlinear permutations. Advances in Cryptology-EUROCRYPT'93, Lecture Notes in Computer Science, 765, Springer-Verlag, New York, pp. 65–76, 1993.
J. Bierbrauer. New semifields, PN and APN functions. Designs, Codes and Cryptography, v. 54, pp. 189–200, 2010.
J. Bierbrauer. Commutative semifields from projection mappings. Designs, Codes and Cryptography, 61(2), pp. 187–196, 2011.
E. Biham and A. Shamir. Differential Cryptanalysis of DES-like Cryptosystems. Journal of Cryptology 4, no. 1, pp. 3–72, 1991.
C. Bracken, Z. Zha. On the Fourier Spectra of the Infinite Families of Quadratic APN Functions. Finite Fields and Their Applications 18(3), pp. 537–546, 2012.
C. Bracken, E. Byrne, N. Markin, G. McGuire. Determining the Nonlinearity of a New Family of APN Functions. Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Lecture Notes in Computer Science, Vol 4851, Springer-Verlag, pp. 72–79, 2007.
C. Bracken, E. Byrne, N. Markin, G. McGuire. On the Walsh Spectrum of a New APN Function. Cryptography and Coding, Lecture Notes in Computer Science, Vol 4887, Springer-Verlag, pp. 92–98, 2007.
C. Bracken, E. Byrne, N. Markin, G. McGuire. New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields and Their Applications 14(3), pp. 703–714, 2008.
C. Bracken, E. Byrne, N. Markin, G. McGuire. On the Fourier spectrum of Binomial APN functions.SIAM journal of Discrete Mathematics, 23(2), pp. 596–608, 2009.
C. Bracken, E. Byrne, N. Markin, G. McGuire. A Few More Quadratic APN Functions. Cryptography and Communications 3(1), pp. 43–53, 2011.
K. A. Browning, J. F. Dillon, R. E. Kibler, M. T. McQuistan. APN Polynomials and Related Codes. Journal of Combinatorics, Information and System Science, Special Issue in honor of Prof. D.K Ray-Chaudhuri on the occasion of his 75th birthday, vol. 34, no. 1–4, pp. 135–159,2009.
K. A. Browning, J. F. Dillon, M. T. McQuistan, A. J. Wolfe. An APN Permutation in Dimension Six. Post-proceedings of the 9-th International Conference on Finite Fields and Their Applications Fq'09, Contemporary Math., AMS, v. 518, pp. 33–42, 2010.
M. Brinkman and G. Leander. On the classification of APN functions up to dimension five. Proceedings of the International Workshop on Coding and Cryptography 2007 dedicated to the memory of Hans Dobbertin, pp. 39–48, Versailles, France, 2007.
L. Budaghyan. The Simplest Method for Constructing APN Polynomials EA-Inequivalent to Power Functions. Proceedings of First International Workshop on Arithmetic of Finite Fields, WAIFI 2007 Lecture Notes in Computer Science, 4547, pp. 177–188, 2007.
L. Budaghyan and C. Carlet. Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. IEEE Trans. Inform. Theory, vol. 54, no. 5, pp. 2354–2357, May 2008.
L. Budaghyan and C. Carlet. On CCZ-equivalence and its use in secondary constructions of bent functions. Preproceedings of International Workshop on Coding and Cryptography WCC 2009, pp. 19–36, 2009.
L. Budaghyan and C. Carlet. CCZ-equivalence of single and multi output Boolean functions. Post-proceedings of the 9-th International Conference on Finite Fields and Their Applications Fq'09, Contemporary Math., AMS, v. 518, pp. 43–54, 2010.
L. Budaghyan and T. Helleseth. New perfect nonlinear multinomials over F p2k for any odd prime p. Proceedings of the International Conference on Sequences and Their Applications SETA 2008, Lecture Notes in Computer Science 5203, pp. 403–414, Lexington, USA, Sep. 2008.
L. Budaghyan and T. Helleseth. On Isotopisms of Commutative Presemifields and CCZ-Equivalence of Functions. Special Issue on Cryptography of International Journal of Foundations of Computer Science, v. 22(6), pp. 1243–1258, 2011. Preprint at http://eprint.iacr.org/2010/507
L. Budaghyan and T. Helleseth. New commutative semifields defined by new PN multinomials. Cryptography and Communications: Discrete Structures, Boolean Functions and Sequences, v. 3(1), pp. 1–16, 2011.
L. Budaghyan, C. Carlet, A. Pott. New Classes of Almost Bent and Almost Perfect Nonlinear Functions. IEEE Trans. Inform. Theory, vol. 52, no. 3, pp. 1141–1152, March 2006.
L. Budaghyan, C. Carlet, G. Leander. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inform. Theory, 54(9), pp. 4218–4229, 2008.
L. Budaghyan, C. Carlet, G. Leander. On inequivalence between known power APN functions. Proceedings of the International Workshop on Boolean Functions: Cryptography and Applications, BFCA 2008, Copenhagen, Denmark, May 2008.
L. Budaghyan, C. Carlet, G. Leander. On a construction of quadratic APN functions. Proceedings of IEEE Information Theory Workshop, ITW'09, pp. 374–378, Taormina, Sicily, Oct. 2009.
L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150–159, April 2009.
L. Budaghyan, C. Carlet, T. Helleseth, A. Kholosha. Generalized Bent Functions and Their Relation to Maiorana-McFarland Class. Proceedings of the IEEE International Symposium on Information Theory, ISIT 2012, Cambridge, MA, USA, 1–6 July 2012.
L. Budaghyan, C. Carlet, T. Helleseth, A. Kholosha, S. Mesnager. Further Results on Niho Bent Functions. IEEE Trans. Inform. Theory, 58(11), pp. 6979–6985, 2012.
A. Canteaut, P. Charpin, H. Dobbertin. Weight divisibility of cyclic codes, highly nonlinear functions on \(\mathbb{F}_{2^m}\), and crosscorrelation of maximum-length sequences. SIAM Journal on Discrete Mathematics, 13(1), pp. 105–138, 2000.
A. Canteaut, P. Charpin and H. Dobbertin. Binary m-sequences with three-valued crosscorrelation: A proof of Welch’s conjecture. IEEE Trans. Inform. Theory, 46 (1), pp. 4–8, 2000.
A. Canteaut, P. Charpin, and G. M. Kyureghyan, “A new class of monomial bent functions,“ Finite Fields Appl., vol. 14, no. 1, pp. 221–241, Jan. 2008.
C. Carlet. Boolean Functions for Cryptography and Error Correcting Codes. Chapter of the monography Boolean Methods and Models, Yves Crama and Peter Hammer eds, Cambridge University Press, pp. 257–397, 2010.
C. Carlet. Vectorial Boolean Functions for Cryptography. Chapter of the monography Boolean Methods and Models, Yves Crama and Peter Hammer eds, Cambridge University Press, pp. 398–469, 2010.
C. Carlet. Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions. Designs, Codes and Cryptography, v. 59(1–3), pp. 89–109, 2011.
C. Carlet and S. Mesnager, “On Dillon’s class H of bent functions, Niho bent functions and o-polynomials,“ J. Combin. Theory Ser. A, vol. 118, no. 8, pp. 2392–2410, Nov. 2011.
C. Carlet, P. Charpin and V. Zinoviev. Codes, bent functions and permutations suitable for DES-like cryptosystems. Designs, Codes and Cryptography, 15(2), pp. 125–156, 1998.
F. Chabaud and S. Vaudenay. Links between differential and linear cryptanalysis, Advances in Cryptology -EUROCRYPT'94, LNCS, Springer-Verlag, New York, 950, pp. 356–365, 1995.
P. Charpin, G. Kyureghyan. On a class of permutation polynomials over \(\mathbb{F}_{2^n}\). Proceedings of SETA 2008, Lecture Notes in Computer Science 5203, pp. 368–376, 2008.
P. Charpin and G. M. Kyureghyan. Cubic monomial bent functions: A subclass of \(\mathcal{M}\). SIAM Journal on Discrete Mathematics, vol. 22, no. 2, pp. 650–665, 2008.
Y. M. Chee, Y. Tan, and X. D. Zhang, “Strongly regular graphs constructed from p-ary bent functions,” J. Algebraic Combin., vol. 34, no. 2, pp.251–266, Sep. 2011.
S. D. Cohen and M. J. Ganley. Commutative semifields, two-dimensional over there middle nuclei. J. Algebra 75, pp. 373–385, 1982.
R. S. Coulter and M. Henderson. Commutative presemifields and semifields. Advances in Math. 217, pp. 282–304, 2008.
R. S. Coulter and R. W. Matthews. Planar functions and planes of Lenz-Barlotti class II. Des., Codes, Cryptogr. 10, pp. 167–184, 1997.
R. S. Coulter, M. Henderson, P. Kosick. Planar polynomials for commutative semifields with specified nuclei. Des. Codes Cryptogr. 44, pp. 275–286, 2007.
T. Cusick and H. Dobbertin. Some new 3-valued crosscorrelation functions of binary m-sequences. IEEE Trans. Inform. Theory, 42, pp.1238–1240, 1996.
J. Daemen andV. Rijmen. AES proposal: Rijndael. http://csrc.nist.gov/encryption/aes/rijndael/Rijndael.pdf, 1999.
P. Dembowski and T. Ostrom. Planes of order n with collineation groups of order n 2. Math. Z. 103, pp. 239–258, 1968.
L. E. Dickson. Linear algebras in which division is always uniquely possible. Trans. Amer. Math. Soc 7, pp. 370–390, 1906.
L. E. Dickson. On commutative linear algebras in which division is always uniquely possible. Trans. Amer. Math. Soc 7, pp. 514–522, 1906.
L. E. Dickson. Linear algebras with associativity not assumed. Duke Math. J. 1, pp. 113–125, 1935.
J. F. Dillon. A survey of bent functions. NSA Technical Journal Special Issue, pp. 191–215, 1972.
J. F. Dillon. Elementary Hadamard Difference sets. Ph. D. Thesis, Univ. of Maryland, 1974.
J. F. Dillon and H. Dobbertin, “New cyclic difference sets with Singer parameters,“ Finite Fields Appl., vol. 10, no. 3, pp. 342–389, Jul. 2004.
C. Ding and J. Yuan. A new family of skew Paley-Hadamard difference sets. J. Comb. Theory Ser. A 133, pp. 1526–1535, 2006.
H. Dobbertin. One-to-One Highly Nonlinear Power Functions on \(GF(2^n)\). Appl. Algebra Eng. Commun. Comput. 9 (2), pp. 139–152, 1998.
H. Dobbertin. Almost perfect nonlinear power functions over \(GF(2^n)\): the Niho case. Inform. and Comput., 151, pp. 57–72, 1999.
H. Dobbertin. Almost perfect nonlinear power functions over \(GF(2^n)\): the Welch case. IEEE Trans. Inform. Theory, 45, pp. 1271–1275, 1999.
H. Dobbertin. Almost perfect nonlinear power functions over \(GF(2^n)\): a new case for n divisible by 5. Proceedings of Finite Fields and Applications FQ5, pp. 113–121, 2000.
H. Dobbertin. Private communication. 2004.
H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke, and P. Gaborit, “Construction of bent functions via Niho power functions,“ J. Combin. Theory Ser. A, vol. 113, no. 5, pp. 779–798, Jul. 2006.
Y. Edel. Quadratic APN functions as subspaces of alternating bilinear forms. Contact Forum Coding Theory and Cryptography III, Belgium (2009), pp. 11–24, 2011.
Y. Edel and A. Pott. A new almost perfect nonlinear function which is not quadratic. Advances in Mathematics of Communications 3, no. 1, pp. 59–81, 2009.
Y. Edel, G. Kyureghyan and A. Pott. A new APN function which is not equivalent to a power mapping. IEEE Trans. Inform. Theory, vol. 52, no. 2, pp. 744–747, Feb. 2006.
M. J. Ganley. Central weak nucleus semifields. European J. Combin.2, pp. 339–347, 1981.
R. Gold. Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inform. Theory, 14, pp. 154–156, 1968.
G. Gong, T. Helleseth, H. Hu, and A. Kholosha. On the dual of certain ternary weakly regular bent functions. IEEE Trans. Inf. Theory, 58(4), pp. 2237–2243, 2012.
T. Helleseth and A. Kholosha. Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory, vol. 52, no. 5, pp. 2018–2032, May 2006.
T. Helleseth and A. Kholosha. On the dual of monomial quadratic p-ary bent functions. Sequences, Subsequences, and Consequences, ser. Lecture Notes in Computer Science, S. Golomb, G. Gong, T. Helleseth, and H.-Y. Song, Eds., vol. 4893. Berlin: Springer-Verlag, 2007, pp. 50–61.
T. Helleseth and A. Kholosha. Sequences, bent functions and Jacobsthal sums. Sequences and Their Applications—SETA 2010, ser. Lecture Notes in Computer Science, C. Carlet and A. Pott, Eds., vol. 6338. Berlin: Springer-Verlag, 2010, pp. 416–429.
T. Helleseth and A. Kholosha. New binomial bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4646–4652, Sep. 2010.
T. Helleseth and A. Kholosha. Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums. Cryptography and Communications, vol. 3, no. 4, pp. 281–291, Dec. 2011.
T. Helleseth and D. Sandberg. Some power mappings with low differential uniformity. Applic. Alg. Eng., Commun. Comput. 8, pp. 363–370, 1997.
T. Helleseth, C. Rong and D. Sandberg. New families of almost perfect nonlinear power mappings. IEEE Trans. in Inf. Theory 45, pp. 475–485, 1999.
T. Helleseth, H. D. L. Hollmann, A. Kholosha, Z. Wang, and Q. Xiang, “Proofs of two conjectures on ternary weakly regular bent functions,” IEEE Trans. Inf. Theory, vol. 55, no. 11, pp. 5272–5283, Nov. 2009.
T. Helleseth, A. Kholosha, and S. Mesnager, “Niho bent functions and Subiaco hyperovals,” in Theory and Applications of Finite Fields, ser. Contemporary Mathematics, M. Lavrauw, G. L. Mullen, S. Nikova, D. Panario, and L. Storme, Eds. Providence, Rhode Island: American Mathematical Society, 2012.
H. Hollmann and Q. Xiang. A proof of the Welch and Niho conjectures on crosscorrelations of binary m-sequences. Finite Fields and Their Applications 7, pp. 253–286, 2001.
X.-D. Hou. Affinity of permutations of \(\mathbb{F}_{2^n}\). Proceedings of the Workshop on the Coding and Cryptography 2003, Augot, Charpin and Kabatianski eds, pp. 273–280, 2003.
X.-D. Hou.'p-Ary and q-ary versions of certain results about bent functions and resilient functions. Finite Fields Appl., vol. 10, no. 4, pp. 566–582, Oct. 2004.
H. Janwa and R. Wilson. Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. Proceedings of AAECC-10, LNCS, vol. 673, Berlin, Springer-Verlag, pp. 180–194, 1993.
T. Kasami. The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes. Inform. and Control, 18, pp. 369–394, 1971.
D. E. Knuth. Finite semifields and projective planes. J. Algebra 2, pp. 182–217, 1965.
P. V. Kumar, R. A. Scholtz, and L. R. Welch, “Generalized bent functions and their properties,” J. Combin. Theory Ser. A, vol. 40, no. 1, pp. 90–107, Sep. 1985.
G. Lachaud and J. Wolfmann. The Weights of the Orthogonals of the Extended Quadratic Binary Goppa Codes. IEEE Trans. Inform. Theory, vol. 36, pp. 686–692, 1990.
P. Langevin, G. Leander. Counting all bent functions in dimension eight 99270589265934370305785861242880. Des. Codes Cryptography59(1–3), pp. 193–205, 2011.
G. Leander. Monomial bent functions. IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 738–743, 2006.
G. Leander and A. Kholosha, “Bent functions with \(2^r\) Niho exponents,” IEEE Trans. Inf. Theory, vol. 52, no. 12, pp. 5529–5532, Dec. 2006.
G. Leander and P. Langevin. On exponents with highly divisible Fourier coefficients and conjectures of Niho and Dobbertin. 2007.
Y. Li, M. Wang. Permutation polynomials EA-equivalent to the inverse function over \(GF(2^n)\). Cryptography and Communications 3(3), pp. 175–186, 2011.
Y. Li, M. Wang. The Nonexistence of Permutations EA-Equivalent to Certain AB Functions. IEEE Trans. Inf. Theory, vol. 59, no. 1, pp. 672–679, 2013.
G. Lunardon, G. Marino, O. Polverion, R. Trombetti. Symplectic spreads and quadric Veroneseans. Manuscript, 2009.
M. Matsui. Linear cryptanalysis method for DES cipher. Advances in Cryptology-EUROCRYPT'93, LNCS, Springer-Verlag, pp. 386–397, 1994.
G. Menichetti. On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field. J. Algebra 47, pp. 400–410, 1977.
K. Minami and N. Nakagawa. On planar functions of elementary abelian p-group type. Submitted.
Y. Niho. Multi-valued cross-correlation functions between two maximal linear recursive sequences. Ph.D. dissertation, Dept. Elec. Eng., Univ. Southern California. [USCEE Rep. 409], 1972.
K. Nyberg. Perfect nonlinear S-boxes. Advances in Cryptography, EUROCRYPT'91, Lecture Notes in Computer Science 547, pp. 378–386, 1992.
K. Nyberg. Differentially uniform mappings for cryptography. Advances in Cryptography, EUROCRYPT'93, Lecture Notes in Computer Science 765, pp. 55–64, 1994.
K. Nyberg. S-boxes and Round Functions with Controllable Linearity and Differential Uniformity. Proceedings of Fast Software Encryption 1994, LNCS 1008, pp. 111–130, 1995.
T. Penttila and B. Williams. Ovoids of parabolic spaces. Geom. Dedicata 82, pp. 1–19, 2000.
A. Pott, Y. Zhou. CCZ and EA equivalence between mappings over finite Abelian groups. Des. Codes Cryptography 66(1–3), pp. 99–109, 2013.
A. Pott, Y. Tan, T. Feng, and S. Ling. Association schemes arising from bent functions. Des. Codes Cryptogr., vol. 59, no. 1–3, pp. 319–331, Apr. 2011.
O. S. Rothaus. On “bent” functions. J. Combin. Theory Ser. A, vol. 20, no. 3, pp. 300–305, 1976.
V. Sidelnikov. On mutual correlation of sequences. Soviet Math. Dokl., 12, pp. 197–201, 1971.
Y. Tan, A. Pott, and T. Feng. Strongly regular graphs associated with ternary bent functions. J. Combin. Theory Ser. A, vol. 117, no. 6, pp. 668–682, Aug. 2010.
E.R. van Dam, D. Fon-Der-Flaass. Codes, graphs, and schemes from nonlinear functions. European J. Combin. 24, 85–98, 2003.
G. Weng. Private communications, 2007.
G. Weng, X. Zeng. Further results on planar DO functions and commutative semifields. Des. Codes Cryptogr. 63, pp. 413–423, 2012.
S. Yoshiara. Equivalence of quadratic APN functions. J. Algebr. Comb. 35, pp. 461–475, 2012.
Y. Yu, M. Wang, Y. Li. A matrix approach for constructing quadratic APN functions. Pre-proceedings of the International Conference WCC 2013, Bergen, Norway, 2013.
Y. Zhou. A note on the isotopism of commutative semifields. Preprint, 2010.
Z. Zha, G. Kyureghyan, X. Wang. Perfect nonlinear binomials and their semifields. Finite Fields and Their Applications 15(2), pp. 125–133, 2009.
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Budaghyan, L. (2014). Generalities. In: Construction and Analysis of Cryptographic Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-12991-4_2
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