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Abstract

In this chapter we give a short description of all the results presented in this monograph.

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References

  1. A. A. Albert. On nonassociative division algebras. Trans. Amer. Math. Soc. 72, pp. 296–309, 1952.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Barlotti. Le possibili configurazioni del sistema delle coppie punto-retta (A; a) per cui un piano grafico risulta (A; a)-transitivo. Boll. Un. Mat. Ital. 12, 212–226, 1957.

    MATH  MathSciNet  Google Scholar 

  3. T. Bending, D. Fon-Der-Flaass. Crooked functions, bent functions and distance-regular graphs. Electron. J. Comb., 5 (R34), 14, 1998.

    Google Scholar 

  4. J. Bierbrauer. Commutative semifields from projection mappings. Designs, Codes and Cryptography, 61(2), pp. 187–196, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Biham and A. Shamir. Differential Cryptanalysis of DES-like Cryptosystems. Journal of Cryptology 4, no. 1, pp. 3–72, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  6. 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.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. Bracken, E. Byrne, N. Markin, G. McGuire. A Few More Quadratic APN Functions. Cryptography and Communications 3(1), pp. 43–53, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  8. 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.

    Google Scholar 

  9. 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.

    Google Scholar 

  10. 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.

    Google Scholar 

  11. 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.

    Article  MATH  MathSciNet  Google Scholar 

  12. 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.

    Google Scholar 

  13. 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.

    Google Scholar 

  14. L. Budaghyan and T. Helleseth. New perfect nonlinear multinomials over \(F\) 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.

    Google Scholar 

  15. 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

  16. 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.

    Article  MATH  MathSciNet  Google Scholar 

  17. 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.

    Google Scholar 

  18. 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.

    Article  MATH  MathSciNet  Google Scholar 

  19. 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.

    Google Scholar 

  20. 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.

    Article  MATH  MathSciNet  Google Scholar 

  21. L. Budaghyan, C. Carlet, T. Helleseth. On bent functions associated to AB functions. Proceedings of IEEE Information Theory Workshop, ITW'11, Paraty, Brazil, Oct. 2011.

    Google Scholar 

  22. 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.

    Google Scholar 

  23. 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.

    Article  MathSciNet  Google Scholar 

  24. 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.

    Article  MATH  MathSciNet  Google Scholar 

  25. 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.

    Article  MATH  MathSciNet  Google Scholar 

  26. 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.

    Google Scholar 

  27. 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.

    Article  MATH  MathSciNet  Google Scholar 

  28. L. E. Dickson. On commutative linear algebras in which division is always uniquely possible. Trans. Amer. Math. Soc 7, pp. 514–522, 1906.

    Article  MATH  MathSciNet  Google Scholar 

  29. J. F. Dillon. Elementary Hadamard Difference sets. Ph. D. Thesis, Univ. of Maryland, 1974.

    Google Scholar 

  30. J. F. Dillon. APN Polynomials and Related Codes. Polynomials over Finite Fields and Applications, Banff International Research Station, Nov. 2006.

    Google Scholar 

  31. 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.

    Article  MATH  MathSciNet  Google Scholar 

  32. Y. Edel. APN functions and dual hyperovals. NATO Advanced Research Workshop, 2008.

    Google Scholar 

  33. 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.

    Article  MATH  MathSciNet  Google Scholar 

  34. R. Gold. Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inform. Theory, 14, pp. 154–156, 1968.

    Article  MATH  Google Scholar 

  35. M. Hall. Projective planes. Trans. Amer. Math. Soc. 54, pp. 229–277, 1943.

    Article  MATH  MathSciNet  Google Scholar 

  36. D. E. Knuth. Finite semifields and projective planes. J. Algebra 2, pp. 182–217, 1965.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  38. H. Lenz. Zur Begrundung der analytischen Geometrie. S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., pp. 17–72, 1954.

    Google Scholar 

  39. G. Lunardon, G. Marino, O. Polverion, R. Trombetti. Symplectic spreads and quadric Veroneseans. Manuscript, 2009.

    Google Scholar 

  40. M. Matsui. Linear cryptanalysis method for DES cipher. Advances in Cryptology-EUROCRYPT'93, LNCS, Springer-Verlag, pp. 386–397, 1994.

    Google Scholar 

  41. K. Nyberg. Differentially uniform mappings for cryptography. Advances in Cryptography, EUROCRYPT'93, Lecture Notes in Computer Science 765, pp. 55–64, 1994.

    Google Scholar 

  42. J. D. Olsen, R. A. Scholtz, L. R. Welch. Bent-function sequences. IEEE Trans. Inform. Theory IT-28, pp. 858–864, 1982.

    Google Scholar 

  43. K. H. Parshall. In pursuit of the finite division algebra theorem and beyond. Arch. Internat. Hist. Sci. 33, pp. 274–299, 1983.

    MATH  MathSciNet  Google Scholar 

  44. A. Pott, Y. Zhou. CCZ and EA equivalence between mappings over finite Abelian groups. Des. Codes Cryptography 66(1–3), pp. 99–109, 2013.

    Article  MATH  MathSciNet  Google Scholar 

  45. 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.

    Article  MATH  MathSciNet  Google Scholar 

  46. 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.

    Article  MATH  MathSciNet  Google Scholar 

  47. J. H. M. Wedderburn. A theorem on finite algebras. Trans. Amer. Math. Soc. 6, pp. 349–352, 1905.

    Article  MATH  MathSciNet  Google Scholar 

  48. S. Yoshiara. Notes on APN functions, semibiplanes and dimensional dual hyperovals. Des. Codes Cryptography 56(2–3), pp. 197–218, 2010.

    Article  MATH  MathSciNet  Google Scholar 

  49. 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.

    Google Scholar 

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Correspondence to Lilya Budaghyan .

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Budaghyan, L. (2014). Introduction. In: Construction and Analysis of Cryptographic Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-12991-4_1

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  • DOI: https://doi.org/10.1007/978-3-319-12991-4_1

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