Exact 2-divisibility of exponential sums associated to boolean functions

1. Arce-Nazario, R.A., Castro, F.N., González, O.E., Medina, L.A., Rubio, I.M.: New families of balanced symmetric functions and a generalization of Cusick, Li and Stănică’s conjecture. Des. Codes Crypt. https://doi.org/10.1007/s10623-017-0351-7 (2017)

2. Ax, J.: Zeroes of polynomials over finite fields. Amer. J. Math. 86, 255–261 (1964)

   Article  MathSciNet  MATH  Google Scholar 

3. Canteaut, A.: On the weight distributions of optimal cosets of the first-order Reed-Muller codes. IEEE Trans. Inform. Theory 47(1), 407–413 (2001)

   Article  MathSciNet  MATH  Google Scholar 

4. Carlet, C., Guillot, P.: A new representation of Boolean functions. In: Applied algebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999), vol. 1719 of Lecture Notes in Comput. Sci., pp. 94–103. Springer, Berlin (1999)

5. Castro, F., Rubio, I.M.: Exact p-divisibility of exponential sums via the covering method. Proc. Amer. Math. Soc. 143(3), 1043–1056 (2015)

   Article  MathSciNet  MATH  Google Scholar 

6. Castro, F.N., González, O.E., Medina, L.A.: A divisibility approach to the open boundary cases of Cusick-Li-Stănică’s conjecture. Cryptogr. Commun. 7(4), 379–402 (2015)

   Article  MathSciNet  MATH  Google Scholar 

7. Castro, F.N., Medina, L.A.: Linear recurrences and asymptotic behavior of exponential sums of symmetric Boolean functions. Electron. J. Combin. 18(2), Paper 8, 21 (2011)

   MATH  Google Scholar 

8. Castro, F.N., Medina, L.A. , Rubio, I.M.: Exact divisibility of exponential sums over the binary field via the covering method. In: Groups, algebras and applications, vol. 537 of Contemp. Math., pp. 129–136. American Mathematical Society, Providence, RI (2011)

9. Castro, F.N., Randriam, H., Rubio, I., Mattson, H.F. Jr.: Divisibility of exponential sums via elementary methods. J. Number Theory 130(7), 1520–1536 (2010)

   Article  MathSciNet  MATH  Google Scholar 

10. Castro, F.N., Rubio, I.M.: Construction of systems of polynomial equations with exact p-divisibility via the covering method. J. Algebra Appl. 13(6), 1450013, 15 (2014)

    Article  MathSciNet  MATH  Google Scholar 

11. Cusick, T.W., Cheon, Y.: Counting balanced Boolean functions in n variables with bounded degree. Experiment. Math. 16(1), 101–105 (2007)

    Article  MathSciNet  MATH  Google Scholar 

12. Cusick, T.W., Li, Yuan, Stănică, P.: Balanced symmetric functions over GF(p). IEEE Trans. Inform. Theory 54(3), 1304–1307 (2008)

    Article  MathSciNet  MATH  Google Scholar 

13. Cusick, T.W., Li, Yuan, Stănică, P.: On a conjecture for balanced symmetric Boolean functions. J. Math. Cryptol. 3(4), 273–290 (2009)

    Article  MathSciNet  MATH  Google Scholar 

14. Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications. Elsevier/Academic Press, Amsterdam (2009)

    MATH  Google Scholar 

15. Gao, G.-P., Liu, W.-F., Zhang, X.-Y.: The degree of balanced elementary symmetric Boolean functions of 4k + 3 variables. IEEE Trans. Inform. Theory 57(7), 4822–4825 (2011)

    Article  MathSciNet  MATH  Google Scholar 

16. Hou, X.-D.: The covering radius of R(1, 9) in R(4, 9). Des. Codes Cryptogr. 8(3), 285–292 (1996)

    Article  MathSciNet  Google Scholar 

17. Hou, X.-D.: On the covering radius of R(1, m) in R(3, m). IEEE Trans. Inform. Theory 42(3), 1035–1037 (1996)

    Article  MathSciNet  MATH  Google Scholar 

18. Hou, X.-D.: GL(m, 2) acting on R(r, m)/R(r − 1, m). Discrete Math. 149(1–3), 99–122 (1996)

    Article  MathSciNet  MATH  Google Scholar 

19. Hou, X.-D.: p-ary and q-ary versions of certain results about bent functions and resilient functions. Finite Fields Appl. 10(4), 566–582 (2004)

    Article  MathSciNet  MATH  Google Scholar 

20. Katz, D.J.: Sharp p-divisibility of weights in abelian codes over \(\mathbb {Z}/p^{d}\mathbb {Z}\). IEEE Trans. Inform. Theory 54(12), 5354–5380 (2008)

    Article  MathSciNet  MATH  Google Scholar 

21. Katz, N.M.: On a theorem of Ax. Amer. J. Math. 93, 485–499 (1971)

    Article  MathSciNet  MATH  Google Scholar 

22. McGuire, G.: An alternative proof of a result on the weight divisibility of a cyclic code using supersingular curves. Finite Fields Appl. 18(2), 434–436 (2012)

    Article  MathSciNet  MATH  Google Scholar 

23. Moreno, O., Castro, F.N., Mattson, H.F. Jr.: Correction to: “Divisibility properties for covering radius of certain cyclic codes” [IEEE Trans. Inform. Theory 49 (2003), no. 12, 3299–3303; mr2045808] by Moreno and Castro. IEEE Trans. Inform. Theory 52(4), 1798–1799 (2006)

    Article  MathSciNet  Google Scholar 

24. Moreno, O., Moreno, C.J.: The MacWilliams-Sloane conjecture on the tightness of the Carlitz-Uchiyama bound and the weights of duals of BCH codes. IEEE Trans. Inform. Theory 40(6), 1894–1907 (1994)

    Article  MathSciNet  MATH  Google Scholar 

25. Su, W., Tang, X., Pott, A.: A note on a conjecture for balanced elementary symmetric Boolean functions. IEEE Trans. Inform. Theory 59(1), 665–671 (2013)

    Article  MathSciNet  MATH  Google Scholar 

26. Ward, H.N.: Weight polarization and divisibility. Discrete Math. 83(2-3), 315–326 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references