Abstract
We study Boolean functions derived from Fermat quotients modulo p using the Legendre symbol. We prove bounds on several complexity measures for these Boolean functions: the nonlinearity, sparsity, average sensitivity, and combinatorial complexity. Our main tools are bounds on character sums of Fermat quotients modulo p.
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Brandstätter, N., Lange, T., Winterhof, A.: On the non-linearity and sparsity of Boolean functions related to the discrete logarithm in finite fields of characteristic two. In: Coding and Cryptography. Lect. Notes Comput. Sci., vol. 3969, pp. 135–143. Springer, Berlin (2006)
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, pp. 257–397. Cambridge University Press (2010)
Carlet, C., Helleseth, T.: Sequences, Boolean functions, and cryptography. In: Boztas, S., et al. (eds.) CRC Handbook of Sequences, Codes and Applications. Chapman and Hall/CRC Press (to appear)
Chen, Z., Ostafe, A., Winterhof, A.: Structure of pseudorandom numbers derived from Fermat quotients. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. Lect. Notes Comput. Sci. vol. 6087, pp. 73–85. Springer, Berlin (2010)
Ernvall, R., Metsänkylä, T.: On the p-divisibility of Fermat quotients. Math. Comput. 66(219), 1353–1365 (1997)
Gomez, D., Winterhof, A.: Multiplicative character sums of Fermat quotients and pseudorandomness. Period. Math. Hung. (to appear)
Granville, A.: Some conjectures related to Fermat’s last theorem. Number Theory, pp. 177–192. W. de Gruyter, NY (1990)
Heath-Brown, D.: An estimate for Heilbronn’s exponential sum. In: Analytic Number Theory, Proc. Conf. in Honor of Heini Halberstam, pp. 541–463. Birkhäuser, Boston (1996)
Lange, T., Winterhof, A.: Interpolation of the discrete logarithm in F q by Boolean functions and by polynomials in several variables modulo a divisor of q − 1. International Workshop on Coding and Cryptography (WCC 2001) (Paris). Discrete Appl. Math. 128(1), 193–206 (2003)
Lange, T., Winterhof, A.: Incomplete character sums over finite fields and their application to the interpolation of the discrete logarithm by Boolean functions. Acta Arith. 101(3), 223–229 (2002)
Ostafe, A., Shparlinski, I.: Pseudorandomness and dynamics of Fermat quotients. SIAM J. Discrete Math. 25, 50–71 (2011)
Rodier, F.: Asymptotic nonlinearity of Boolean functions. Des. Codes Cryptogr. 40(1), 59–70 (2006)
Shparlinski, I.E.: Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness. Birkhäuser (2003)
Shparlinski, I.E.: Character sums of Fermat quotients. Quart. J. Math. (to appear)
Shparlinski, I.E.: Bounds of multiplicative character sums with Fermat quotients of primes. Bull. Aust. Math. Soc. (to appear)
Acknowledgements
The authors wish to thank Igor Shparlinski for pointing to the problem of estimating the nonlinearity of these Boolean functions. This work was written during a visit of the first author to RICAM. He wishes to thank the Austrian Academy of Sciences for hospitality and financial support.
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Aly, H., Winterhof, A. Boolean functions derived from Fermat quotients. Cryptogr. Commun. 3, 165–174 (2011). https://doi.org/10.1007/s12095-011-0043-5
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DOI: https://doi.org/10.1007/s12095-011-0043-5