Abstract
We determine the linear complexity of a family of p 2-periodic binary threshold sequences derived from Fermat quotients modulo an odd prime p, where p satisfies \({2^{p-1} \not\equiv 1 ({\rm mod}\, {p^2})}\) . The linear complexity equals p 2 − p or p 2 − 1, depending whether \({p \equiv 1}\) or 3 (mod 4). Our research extends the results from previous work on the linear complexity of the corresponding binary threshold sequences when 2 is a primitive root modulo p 2. Moreover, we present a partial result on their linear complexities for primes p with \({2^{p-1} \equiv 1 ({\rm mod} \,{p^2})}\) . However such so called Wieferich primes are very rare.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agoh T., Dilcher K., Skula L.: Fermat quotients for composite moduli. J. Number Theory 66(1), 29–50 (1997)
Aly H., Meidl W., Winterhof A.: On the k-error linear complexity of cyclotomic sequences. J. Math. Crypt 1(3), 283–296 (2007)
Aly H., Winterhof A.: On the k-error linear complexity over \({\mathbb{F}_p}\) of Legendre and Sidelnikov sequences. Desi. Codes Cryptogr. 40(3), 369–374 (2006)
Chen Z., Hu L., Du X.: Linear complexity of some binary sequences derived from Fermat quotients. China Commun. (2012) (to appear).
Chen Z., Ostafe A., Winterhof A.: Structure of pseudorandom numbers derived from Fermat quotients. In: Proc. of WAIFI 2010, LNCS, vol.6087, pp.73–85. Springer-Verlag, Heidelberg (2010).
Chen Z., Winterhof A.: Additive character sums of polynomial quotients. Preprint (2011).
Crandall R., Dilcher K., Pomerance C.: A search for Wieferich and Wilson primes. Math. Comp. 66(217), 433–449 (1997)
Ding C.: Autocorrelation values of generalized cyclotomic sequences of order two. IEEE Trans. Inform. Theory 44(4), 1699–1702 (1998)
Du X., Klapper A., Chen Z.: Linear complexity of pseudorandom sequences generated by Fermat quotients and their generalizations. Inform. Process. Lett. 112(6), 233–237 (2012)
Ernvall R., Metsänkyl"a T.: On the p-divisibility of Fermat quotients. Math. Comp 66(219), 1353–1365 (1997)
Gomez D., Winterhof A.: Multiplicative character sums of Fermat quotients and pseudorandom sequences. Period. Math. Hungar. (2011). (to appear)
Granville A.: Some conjectures related to Fermat’s last theorem. In: Number theory, pp. 177–192. Walter de Gruyter, NY (1990).
Lidl R., Niederreiter H.: Finite fields. Addison-Wesley, Reading (1983).
Meidl W.: How many bits have to be changed to decrease the linear complexity. Des. Codes Cryptogr 33(2), 109–122 (2004)
Ostafe A., Shparlinski I.E.: Pseudorandomness and dynamics of Fermat quotients. SIAM J. Discr. Math. 25(1), 50–71 (2011)
Shanks D.: Solved and Unsolved Problems in Number Theory. Chelsea Publishing Company, New York (1978) Second Edition
Shparlinski I.E.: Character sums with Fermat quotient. Quart. J. Math. Oxf 62(4), 1031–1043 (2011)
Shparlinski I.E.: Bounds of multiplicative character sums with Fermat quotients of primes. Bull. Aust. Math. Soc 83(3), 456–462 (2011)
Shparlinski I.E.: On the value set of Fermat quotients. Proc. Amer. Math. Soc. (2011) (to appear).
Shparlinski I.E.: Fermat quotients: exponential sums, value set and primitive roots. Bull. Lond. Math. Soc. 43(6), 1228–1238 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Winterhof.
Rights and permissions
About this article
Cite this article
Chen, Z., Du, X. On the linear complexity of binary threshold sequences derived from Fermat quotients. Des. Codes Cryptogr. 67, 317–323 (2013). https://doi.org/10.1007/s10623-012-9608-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-012-9608-3