Abstract
We describe the trace representations of two families of binary sequences derived from the Fermat quotients modulo an odd prime p (one is the binary threshold sequences and the other is the Legendre Fermat quotient sequences) by determining the defining pairs of all binary characteristic sequences of cosets, which coincide with the sets of pre-images modulo p 2 of each fixed value of Fermat quotients. From the defining pairs, we can obtain an earlier result of linear complexity for the binary threshold sequences and a new result of linear complexity for the Legendre Fermat quotient sequences under the assumption of 2p−1 ≢ 1 mod p 2.
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References
Agoh T, Dilcher K, Skula L. Fermat quotients for composite moduli. J Number Theory, 1997, 66: 29–50
Aly H, Winterhof A. Boolean functions derived from Fermat quotients. Cryptogr Commun, 2011, 3: 165–174
Bourgain J, Ford K, Konyagin S, et al. On the divisibility of Fermat quotients. Michigan Math J, 2010, 59: 313–328
Chang M C. Short character sums with Fermat quotients. Acta Arith, 2012, 152: 23–38
Chen Z X, Du X N. On the linear complexity of binary threshold sequences derived from Fermat quotients. Des Codes Cryptogr, 2013, 67: 317–323
Chen ZX, Gómez-Pérez D. Linear complexity of binary sequences derived from polynomial quotients. In: Proceedings of Sequences and Their Applications-SETA 2012, Lecture Notes in Computer Science 7280. Berlin: Springer, 2012. 181–189
Chen Z X, Ostafe A, Winterhof A. Structure of pseudorandom numbers derived from Fermat quotients. In: Proceedings of Arithmetic of Finite Fields-WAIFI 2010, Lecture Notes in Computer Science 6087. Berlin: Springer, 2010. 73–85
Chen Z X, Winterhof A. Additive character sums of polynomial quotients. Contemp Math, 2012, 579: 67–73
Chen Z X, Winterhof A. On the distribution of pseudorandom numbers and vectors derived from Euler-Fermat quotients. Int J Number Theory, 2012, 8: 631–641
Chen Z X, Winterhof A. Interpolation of Fermat quotients. SIAM J Discr Math, 2014, 28: 1–7
Du X N, Chen Z X, Hu L. Linear complexity of binary sequences derived from Euler quotients with prime-power modulus. Inform Process Lett, 2012, 112: 604–609
Du X N, Klapper A, Chen Z X. Linear complexity of pseudorandom sequences generated by Fermat quotients and their generalizations. Inform Process Lett, 2012, 112: 233–237
Ernvall R, Metsänkylä T. On the p-divisibility of Fermat quotients. Math Comp, 1997, 66: 1353–1365
Gómez-Pérez D, Winterhof A. Multiplicative character sums of Fermat quotients and pseudorandom sequences. Period Math Hungar, 2012, 64: 161–168
Ostafe A, Shparlinski I E. Pseudorandomness and dynamics of Fermat quotients. SIAM J Discr Math, 2011, 25: 50–71
Sha M. The arithmetic of Carmichael quotients. arXiv:1108.2579, 2011
Shkredov I D. On Heilbronn’s exponential sum. Quart J Math, 2013, 64: 1221–1230
Shparlinski I E. Character sums with Fermat quotients. Quart J Math, 2011, 62: 1031–1043
Shparlinski I E. Bounds of multiplicative character sums with Fermat quotients of primes. Bull Aust Math Soc, 2011, 83: 456–462
Shparlinski I E. On the value set of Fermat quotients. Proc Amer Math Soc, 2012, 140: 1199–1206
Shparlinski I E. Fermat quotients: Exponential sums, value set and primitive roots. Bull Lond Math Soc, 2011, 43: 1228–1238
Shparlinski I E, Winterhof A. Distribution of values of polynomial Fermat quotients. Finite Fields Appl, 2013, 19: 93–104
Chen Z X, Hu L, Du X N. Linear complexity of some binary sequences derived from Fermat quotients. China Commun, 2012, 9: 105–108
Golomb S W, Gong G. Signal Design for Good Correlation. Cambridge: Cambridge University Press, 2005
Lidl R, Niederreiter H. Finite Fields. 2nd ed. Cambridge: Cambridge University Press, 1997
Dai Z D, Gong G, Song H Y. Trace representation and linear complexity of binary e-th residue sequences. In: Proceedings of International Workshop on Coding and Cryptography, Versailles, 2003. 121–133
Dai Z D, Gong G, Song H Y. A trace representation of binary Jacobi sequences. Discrete Math, 2009, 309: 1517–1527
Dai Z D, Gong G, Song H Y, et al. Trace representation and linear complexity of binary e-th power residue sequences of period p. IEEE Trans Inform Theory, 2011, 57: 1530–1547
Du X N, Yan T J, Xiao G Z. Trace representation of some generalized cyclotomic sequences of length pq. Inform Sci, 2008, 178: 3307–3316
Helleseth T, Kim S H, No J S. Linear complexity over \(\mathbb{F}_p \) and trace representation of Lempel-Cohn-Eastman sequences. IEEE Trans Inform Theory, 2003, 49: 1548–1552
Kim J H, Song H Y. Trace representation of Legendre sequences. Des Codes Cryptogr, 2001, 24: 343–348
Kim J H, Song H Y, Gong G. Trace representation of Hall’s sextic residue sequences of period p ≡ 7 (mod 8). In: No J S, Song H Y, Helleseth T, et al, eds. Mathematical Properties of Sequences and Other Combinatorial Structures. Berlin: Springer, 2003. 23–32
No J S, Lee H K, Chung H, et al. Trace representation of Legendre sequences of Mersenne prime period. IEEE Trans Inform Theory, 1996, 42: 2254–2255
Jungnickel D. Finite Fields: Structure and Arithmetics. Mannheim: Bibliographisches Institute, 1993
Massey J L. Shift register synthesis and BCH decoding. IEEE Trans Inform Theory, 1969, 15: 122–127
Blahut R E. Transform techniques for error control codes. IBM J Res Develop, 1979, 23: 299–315
Crandall R, Dilcher K, Pomerance C. A search for Wieferich and Wilson primes. Math Comp, 1997, 66: 433–449
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Chen, Z. Trace representation and linear complexity of binary sequences derived from Fermat quotients. Sci. China Inf. Sci. 57, 1–10 (2014). https://doi.org/10.1007/s11432-014-5092-x
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DOI: https://doi.org/10.1007/s11432-014-5092-x