Abstract
Two pseudorandom number generators are devised based on the projective linear group over \(\mathbb{F}_{p^n}\), outputting balanced sequences on \(\mathbb{F}_{p}\) meeting some statistical randomness properties. Sequences generated by the first generator have least period p n + 1 if p n ≥ 7, and linear complexity at least p n − p n − 1. Furthermore, autocorrelation of such sequences oscillates within a low amplitude except for the trivial peaks. If p n ∉ {2,4,8,16}, sequences generated by the second generator have least period p n , linear complexity at least p n − 1 + 1, and good k-error linear complexity when p = 2. If p = 2 and 2n is large enough, then for a binary sequence generated by either generator, a randomly chosen 2-tuple is almost uniformly distributed in {00,01,10,11}, the probability that a randomly chosen 3-tuple is a run of length one is approximately 1/4. For such a binary sequence \(\vec{s}\) and integers 0 < i 1 < i 2 < ⋯ < i k ≤ m, s(t) + s(t + i 1) + s(t + i 2) + ⋯ + s(t + i k ) is equal to 0 or 1 at almost the same probability when m is far less than 2n/2.
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References
Blackburn, S.R., Etzion, T.: Permutation polynomials, de Bruijn sequences, and linear complexity. J. Combin. Theory Ser. A 76, 55–82 (1996)
Blackburn, S.R., Gomez-Perez, D., Gutierrez, J., Shparlinski, I.E.: Predicting nonlinear pseudorandom number generators. Math. Comp. 74, 1471–1494 (2005)
Bombieri, E.: On exponential sums in finite fields. Amer. J. of Math. 88, 11–32 (1966)
De Cannière, C., Preneel, B.: Trivium. In: Robshaw, M., Billet, O. (eds.) New Stream Cipher Designs. LNCS, vol. 4986, pp. 244–266. Springer, Heidelberg (2008)
Chou, W.S.: On inversive maximal period polynomials over finite fields. Appl. Algebra Engrg. Comm. Comput. 6, 245–250 (1995)
Cohen, H., Frey, G., et al.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC Taylor & Francis Group, Boca Raton (2006)
Cusick, T.W., Ding, C., Renvall, A.: Stream Ciphers and Number Theory. North-Holland Mathematical Library, vol. 55. Elsevier, North-Holland (1998)
Ding, C.: Lower bounds on the weight complexities of cascaded binary sequences. In: Seberry, J., Pieprzyk, J.P. (eds.) AUSCRYPT 1990. LNCS, vol. 453, pp. 39–43. Springer, Heidelberg (1990)
Ding, C., Xiao, G., Shan, W.: The Stability Theory of Stream Ciphers. LNCS, vol. 561. Springer, Heidelberg (1991)
Eichenauer, J., Lehn, J.: A nonlinear congruential pseudorandom number generator. Statist. Hefte 27(4), 315–326 (1986)
Eichenauer-Herrmann, J., Grothe, H.: A new inversive congruential pseudorandom number generator with power of two modulus. ACM Transactions on Modeling and Computer Simulation (TOMACS) 2(1), 1–11 (1992)
Eichenauer-Herrmann, J., Niederreiter, H.: Digital Inversive Pseudorandom Numbers. ACM Trans. Model. Comput. Simul. 4(4), 339–349 (1994)
Gong, G., Youssef, A.M.: Cryptographic properties of the Welsh-Gong transformation sequence generators. IEEE Transactions on Information Theory 48(11), 2837–2846 (2002)
Niederreiter, H.: Random number generation and quasi-Monte Carlo methods. SIAM, Philadelphia (1992)
Niederreiter, H., Rivat, J.: On the correlation of pseudorandom numbers generated by inversive methods. Monatshefte für Mathematik 153(3), 251–264 (2008)
Niederreiter, H., Shparlinski, I.E.: Recent advances in the theory of nonlinear pseudorandom number generators. In: Proc. Conf. Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 86–102. Springer, Berlin (2002)
Niederreiter, H., Shparlinski, I.E.: Dynamical systems generated by rational functions. In: Fossorier, M., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 6–17. Springer, Heidelberg (2003)
Niederreiter, H., Winterhof, A.: Lattice structure and linear complexity of nonlinear pseudorandom numbers. Appl. Algebra Eng. Commun. Comput. 13(4), 319–326 (2002)
Niederreiter, H., Winterhof, A.: Incomplete exponential sums over finite fields and their applications to new inversive pseudorandom number. Acta Arith. 93, 387–399 (2000)
Paterson, K.G.: Perfect factors in the de Bruijn graph. Designs, Codes and Cryptography 5, 115–138 (1995)
Si, W., Ding, C.: A simple stream cipher with proven properties. Cryptography and Communications 4, 79–104 (2012)
Weil, A.: On some exponential sums. Mathematics, Proc. Natl. Acad. Sci. USA 34, 204–207 (1948)
Winterhof, A.: Recent results on recursive nonlinear pseudorandom number generators. In: Carlet, C., Pott, A. (eds.) SETA 2010. LNCS, vol. 6338, pp. 113–124. Springer, Heidelberg (2010)
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Wang, L., Hu, Z. (2013). New Sequences of Period p n and p n + 1 via Projective Linear Groups. In: Kutyłowski, M., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2012. Lecture Notes in Computer Science, vol 7763. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38519-3_20
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DOI: https://doi.org/10.1007/978-3-642-38519-3_20
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