Construction of pseudorandom binary lattices using elliptic curves
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Abstract:
In an earlier paper, Hubert, Mauduit and Sárközy introduced and studied the notion of pseudorandomness of binary lattices. Later constructions were given by using characters and the notion of a multiplicative inverse over finite fields. In this paper a further large family of pseudorandom binary lattices is constructed by using elliptic curves.References
- Zhixiong Chen, Elliptic curve analogue of Legendre sequences, Monatsh. Math. 154 (2008), no. 1, 1–10. MR 2395518, DOI 10.1007/s00605-008-0520-x
- Zhixiong Chen, Shengqiang Li, and Guozhen Xiao, Construction of pseudo-random binary sequences from elliptic curves by using discrete logarithm, Sequences and their applications—SETA 2006, Lecture Notes in Comput. Sci., vol. 4086, Springer, Berlin, 2006, pp. 285–294. MR 2444692, DOI 10.1007/11863854_{2}4
- A. Enge: Elliptic Curves and Their Application to Cryptography: An Introduction, Kluwer Academic Publisher, Dordrecht, 1999.
- Louis Goubin, Christian Mauduit, and András Sárközy, Construction of large families of pseudorandom binary sequences, J. Number Theory 106 (2004), no. 1, 56–69. MR 2049592, DOI 10.1016/j.jnt.2003.12.002
- P. Hubert, C. Mauduit, and A. Sárközy, On pseudorandom binary lattices, Acta Arith. 125 (2006), no. 1, 51–62. MR 2275217, DOI 10.4064/aa125-1-5
- David R. Kohel and Igor E. Shparlinski, On exponential sums and group generators for elliptic curves over finite fields, Algorithmic number theory (Leiden, 2000) Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 395–404. MR 1850620, DOI 10.1007/10722028_{2}4
- Huaning Liu, New pseudorandom sequences constructed by quadratic residues and Lehmer numbers, Proc. Amer. Math. Soc. 135 (2007), no. 5, 1309–1318. MR 2276639, DOI 10.1090/S0002-9939-06-08630-8
- Huaning Liu, A large family of pseudorandom binary lattices, Proc. Amer. Math. Soc. 137 (2009), no. 3, 793–803. MR 2457416, DOI 10.1090/S0002-9939-08-09706-2
- Huaning Liu, Tao Zhan, and Xiaoyun Wang, Large families of elliptic curve pseudorandom binary sequences, Acta Arith. 140 (2009), no. 2, 135–144. MR 2558449, DOI 10.4064/aa140-2-3
- Christian Mauduit and András Sárközy, On large families of pseudorandom binary lattices, Unif. Distrib. Theory 2 (2007), no. 1, 23–37. MR 2318530
- Christian Mauduit and András Sárközy, On finite pseudorandom binary sequences. I. Measure of pseudorandomness, the Legendre symbol, Acta Arith. 82 (1997), no. 4, 365–377. MR 1483689, DOI 10.4064/aa-82-4-365-377
- C. Mauduit and A. Sárközy, Construction of pseudorandom binary sequences by using the multiplicative inverse, Acta Math. Hungar. 108 (2005), no. 3, 239–252. MR 2162562, DOI 10.1007/s10474-005-0222-y
- Christian Mauduit and András Sárközy, Construction of pseudorandom binary lattices by using the multiplicative inverse, Monatsh. Math. 153 (2008), no. 3, 217–231. MR 2379668, DOI 10.1007/s00605-007-0479-z
- László Mérai, Construction of large families of pseudorandom binary sequences, Ramanujan J. 18 (2009), no. 3, 341–349. MR 2495552, DOI 10.1007/s11139-008-9131-3
- László Mérai, A construction of pseudorandom binary sequences using both additive and multiplicative characters, Acta Arith. 139 (2009), no. 3, 241–252. MR 2545928, DOI 10.4064/aa139-3-3
- L. Mérai, Construction of pseudorandom binary sequences over elliptic curves using multiplicative characters, submitted.
- László Mérai, Construction of pseudorandom binary lattices based on multiplicative characters, Period. Math. Hungar. 59 (2009), no. 1, 43–51. MR 2544619, DOI 10.1007/s10998-009-9043-z
- András Sárközy, On finite pseudorandom binary sequences and their applications in cryptography, Tatra Mt. Math. Publ. 37 (2007), 123–136. MR 2553412
- A. Sárközy, A finite pseudorandom binary sequence, Studia Sci. Math. Hungar. 38 (2001), 377–384. MR 1877793, DOI 10.1556/SScMath.38.2001.1-4.28
- Arne Winterhof, Some estimates for character sums and applications, Des. Codes Cryptogr. 22 (2001), no. 2, 123–131. MR 1813781, DOI 10.1023/A:1008300619004
Additional Information
- László Mérai
- Affiliation: Alfréd Rényi Institute of Mathematics, Budapest, Pf. 127, H-1364 Hungary
- Email: merai@cs.elte.hu
- Received by editor(s): February 5, 2010
- Published electronically: September 30, 2010
- Additional Notes: This research was partially supported by the Hungarian National Foundation for Scientific Research, Grant No. K67676, and by the Momentum Fund of the Hungarian Academy of Sciences.
- Communicated by: Jim Haglund
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 407-420
- MSC (2010): Primary 11K45
- DOI: https://doi.org/10.1090/S0002-9939-2010-10631-7
- MathSciNet review: 2736325