# Maximum-Order Complexity and Correlation Measures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

## 2. Proof of Theorem 1

**Proof.**

## 3. Further Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Gyarmati, K. Measures of pseudorandomness. Finite Fields and Their Applications. In Radon Series on Computational and Applied Mathematics; De Gruyter: Berlin, Germany, 2013; Volume 11, pp. 43–64. [Google Scholar]
- Meidl, W.; Winterhof, A. Linear complexity of sequences and multisequences, Section 10.4 of the Handbook of Finite Fields. In Discrete Mathematics and its Applications (Boca Raton); Mullen, G.L., Panario, D., Eds.; CRC Press: Boca Raton, FL, USA, 2013; pp. 324–336. [Google Scholar]
- Niederreiter, H. Linear complexity and related complexity measures for sequences. In Progress in Cryptology-INDOCRYPT 2003; Lecture Notes in Computer Science, 2904; Springer: Berlin, Germany, 2003; pp. 1–17. [Google Scholar]
- Sárközy, A. On finite pseudorandom binary sequences and their applications in cryptography. Tatra Mt. Math. Publ.
**2007**, 37, 123–136. [Google Scholar] - Topuzoğlu, A.; Winterhof, A. Pseudorandom sequences. In Topics in Geometry, Coding Theory and Cryptography; Algebra Applications, 6; Springer: Dordrecht, The Netherlands, 2007; pp. 135–166. [Google Scholar]
- Winterhof, A. Linear complexity and related complexity measures. In Selected Topics in Information and Coding Theory; Series on Coding Theory and Cryptology, 7; World Science Publishing: Hackensack, NJ, USA, 2010; pp. 3–40. [Google Scholar]
- Mauduit, C.; Sárközy, A. On finite pseudorandom binary sequences. I. Measure of pseudorandomness, the Legendre symbol. Acta Arith.
**1997**, 82, 365–377. [Google Scholar] - Brandstätter, N.; Winterhof, A. Linear complexity profile of binary sequences with small correlation measure. Period. Math. Hung.
**2006**, 52, 1–8. [Google Scholar] [CrossRef] - Shparlinski, I. Cryptographic Applications of Analytic Number Theory. Complexity Lower Bounds and Pseudorandomness; Progress in Computer Science and Applied Logic, 22; Birkhäuser Verlag: Basel, Switzerland, 2003. [Google Scholar]
- Jansen, C.J.A. Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods. Ph.D. Thesis, Technische Universiteit Delft, Delft, The Netherlands, 1989; p. 195. [Google Scholar]
- Jansen, C.J.A. The maximum order complexity of sequence ensembles. In Advances in Cryptology— EUROCRYPT’91, LNCS 547; Davies, D.W., Ed.; Springer: Berlin/Heidelberg, Germany, 1991; pp. 153–159. [Google Scholar]
- Niederreiter, H.; Xing, C. Sequences with high nonlinear complexity. IEEE Trans. Inf. Theory
**2014**, 60, 6696–6701. [Google Scholar] [CrossRef] - Alon, N.; Kohayakawa, Y.; Mauduit, C.; Moreira, C.G.; Rödl, V. Measures of pseudorandomness for finite sequences: Typical values. Proc. Lond. Math. Soc.
**2007**, 95, 778–812. [Google Scholar] [CrossRef] - Chen, Z.; Winterhof, A. Linear complexity profile of m-ary pseudorandom sequences with small correlation measure. Indag. Math.
**2009**, 20, 631–640. [Google Scholar] [CrossRef] - Mauduit, C.; Sárközy, A. On finite pseudorandom sequences of k symbols. Indag. Math.
**2002**, 13, 89–101. [Google Scholar] [CrossRef] - He, J.J.; Panario, D.; Wang, Q.; Winterhof, A. Linear complexity profile and correlation measure of interleaved sequences. Cryptogr. Commun.
**2015**, 7, 497–508. [Google Scholar] [CrossRef] - Brandstätter, N.; Winterhof, A. Some notes on the two-prime generator of order 2. IEEE Trans. Inf. Theory
**2005**, 5, 3654–3657. [Google Scholar] [CrossRef] - Rivat, J.; Sárközy, A. Modular constructions of pseudorandom binary sequences with composite moduli. Period. Math. Hung.
**2005**, 51, 75–107. [Google Scholar] [CrossRef] - Goubin, L.; Mauduit, C.; Sárközy, A. Construction of large families of pseudorandom binary sequences. J. Number Theory
**2004**, 106, 56–69. [Google Scholar] [CrossRef] - Mérai, L.; Yayla, O. Improving results on the pseudorandomness of sequences generated via the additive order of a finite field. Discret. Math.
**2015**, 338, 2020–2025. [Google Scholar] [CrossRef] - Sárközy, A.; Winterhof, A. Measures of pseudorandomness for binary sequences constructed using finite fields. Discret. Math.
**2009**, 309, 1327–1333. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Işık, L.; Winterhof, A.
Maximum-Order Complexity and Correlation Measures. *Cryptography* **2017**, *1*, 7.
https://doi.org/10.3390/cryptography1010007

**AMA Style**

Işık L, Winterhof A.
Maximum-Order Complexity and Correlation Measures. *Cryptography*. 2017; 1(1):7.
https://doi.org/10.3390/cryptography1010007

**Chicago/Turabian Style**

Işık, Leyla, and Arne Winterhof.
2017. "Maximum-Order Complexity and Correlation Measures" *Cryptography* 1, no. 1: 7.
https://doi.org/10.3390/cryptography1010007