## Abstract

Pseudorandom sequences are important for many applications in communication systems, in coding theory, and in the design of stream ciphers. Maximum-length linear sequences (or *m*-sequences) are popular in sequence designs due to their long period and excellent pseudorandom properties. In code-division multiple-access (CDMA) applications, there is a demand for large families of sequences having good correlation properties. The best families of sequences in these applications frequently use *m*-sequences in their constructions. Therefore, the problem of determining the correlation properties of *m*-sequences has received a lot of attention since the 1960s, and many interesting theoretical results of practical interest have been obtained. The cross-correlation of *m*-sequences is also related to other important problems, such as almost perfect nonlinear functions (APN) and almost bent functions (AB), and to the nonlinearity of S-boxes in many block ciphers including AES. This chapter gives an updated survey of the cross-correlation of *m*-sequences and describes some of the most important open problems that still remain in this area.

## Access this chapter

Tax calculation will be finalised at checkout

Purchases are for personal use only

### Similar content being viewed by others

## References

C.Â Bracken, Designs, Codes,

*Spin Models and the Walsh Transform*, Ph.D. thesis, Department of Mathematics, National University Ireland (NUI), Maynooth, 2004 In this Ph.D. thesis one can find a nice proof of the number of solutions of a linearized polynomial playing an important role in the proof of the 3-valued crosscorrelation with the Kasamiâ€“Welch exponent \(d = 2^{2k} - 2^{k} + 1\).A.Â Canteaut, P.Â Charpin, H.Â Dobbertin, Binary

*m*-sequences with three-valued crosscorrelation: a proof of Welchâ€™s conjecture. IEEE Trans. Inf. Theory**46**(1), 4â€“8 (2000) The more than 30â€‰year old conjecture by Welch on a decimation with 3-valued crosscorrelation between two*m*-sequences is proved in this paper.F. Chabaud, S. Vaudenay, Links between differential and linear cryptanalysis, in

*Advances in Cryptology-EUROCRYPTâ€™94*(Springer, New York, 1995), pp. 356â€“365 The paper gives important relations between differential and linear analysis and shows in particular that AB functions are APN functions.T.W. Cusick, H.Â Dobbertin, Some new three-valued crosscorrelation functions for binary

*m*-sequences. IEEE Trans. Inf. Theory**42**(4), 1238â€“1240 (1996) The authors prove two conjectures due to Niho on two decimation that (for*n*even) give 3-valued crosscorrelation.H.Â Dobbertin, One-to-one highly nonlinear power functions on GF(2

^{n}). Appl. Algebra Eng. Commun. Comput.**9**(2), 139â€“152 (1998) The author finds a new decimation with 4-valued crosscorrelation, the first new one since Nihoâ€™s Ph.D. thesis from 1972.H.Â Dobbertin, Almost perfect nonlinear power functions on GF(2

^{n}): the Niho case. Inf. Comput.**151**(1â€“2), 57â€“72 (1999) The author shows that two decimations conjectured by Niho to have 3-valued crosscorrelation for odd*m*give almost perfect nonlinear functions. This was an important step in order to later complete the proof of these conjectures in [16].H.Â Dobbertin, P.Â Felke, T.Â Helleseth, P.Â Rosendahl, Niho type cross-correlation functions via Dickson polynomials and Kloosterman sums. IEEE Trans. Inf. Theory

**52**(2), 613â€“627 (2006) Dickson polynomials were used for the first time to find the crosscorrelation between*m*-sequences. The paper also settled the correlation distribution of many new decimations with 4-valued crosscorrelation.H.Â Dobbertin, T.Â Helleseth, P. Vijay Kumar, H.Â Martinsen, Ternary

*m*-sequences with three-valued crosscorrelation function: two new decimations of Welch and Niho type. IEEE Trans. Inf. Theory**47**(4), 1473â€“1481 (2001) The importance of this paper is that is found the first new nonbinary decimations with three values since the constructions 30â€‰years earlier by Trachtenberg.R.Â Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory

**14**(1), 154â€“156 (1968) This pioneering paper defined the Gold decimation and proved that it had a 3-valued crosscorrelation and determined the complete correlation distribution. This was the basis for the important Gold sequences.S.W. Golomb,

*Shift Register Sequences*(Holden-Day, San Francisco, 1967) This is a classical book on linear and nonlinear recursions.S.W. Golomb, Theory of transformation groups of polynomials over GF(2) with applications to linear shift register sequences. Inf. Sci.

**1**(1), 87â€“109 (1968) The author states (without proof) that the crosscorrelation between binary*m*-sequences takes on at least three values. The Welch conjecture that two special decimations have 3-valued crosscorrelation was published here for the first time.T.Â Helleseth, Some results about the cross-correlation function between two maximal linear sequences. Discrete Math.

**16**(3), 209â€“232 (1976) This paper contains many basic results on the crosscorrelations of*m*-sequences. The first nonbinary decimation is found giving a four-valued crosscorrelation between two*m*-sequences. The distributions of several decimations are completely settled. The âˆ’ 1 conjecture is stated in this paper.T.Â Helleseth, Crosscorrelation of

*m*-sequences, exponential sums and Dickson polynomials. IEICE Trans. Fundamentals**E93A**(11), 2212â€“2219 (2010) Presents a survey on the crosscorrelation between binary*m*-sequences having at most 5-valued crosscorrelation with a focus on the many connections between exponential sums and Dickson polynomials.T.Â Helleseth, P.V. Kumar, Sequences with low correlation, in

*Handbook in Coding Theory*, eds. by V.S. Pless, W.C. Huffman, ch.â€‰21 (Elsevier Science B.V., Amsterdam, 1998), pp.1765â€“1853 This is a survey of sequences with low correlation that contains constructions and analysis of many important sequence families and some of their relations to coding theory.T.Â Helleseth, P.Â Rosendahl, New pairs of

*m*-sequences with 4-level cross-correlation. Finite Fields Appl.**11**(4), 674â€“683 (2005) This paper introduced new decimations with 4-valued cross correlation.H.D.L. Hollmann, Q.Â Xiang, A proof of the Welch and Niho conjectures on cross-correlations of binary

*m*-sequences. Finite Fields Appl.**7**(2), 253â€“286 (2001) This paper completed the proof of two decimations, for odd*m*, that were conjectured by Niho to lead to 3-valued crosscorrelation.A.Â Johansen, T.Â Helleseth, A family of

*m*-sequences with five-valued cross correlation. IEEE Trans. Inf. Theory**55**(2), 880â€“887 (2009) The distribution of the crosscorrelation of pairs of*m*-sequences with decimations giving five-valued crosscorrelation was found using techniques involving Dickson polynomials.A.Â Johansen, T.Â Helleseth, A.Â Kholosha, Further results on

*m*-sequences with five-valued cross correlation. IEEE Trans. Inf. Theory**55**(12), 5792â€“5802 (2009) This paper extends the results in [17] to other decimations with five-valued crosscorrelation. Some results depend on open conjectures on some exponential sums.T.Â Kasami, The weight enumerators for several classes of subcodes of the 2nd order binary Reedâ€“Muller codes. Inf. Control

**18**(4), 369â€“394 (1971) The author determined the weight enumerator of some subcodes of the 2nd order Reedâ€“Muller. A consequence of these results is a proof of the Kasamiâ€“Welch decimation leading to 3-valued crosscorrelation. This decimation was also proved by Welch (unpublished).D.Â Katz, Weil sums of binomials, three-level cross-correlation and a conjecture by Helleseth. J. Combin. Theory A

**119**(8), 1644â€“1659 (2012) The paper gives a solution of the conjecture of Helleseth that for*n*â€‰=â€‰2^{i}and*p*â€‰=â€‰2 the crosscorrelation takes on at least 4 values.G. Lachaud, J. Wolfmann, The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inf. Theory

**36**(3), 686â€“692 (1990) The paper shows that the Kloosterman sums takes on all possible values \(\equiv -1\pmod 4\) within its bound.J.Â Lahtonen, G.Â McGuire, H.N. Ward, Gold and Kasamiâ€“Welch functions, quadratic forms, and bent functions. Adv. Math. Commun.

**1**(2), 243â€“250 (2007) Provides a local result on*C*_{ d }(0) for the Kasamiâ€“Welch decimation.Y.Â Niho,

*Multi-valued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequences*, Ph.D. thesis, University of Southern California, Los Angeles, 1972 This thesis gave the complete crosscorrelation distribution of several decimations with 4-valued cross correlation. Furthermore, many conjectures on the cross correlation distribution of sequences with few values in the crosscorrelation were given. This Ph.D. thesis had a significant influence on later research on the crosscorrelation.D. Sarwate, M. Pursley, Crosscorrelation properties of pseudorandom and related sequences. Proc. IEEE,

**68**(5), 593â€“619 (1980) This is a classical and excellent survey of the crosscorrelation between*m*-sequences.H.M. Trachtenberg,

*On the Cross-Correlation Functions of Maximal Linear Recurring Sequences*, Ph.D. thesis, University of Southern California, Los Angeles, 1970 The main result is the two families of decimation giving three-valued crosscorrelation. These are the only decimations that work for all nonbinary*m*-sequences.T. Zhang, S. Li, T. Feng, G. Ge, Some new results on the cross correlation of

*m*-sequences. arXiv:1309.7734 [cs.IT] This recent paper gives new ternary decimations with four-valued crosscorrelation.

## Acknowledgements

This research was supported by the Norwegian Research Council.

## Author information

### Authors and Affiliations

### Corresponding author

## Editor information

### Editors and Affiliations

## Rights and permissions

## Copyright information

Â© 2014 Springer International Publishing Switzerland

## About this chapter

### Cite this chapter

Helleseth, T. (2014). Open Problems on the Cross-correlation of *m*-Sequences.
In: KoÃ§, Ã‡. (eds) Open Problems in Mathematics and Computational Science. Springer, Cham. https://doi.org/10.1007/978-3-319-10683-0_8

### Download citation

DOI: https://doi.org/10.1007/978-3-319-10683-0_8

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-319-10682-3

Online ISBN: 978-3-319-10683-0

eBook Packages: Computer ScienceComputer Science (R0)