Pseudo Random Binary Sequence Based on Cyclic Difference Set
Abstract
:1. Introduction
2. Preliminaries
2.1. Notation and Convention
 $p:$ a prime number.
 ${\mathbb{F}}_{{p}^{}}:$ prime field of p elements.
 ${\mathbb{F}}_{{p}^{m}}:$ finite field of ${p}^{m}$ elements where m is a nonnegative integer and $m\ge 2$.
 ${{\mathbb{F}}^{\ast}}_{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{p}^{m}}:{\mathbb{F}}_{{p}^{m}}\left\{0\right\}$.
 $f\left(x\right):$ a primitive polynomial of degree m in characteristic field ${\mathbb{F}}_{{p}^{m}}$.
 $\omega :$ a primitive element of primitive polynomial, $f\left(x\right)$.
 $\lambda :$ period of sequence.
2.2. Primitive Polynomial
 ①
 ${x}^{{p}^{m}1}\equiv 1\left(\mathrm{mod}\phantom{\rule{3.33333pt}{0ex}}f\left(x\right)\right)$,
 ②
 ${x}^{k}\not\equiv 1\left(\mathrm{mod}\phantom{\rule{3.33333pt}{0ex}}f\left(x\right)\right)$ for $1\le k\le {p}^{m}2$.
2.3. Quadratic Residue and Quadratic Nonresidue
2.4. Linear Complexity
3. Proposal of MK Sequence
3.1. Cyclic Difference Set
3.2. Generation Algorithm
Algorithm 1 Proposed Algorithm for MK Sequence. 

4. Experimental Results
4.1. Randomness Analysis
4.2. Linear Complexity Analysis
4.3. Result of Uniformity
4.4. Evaluation by Comparison
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
 Lambić, D.; Nikolić, M. Pseudorandom number generator based on discretespace chaotic map. Nonlinear Dyn. 2017, 90, 223–232. [Google Scholar] [CrossRef]
 Akhter, F.; Nogami, Y.; Kusaka, T.; Taketa, Y.; Tatara, T. Binary sequence generated by alternative trace map function and its properties. In Proceedings of the 2019 Seventh International Symposium on Computing and Networking Workshops (CANDARW), Nagasaki, Japan, 26–29 November 2019; pp. 408–411. [Google Scholar]
 Akhter, F.; Al Mamun, M.S. Pseudo random binary sequence: A new approach over finite field and its properties. In Proceedings of the 2017 International Conference on Electrical, Computer and Communication Engineering (ECCE), Cox’s Bazar, Bangladesh, 16–18 February 2017; pp. 676–680. [Google Scholar]
 Šajić, S.; Maletić, N.; Todorović, B.M.; Šunjevarić, M. Random binary sequences in telecommunications. J. Electr. Eng. 2013, 64, 230–237. [Google Scholar] [CrossRef] [Green Version]
 Pasqualini, L.; Parton, M. Pseudo random number generation: A reinforcement learning approach. Procedia Comput. Sci. 2020, 170, 1122–1127. [Google Scholar] [CrossRef]
 Golomb, S.W. Shift Register Sequences; Aegean Park Press: Walnut Creek, CA, USA, 1967. [Google Scholar]
 Gold, R. Optimal binary sequences for spread spectrum multiplexing (Corresp.). IEEE Trans. Inf. Theory 1967, 13, 619–621. [Google Scholar] [CrossRef] [Green Version]
 Kasami, T. Weight Distribution Formula for Some Class of Cyclic Codes; Report No. R285; Coordinated Science Laboratory, University of Illinois: Champaign, IL, USA, 1966. [Google Scholar]
 Nogami, Y.; Tada, K.; Uehara, S. A geometric sequence binarized with Legendre symbol over odd characteristic field and its properties. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 2014, 97, 2336–2342. [Google Scholar] [CrossRef]
 Yu, N.Y.; Gong, G. New construction of Mary sequence families with low correlation from the structure of Sidelnikov sequences. IEEE Trans. Inf. Theory 2010, 56, 4061–4070. [Google Scholar] [CrossRef]
 Su, M.; Winterhof, A. Autocorrelation of Legendre–Sidelnikov Sequences. IEEE Trans. Inf. Theory 2010, 56, 1714–1718. [Google Scholar] [CrossRef]
 Kim, Y.T.; San Kim, D.; Song, H.Y. New MAry Sequence families with low correlation from the array structure of Sidelnikov sequences. IEEE Trans. Inf. Theory 2014, 61, 655–670. [Google Scholar]
 Zierler, N. Legendre Sequences; Technical Report; Massachusetts Institute of Technology, Lincoln Laboratory: Lincoln, NE, USA, 1958. [Google Scholar]
 No, J.S.; Lee, H.K.; Chung, H.; Song, H.Y.; Yang, K. Trace representation of Legendre sequences of Mersenne prime period. IEEE Trans. Inf. Theory 1996, 42, 2254–2255. [Google Scholar]
 Ding, C.; Hesseseth, T.; Shan, W. On the linear complexity of Legendre sequences. IEEE Trans. Inf. Theory 1998, 44, 1276–1278. [Google Scholar] [CrossRef]
 Rukhin, A.; Soto, J.; Nechvatal, J.; Smid, M.; Barker, E. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications; Technical Report; Booz Allen & Hamilton inc Greensboro Drive: McLean, VA, USA, 2001. [Google Scholar]
 Bassham, L.E., III; Rukhin, A.L.; Soto, J.; Nechvatal, J.R.; Smid, M.E.; Barker, E.B.; Leigh, S.D.; Levenson, M.; Vangel, M.; Banks, D.L.; et al. Sp 80022 rev. 1a. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications; National Institute of Standards & Technology: Gaithersburg, MD, USA, 2010. [Google Scholar]
 Koblitz, N. A Course in Number Theory and Cryptography; Springer Science & Business Media: Berlin, Germany, 1994; Volume 114. [Google Scholar]
 Lehmer, D. On Euler’s totient function. Bull. Am. Math. Soc. 1932, 38, 745–751. [Google Scholar] [CrossRef] [Green Version]
 Massey, J. Shiftregister synthesis and BCH decoding. IEEE Trans. Inf. Theory 1969, 15, 122–127. [Google Scholar] [CrossRef] [Green Version]
 Cohen, S.D. Generators in cyclic difference sets. J. Comb. Theory Ser. A 1989, 51, 227–236. [Google Scholar] [CrossRef] [Green Version]
 Xia, B. Cyclotomic difference sets in finite fields. Math. Comput. 2018, 87, 2461–2482. [Google Scholar] [CrossRef] [Green Version]
 Dillon, J.F.; Dobbertin, H. New cyclic difference sets with Singer parameters. Finite Fields Their Appl. 2004, 10, 342–389. [Google Scholar] [CrossRef]
 Polhill, J. Generalizations of partial difference sets from cyclotomy to nonelementary abelian pgroups. Electron. J. Comb. 2008, 15, R125. [Google Scholar] [CrossRef] [Green Version]
 MurilloEscobar, M.; CruzHernández, C.; CardozaAvendaño, L.; MéndezRamírez, R. A novel pseudorandom number generator based on pseudorandomly enhanced logistic map. Nonlinear Dyn. 2017, 87, 407–425. [Google Scholar] [CrossRef]
 Marsaglia, G. DIEHARD Test Suite. 1998, Volume 8. Available online: http://www.Stat.Fsu.Edu/pub/diehard (accessed on 20 March 2014).
 Gustafson, H.; Dawson, E.; Nielsen, L.; Caelli, W. A computer package for measuring the strength of encryption algorithms. Comput. Secur. 1994, 13, 687–697. [Google Scholar] [CrossRef]
 Knuth, G. The Art of Computer Programming, Seminumerical Algorithms—Volume 2: Addition; Wesley: Reading, MA, USA, 1998. [Google Scholar]
 Sulak, F.; Uğuz, M.; Kocak, O.; Doğanaksoy, A. On the independence of statistical randomness tests included in the NIST test suite. Turk. J. Electr. Eng. Comput. Sci. 2017, 25, 3673–3683. [Google Scholar] [CrossRef]
 Patidar, V.; Sud, K.K.; Pareek, N.K. A pseudo random bit generator based on chaotic logistic map and its statistical testing. Informatica 2009, 33, 441–452. [Google Scholar]
 Sỳs, M.; Matyáš, V. Randomness testing: Result interpretation and speed. In The New Codebreakers; Springer: New York, NY, USA, 2016; pp. 389–395. [Google Scholar]
 Hu, H.; Liu, L.; Ding, N. Pseudorandom sequence generator based on the Chen chaotic system. Comput. Phys. Commun. 2013, 184, 765–768. [Google Scholar] [CrossRef]
 Yang, L.; XiaoJun, T. A new pseudorandom number generator based on a complex number chaotic equation. Chin. Phys. B 2012, 21, 090506. [Google Scholar]
 Liu, L.; Miao, S.; Hu, H.; Deng, Y. Pseudorandom bit generator based on nonstationary logistic maps. IET Inf. Secur. 2016, 10, 87–94. [Google Scholar] [CrossRef]
 Helleseth, T. Golomb’s randomness postulates. In Encyclopedia of Cryptography and Security; van Tilborg, H.C.A., Jajodia, S., Eds.; Springer: Boston, MA, USA, 2011; pp. 516–517. [Google Scholar] [CrossRef]
 Doğanaksoy, A.; Sulak, F.; Uğuz, M.; Şeker, O.; Akcengiz, Z. New statistical randomness tests based on length of runs. Math. Probl. Eng. 2015, 2015. [Google Scholar] [CrossRef] [Green Version]
 Ers, H.W. On the significance of golomb’s randomness postulates in cryptography. Philips J. Res 1988, 43, 185–222. [Google Scholar]
 Schwabe, P.; Stoffelen, K. All the AES you need on CortexM3 and M4. In International Conference on Selected Areas in Cryptography; Springer: Cham, Switzerland, 2016; pp. 180–194. [Google Scholar]
 De Santis, F.; Schauer, A.; Sigl, G. ChaCha20Poly1305 authenticated encryption for highspeed embedded IoT applications. In Proceedings of the Design, Automation & Test in Europe Conference & Exhibition (DATE), Lausanne, Switzerland, 27–31 March 2017; pp. 692–697. [Google Scholar]
 Gao, X. Comparison analysis of Ding’s RLWEbased key exchange protocol and NewHope variants. Adv. Math. Commun. 2019, 13, 221. [Google Scholar] [CrossRef] [Green Version]
Statistical Test  Portion of Successful Sequences $\ge 0.01$  Result 

Frequency  0.997  ◯ 
Block frequency  0.991  ◯ 
Cumulative sums (1)  0.997  ◯ 
Cumulative sums (2)  0.996  ◯ 
Runs  0.995  ◯ 
Longest run  0.992  ◯ 
Rank  0.991  ◯ 
Fast fourier transform  0.989  ◯ 
Nonoverlapping template  max: 0.997  ◯ 
min: 0.983  ◯  
Overlapping template  0.986  ◯ 
Maurer’s universal statistical  0.990  ◯ 
Approximate entropy  0.984  ◯ 
Random excursions  max: 1.000  ◯ 
min: 0.974  ◯  
Random Excursions Variant  max: 1.000  ◯ 
min: 0.983  ◯  
Serial (1)  0.987  ◯ 
Serial (2)  0.989  ◯ 
Linear complexity  0.984  ◯ 
$\mathit{p},\mathit{m}$  Length of Sequence  Linear Complexity 

$p=5,m=3$  124  62 
$p=7,m=3$  342  171 
$p=7,m=5$  16,806  8403 
$p=11,m=5$  161,050  80,525 
$p=101,m=3$  1,030,300  515,150 
$p=467,m=3$  101,847,562  50,923,781 
Pattern Length  Bit Pattern  # of Appearance 

1  0  1562 
1  1562  
2  00  781 
01  781  
10  781  
11  781  
3  000  385 
001  396  
010  396  
011  385  
100  396  
101  385  
110  385  
111  396 
Pattern Length  Bit Pattern  # of Appearance 

1  0  50,923,781 
1  50,923,781  
2  00  25,461,890 
01  25,461,891  
10  25,461,891  
11  25,461,890  
3  000  12,731,725 
001  12,730,165  
010  12,730,165  
011  12,731,726  
100  12,730,165  
101  12,731,726  
110  12,731,726  
111  12,730,164 
Length of Sequence  Linear Complexity  

Proposed Sequence  NTU Sequence  
$p=5,m=3$  62  62 
$p=7,m=3$  171  114 
$p=7,m=5$  8403  5602 
$p=11,m=5$  80,525  32,210 
$p=101,m=3$  515,150  20,606 
$p=463,m=3$  50,923,781  437,114 
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Mamun, M.S.A.; Akhter, F. Pseudo Random Binary Sequence Based on Cyclic Difference Set. Symmetry 2020, 12, 1202. https://doi.org/10.3390/sym12081202
Mamun MSA, Akhter F. Pseudo Random Binary Sequence Based on Cyclic Difference Set. Symmetry. 2020; 12(8):1202. https://doi.org/10.3390/sym12081202
Chicago/Turabian StyleMamun, Md. Selim Al, and Fatema Akhter. 2020. "Pseudo Random Binary Sequence Based on Cyclic Difference Set" Symmetry 12, no. 8: 1202. https://doi.org/10.3390/sym12081202