Research Article
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Year 2025, Volume: 46 Issue: 4, 891 - 901, 30.12.2025

Bugce Eminaga Tatlicioglu , Hatice Aktöre

https://doi.org/10.17776/csj.1557049

Abstract

References

  • [1] Diffie W., Hellman M., New directions in cryptography, IEEE Trans. Inf. Theory, 22 (6) (1976) 644-654.
  • [2] Rivest R.L., Shamir A., Adleman L., A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 21 (2) (1978) 120-126.
  • [3] Rivest R.L., Shamir A., Adleman L., A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 26 (1) (1983) 96-99.
  • [4] Lin T., Chua L., A new class of pseudo-random number generator based on chaos in digital filters, Int. J. Circuit Theory Appl., 21 (5) (1993) 473-480.
  • [5] Lücher M., A portable high-quality random number generator for lattice field theory simulations, Comput. Phys. Commun., 79 (1) (1994) 100-110.
  • [6] Zhang S.C., Zhou S.M., A pseudo-random number generator, Math. Practice Theory, 1 (1992) 33-34.
  • [7] Bonilla L.L., Alvaro M., Carretero M., Chaos-based true random number generators, J. Math. Ind., 7 (2017) 1-17.
  • [8] Teh J.S., Samsudin A., Al-Mazrooie M., Akhavan A., GPUs and chaos: a new true random number generator, Nonlinear Dyn., 82 (4) (2015) 1913-1922.
  • [9] Avaroglu E., Koyuncu I., Ozer A.B., Turk M., Hybrid pseudo-random number generator for cryptographic systems, Nonlinear Dyn., 82 (1-2) (2015) 239-248.
  • [10] Lynnyk V., Sakamoto N., Čelikovský S., Pseudo random number generator based on the generalized Lorenz chaotic system, IFAC-Pap. OnLine, 48 (18) (2015) 257-261.
  • [11] Cho K., Miyano T., Design and test of pseudorandom number generator using a star network of Lorenz oscillators, Int. J. Bifurcation Chaos, 27 (12) (2017) 1750184.
  • [12] Ebrahimzadeh R., Jampour M., Chaotic genetic algorithm based on Lorenz chaotic system for optimization problems, Int. J. Intell. Syst. Appl., 5 (5) (2013) 19-24.
  • [13] Abdelhaleem S.H., Abd-El-Hafiz S.K., Radwan A.G., Analysis and guidelines for different designs of pseudo random number generators, IEEE Access, (2024).
  • [14] Zhao X.-Q., Dynamical systems in population biology, Springer, Berlin, (2013).
  • [15] Tél T., de Moura A., Grebogi C., Károlyi G., Chemical and biological activity in open flows: A dynamical system approach, Phys. Rep., 413 (2) (2005) 91-196.
  • [16] Rössler O.E., Chaos and chemistry. In: Nonlinear phenomena in chemical dynamics, Springer, Berlin, (1981) 79-87.
  • [17] Rössler O.E., An equation for continuous chaos, Phys. Lett. A, 57 (5) (1976) 397-398.
  • [18] Lorenz E.N., Deterministic nonperiodic flow, J. Atmos. Sci., 20 (2) (1963) 130-143.
  • [19] Arnold V.I., Mathematical methods of classical mechanics, Springer, Berlin, (1989).
  • [20]Kyrtsou C., Labys W.C., Evidence for chaotic dependence between US inflation and commodity prices, J. Macroecon., 28 (1) (2006) 256-266.
  • [21] Kyrtsou C., Vorlow C.E., Complex dynamics in macroeconomics: A novel approach. In: New trends in macroeconomics, Springer, Berlin, (2005) 223-238.
  • [22] Peters E.E., Fractal market analysis: applying chaos theory to investment and economics, John Wiley & Sons, New York, (1994).
  • [23]Poincaré H., Sur le problème des trois corps et les équations de la dynamique, Acta Math., 13 (1) (1890) A3-A270.
  • [24]Fatou P., Sur les substitutions rationnelles, C. R. Acad. Sci. Paris, 164 (1917) 806-808.
  • [25]Julia G., Mémoire sur l’itération des fonctions rationnelles, J. Math. Pures Appl., 1 (1918) 47-246.
  • [26]Birkhoff G.D., On the periodic motions of dynamical systems, Acta Math., 50 (1) (1927) 359-379.
  • [27] Kolmogorov A.N., The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR, 30 (1941) 299-303.
  • [28]Kolmogorov A.N., Preservation of conditionally periodic movements with small change in the Hamilton function. In: Casati G., Ford J., (Eds.), Stochastic behavior in classical and quantum Hamiltonian systems, Lecture Notes in Physics, Vol. 93, Springer, Berlin, (1979) 51-56.
  • [29]Kolmogorov A.N., Dissipation of energy in the locally isotropic turbulence, Proc. R. Soc. Lond. A, 434 (1991) 15-17.
  • [30] Cartwright M., Littlewood J., On non-linear differential equations of the second order: II. The equation y+kf(y,y+g(y,k)=p(t)=p1(t)+kp2(t); k>0, f(y)≠1, Ann. Math., (1947) 472-494.
  • [31] Smale S., Morse inequalities for a dynamical system, Bull. Am. Math. Soc., 66 (1) (1960) 43-49.
  • [32] Smale S., On gradient dynamical systems, Ann. Math., (1961) 199-206.
  • [33]Shaw R., Strange attractors, chaotic behavior and information flow, Z. Naturforsch. A, 36 (1981) 80-112.
  • [34]Chen G., Ueta T., Yet another chaotic attractor, Int. J. Bifurcation Chaos, 9 (7) (1999) 1465-1466.
  • [35]Lü J., Chen G., A new chaotic attractor coined, Int. J. Bifurcation Chaos, 12 (3) (2002) 659-661.
  • [36]Yang Q., Wei Z., Chen G., An unusual 3D autonomous quadratic chaotic system with two stable node-foci, Int. J. Bifurcation Chaos, 20 (4) (2010) 1061-1083.
  • [37] Pehlivan I., Uyaroğlu Y., A new chaotic attractor from general Lorenz system family and its electronic experimental implementation, Turk. J. Electr. Eng. Comput. Sci., 18 (2) (2010) 171-184.
  • [38] Cuomo K.M., Oppenheim A.V., Circuit implementation of synchronized chaos with applications to communications, Phys. Rev. Lett., 71 (1) (1993) 65-68.
  • [39]Yu S., Lü J., Tang W.K., Chen G., A general multiscroll Lorenz system family and its realization via digital signal processors, Chaos, 16 (3) (2006) 033126.
  • [40]Lü J.G., Chaotic dynamics of the fractional-order Lü system and its synchronization, Phys. Lett. A, 354 (4) (2006) 305-311.
  • [41] Bassham L.E., Rukhin A.L., Soto J., Nechvatal J.R., Smid M.E., Barker E.B., et al., A statistical test suite for random and pseudorandom number generators for cryptographic applications, NIST Special Publication 800-22 Rev. 1a, NIST, Gaithersburg, (2010).
  • [42] Hahn W., Baartz A.P., Stability of motion, Springer, Berlin, (1967).
  • [43] Argyris J., Faust G., Haase M., Friedrich R., An exploration of dynamical systems and chaos, 2nd ed., Springer, Berlin, (2015).
  • [44]Kaplan J.L., Yorke J.A., Chaotic behavior of multidimensional difference equations. In: Functional differential equations and approximation of fixed points, Springer, Berlin, (1979) 204-227.
  • [45]Chlouverakis K.E., Sprott J., A comparison of correlation and Lyapunov dimensions, Physica D, 200 (1-2) (2005) 156-165.
  • [46]Brown R.G., Eddelbuettel D., Bauer D., Dieharder, Duke University Physics Department, Durham, NC, (2018).
  • [47]Bassham L.E., Rukhin A.L., Soto J., Nechvatal J.R., Smid M.E., Leigh S.D., et al., A statistical test suite for random and pseudorandom number generators for cryptographic applications, NIST SP 800-22 Rev. 1a, NIST, Gaithersburg, (2010).
  • [48]Turan M.S., Barker E., Kelsey J., McKay K., Baish M., Boyle M., Recommendation for the entropy sources used for random bit generation, NIST Special Publication 800-90B (Draft), NIST, Gaithersburg, (2016).
  • [49]Barker E., Roginsky A., Davis R., Recommendation for cryptographic key generation, NIST Special Publication 800-133 Rev. 1, NIST, Gaithersburg, (2012).
  • [50]Walker J., ENT: a pseudorandom number sequence test program. Available at: http://www.fourmilab.ch/random/. Retrieved March 2, 2008.

Design and Analysis of a Quadratic Lorenz-like Attractor for Pseudo-Random Number Generation

Year 2025, Volume: 46 Issue: 4, 891 - 901, 30.12.2025

Bugce Eminaga Tatlicioglu , Hatice Aktöre

https://doi.org/10.17776/csj.1557049

Abstract

This study introduces and analyzes a novel quadratic Lorenz-like attractor and demonstrates its application as a pseudo-random number generator (PRNG). A theoretical and numerical investigation of the system dynamics is conducted, including equilibrium point analysis, Jacobian eigenvalues, dissipativity, and Lyapunov exponents. The results indicate that the quadratic modification alters the system dynamics, producing stronger chaotic intensity than the classical Lorenz system. This is supported by larger positive Lyapunov exponents and a non-integer Lyapunov dimension (≈2.40), which reflects a more complex attractor geometry. Numerical simulations, time series, phase projections, and bifurcation analysis further illustrate the rich nonlinear dynamics of the system. To validate its applicability, the chaotic sequences generated by the attractor were subjected to standard statistical evaluations, including the NIST SP800-22, Diehard, and ENT test suites. A dataset of 18 million samples was employed, significantly exceeding the minimum requirement for meaningful validation, and the results confirm that the proposed generator produces high-quality randomness. The importance of this work lies in demonstrating that the quadratic Lorenz-like attractor offers both strong chaotic intensity and relatively simple structure, making it a promising candidate for cryptographic applications, secure communications, and other domains requiring reliable pseudo-random number generation.

Keywords

Chaos Theory Random number generation Dynamical system Lorenz attractor

Ethical Statement

no ethıcal

References

  • [1] Diffie W., Hellman M., New directions in cryptography, IEEE Trans. Inf. Theory, 22 (6) (1976) 644-654.
  • [2] Rivest R.L., Shamir A., Adleman L., A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 21 (2) (1978) 120-126.
  • [3] Rivest R.L., Shamir A., Adleman L., A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 26 (1) (1983) 96-99.
  • [4] Lin T., Chua L., A new class of pseudo-random number generator based on chaos in digital filters, Int. J. Circuit Theory Appl., 21 (5) (1993) 473-480.
  • [5] Lücher M., A portable high-quality random number generator for lattice field theory simulations, Comput. Phys. Commun., 79 (1) (1994) 100-110.
  • [6] Zhang S.C., Zhou S.M., A pseudo-random number generator, Math. Practice Theory, 1 (1992) 33-34.
  • [7] Bonilla L.L., Alvaro M., Carretero M., Chaos-based true random number generators, J. Math. Ind., 7 (2017) 1-17.
  • [8] Teh J.S., Samsudin A., Al-Mazrooie M., Akhavan A., GPUs and chaos: a new true random number generator, Nonlinear Dyn., 82 (4) (2015) 1913-1922.
  • [9] Avaroglu E., Koyuncu I., Ozer A.B., Turk M., Hybrid pseudo-random number generator for cryptographic systems, Nonlinear Dyn., 82 (1-2) (2015) 239-248.
  • [10] Lynnyk V., Sakamoto N., Čelikovský S., Pseudo random number generator based on the generalized Lorenz chaotic system, IFAC-Pap. OnLine, 48 (18) (2015) 257-261.
  • [11] Cho K., Miyano T., Design and test of pseudorandom number generator using a star network of Lorenz oscillators, Int. J. Bifurcation Chaos, 27 (12) (2017) 1750184.
  • [12] Ebrahimzadeh R., Jampour M., Chaotic genetic algorithm based on Lorenz chaotic system for optimization problems, Int. J. Intell. Syst. Appl., 5 (5) (2013) 19-24.
  • [13] Abdelhaleem S.H., Abd-El-Hafiz S.K., Radwan A.G., Analysis and guidelines for different designs of pseudo random number generators, IEEE Access, (2024).
  • [14] Zhao X.-Q., Dynamical systems in population biology, Springer, Berlin, (2013).
  • [15] Tél T., de Moura A., Grebogi C., Károlyi G., Chemical and biological activity in open flows: A dynamical system approach, Phys. Rep., 413 (2) (2005) 91-196.
  • [16] Rössler O.E., Chaos and chemistry. In: Nonlinear phenomena in chemical dynamics, Springer, Berlin, (1981) 79-87.
  • [17] Rössler O.E., An equation for continuous chaos, Phys. Lett. A, 57 (5) (1976) 397-398.
  • [18] Lorenz E.N., Deterministic nonperiodic flow, J. Atmos. Sci., 20 (2) (1963) 130-143.
  • [19] Arnold V.I., Mathematical methods of classical mechanics, Springer, Berlin, (1989).
  • [20]Kyrtsou C., Labys W.C., Evidence for chaotic dependence between US inflation and commodity prices, J. Macroecon., 28 (1) (2006) 256-266.
  • [21] Kyrtsou C., Vorlow C.E., Complex dynamics in macroeconomics: A novel approach. In: New trends in macroeconomics, Springer, Berlin, (2005) 223-238.
  • [22] Peters E.E., Fractal market analysis: applying chaos theory to investment and economics, John Wiley & Sons, New York, (1994).
  • [23]Poincaré H., Sur le problème des trois corps et les équations de la dynamique, Acta Math., 13 (1) (1890) A3-A270.
  • [24]Fatou P., Sur les substitutions rationnelles, C. R. Acad. Sci. Paris, 164 (1917) 806-808.
  • [25]Julia G., Mémoire sur l’itération des fonctions rationnelles, J. Math. Pures Appl., 1 (1918) 47-246.
  • [26]Birkhoff G.D., On the periodic motions of dynamical systems, Acta Math., 50 (1) (1927) 359-379.
  • [27] Kolmogorov A.N., The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR, 30 (1941) 299-303.
  • [28]Kolmogorov A.N., Preservation of conditionally periodic movements with small change in the Hamilton function. In: Casati G., Ford J., (Eds.), Stochastic behavior in classical and quantum Hamiltonian systems, Lecture Notes in Physics, Vol. 93, Springer, Berlin, (1979) 51-56.
  • [29]Kolmogorov A.N., Dissipation of energy in the locally isotropic turbulence, Proc. R. Soc. Lond. A, 434 (1991) 15-17.
  • [30] Cartwright M., Littlewood J., On non-linear differential equations of the second order: II. The equation y+kf(y,y+g(y,k)=p(t)=p1(t)+kp2(t); k>0, f(y)≠1, Ann. Math., (1947) 472-494.
  • [31] Smale S., Morse inequalities for a dynamical system, Bull. Am. Math. Soc., 66 (1) (1960) 43-49.
  • [32] Smale S., On gradient dynamical systems, Ann. Math., (1961) 199-206.
  • [33]Shaw R., Strange attractors, chaotic behavior and information flow, Z. Naturforsch. A, 36 (1981) 80-112.
  • [34]Chen G., Ueta T., Yet another chaotic attractor, Int. J. Bifurcation Chaos, 9 (7) (1999) 1465-1466.
  • [35]Lü J., Chen G., A new chaotic attractor coined, Int. J. Bifurcation Chaos, 12 (3) (2002) 659-661.
  • [36]Yang Q., Wei Z., Chen G., An unusual 3D autonomous quadratic chaotic system with two stable node-foci, Int. J. Bifurcation Chaos, 20 (4) (2010) 1061-1083.
  • [37] Pehlivan I., Uyaroğlu Y., A new chaotic attractor from general Lorenz system family and its electronic experimental implementation, Turk. J. Electr. Eng. Comput. Sci., 18 (2) (2010) 171-184.
  • [38] Cuomo K.M., Oppenheim A.V., Circuit implementation of synchronized chaos with applications to communications, Phys. Rev. Lett., 71 (1) (1993) 65-68.
  • [39]Yu S., Lü J., Tang W.K., Chen G., A general multiscroll Lorenz system family and its realization via digital signal processors, Chaos, 16 (3) (2006) 033126.
  • [40]Lü J.G., Chaotic dynamics of the fractional-order Lü system and its synchronization, Phys. Lett. A, 354 (4) (2006) 305-311.
  • [41] Bassham L.E., Rukhin A.L., Soto J., Nechvatal J.R., Smid M.E., Barker E.B., et al., A statistical test suite for random and pseudorandom number generators for cryptographic applications, NIST Special Publication 800-22 Rev. 1a, NIST, Gaithersburg, (2010).
  • [42] Hahn W., Baartz A.P., Stability of motion, Springer, Berlin, (1967).
  • [43] Argyris J., Faust G., Haase M., Friedrich R., An exploration of dynamical systems and chaos, 2nd ed., Springer, Berlin, (2015).
  • [44]Kaplan J.L., Yorke J.A., Chaotic behavior of multidimensional difference equations. In: Functional differential equations and approximation of fixed points, Springer, Berlin, (1979) 204-227.
  • [45]Chlouverakis K.E., Sprott J., A comparison of correlation and Lyapunov dimensions, Physica D, 200 (1-2) (2005) 156-165.
  • [46]Brown R.G., Eddelbuettel D., Bauer D., Dieharder, Duke University Physics Department, Durham, NC, (2018).
  • [47]Bassham L.E., Rukhin A.L., Soto J., Nechvatal J.R., Smid M.E., Leigh S.D., et al., A statistical test suite for random and pseudorandom number generators for cryptographic applications, NIST SP 800-22 Rev. 1a, NIST, Gaithersburg, (2010).
  • [48]Turan M.S., Barker E., Kelsey J., McKay K., Baish M., Boyle M., Recommendation for the entropy sources used for random bit generation, NIST Special Publication 800-90B (Draft), NIST, Gaithersburg, (2016).
  • [49]Barker E., Roginsky A., Davis R., Recommendation for cryptographic key generation, NIST Special Publication 800-133 Rev. 1, NIST, Gaithersburg, (2012).
  • [50]Walker J., ENT: a pseudorandom number sequence test program. Available at: http://www.fourmilab.ch/random/. Retrieved March 2, 2008.
There are 50 citations in total.

Details

Primary Language English
Subjects Dynamical Systems in Applications, Applied Mathematics (Other)
Journal Section Research Article
Authors

Bugce Eminaga Tatlicioglu 0000-0001-8854-4464

Hatice Aktöre 0000-0003-2455-3681

Submission Date September 27, 2024
Acceptance Date November 27, 2025
Publication Date December 30, 2025
Published in Issue Year 2025 Volume: 46 Issue: 4

Cite

APA Eminaga Tatlicioglu, B., & Aktöre, H. (2025). Design and Analysis of a Quadratic Lorenz-like Attractor for Pseudo-Random Number Generation. Cumhuriyet Science Journal, 46(4), 891-901. https://doi.org/10.17776/csj.1557049

Editor