Abstract

We propose and theoretically demonstrate an all-optical method for directly generating all-optical random numbers from pulse amplitude chaos produced by a mode-locked fiber ring laser. Under an appropriate pump intensity, the mode-locked laser can experience a quasi-periodic route to chaos. Such a chaos consists of a stream of pulses with a fixed repetition frequency but random intensities. In this method, we do not require sampling procedure and external triggered clocks but directly quantize the chaotic pulses stream into random number sequence via an all-optical flip-flop. Moreover, our simulation results show that the pulse amplitude chaos has no periodicity and possesses a highly symmetric distribution of amplitude. Thus, in theory, the obtained random number sequence without post-processing has a high-quality randomness verified by industry-standard statistical tests.

© 2012 OSA

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    [CrossRef]
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2011 (1)

T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011).
[CrossRef]

2010 (7)

2009 (3)

H. Zhang, D. Y. Tang, L. M. Zhao, X. Wu, and H. Y. Tam, “Dissipative vector solitons in a dispersionmanaged cavity fiber laser with net positive cavity dispersion,” Opt. Express 17(2), 455–460 (2009).
[CrossRef] [PubMed]

H. Zhang, D. Y. Tang, L. M. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A 80(4), 045803 (2009).
[CrossRef]

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[CrossRef] [PubMed]

2008 (4)

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

H. Zhang, D. Y. Tang, L. M. Zhao, and N. Xiang, “Coherent energy exchange between components of a vector soliton in fiber lasers,” Opt. Express 16(17), 12618–12623 (2008).
[PubMed]

K. Huybrechts, W. D'Oosterlinck, G. Morthier, and R. Baets, “Proposal for an All-Optical Flip-Flop Using a Single Distributed Feedback Laser Diode,” IEEE Photon. Technol. Lett. 20(1), 18–20 (2008).
[CrossRef]

K. Huybrechts, G. Morthier, and R. Baets, “Fast all-optical flip-flop based on a single distributed feedback laser diode,” Opt. Express 16(15), 11405–11410 (2008).
[CrossRef] [PubMed]

2005 (1)

D. Y. Tang, L. M. Zhao, and F. Lin, “Numerical studies of routes to chaos in passively mode-locked fiber soliton ring lasers with dispersion-managed cavity,” Europhys. Lett. 71(1), 56–62 (2005).
[CrossRef]

2004 (1)

2003 (1)

M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a Smart Card IC,” IEEE Trans. Comput. 52(4), 403–409 (2003).
[CrossRef]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

2001 (2)

T. Stojanovski and L. Kocarev, “Chaos-based random number generators - Part I: analysis [cryptography],” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48, 281–288 (2001).
[CrossRef]

T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators - Part II: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48(3), 382–385 (2001).
[CrossRef]

2000 (1)

C. S. Petrie and J. A. Connelly, “A noise-based IC random number generator for applications in cryptography,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 47(5), 615–621 (2000).
[CrossRef]

1999 (1)

R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Comm. 17(4), 539–550 (1999).
[CrossRef]

1994 (1)

D. Prichard and J. Theiler, “Generating surrogate data for time series with several simultaneously measured variables,” Phys. Rev. Lett. 73(7), 951–954 (1994).
[CrossRef] [PubMed]

1990 (1)

G. M. Bernstein and M. A. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circ. Syst. 37(9), 1157–1164 (1990).
[CrossRef]

1989 (1)

D. S. Ornstein, “Ergodic theory, randomness, and “chaos”,” Science 243(4888), 182–187 (1989).
[CrossRef] [PubMed]

1983 (1)

P. Grassberger and I. Procaccia, “Characterization of Strange Attractors,” Phys. Rev. Lett. 50(5), 346–349 (1983).
[CrossRef]

1951 (1)

J. Von Neumann, “Various techniques used in connection with random digits,” Appl. Math. Series 12, 36–38 (1951).

Aida, H.

Amano, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Argyris, A.

Aviad, Y.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[CrossRef]

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[CrossRef] [PubMed]

Baets, R.

K. Huybrechts, W. D'Oosterlinck, G. Morthier, and R. Baets, “Proposal for an All-Optical Flip-Flop Using a Single Distributed Feedback Laser Diode,” IEEE Photon. Technol. Lett. 20(1), 18–20 (2008).
[CrossRef]

K. Huybrechts, G. Morthier, and R. Baets, “Fast all-optical flip-flop based on a single distributed feedback laser diode,” Opt. Express 16(15), 11405–11410 (2008).
[CrossRef] [PubMed]

Bernstein, G. M.

G. M. Bernstein and M. A. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circ. Syst. 37(9), 1157–1164 (1990).
[CrossRef]

Bogris, A.

Bucci, M.

M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a Smart Card IC,” IEEE Trans. Comput. 52(4), 403–409 (2003).
[CrossRef]

Cohen, E.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[CrossRef]

Connelly, J. A.

C. S. Petrie and J. A. Connelly, “A noise-based IC random number generator for applications in cryptography,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 47(5), 615–621 (2000).
[CrossRef]

Davis, P.

T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011).
[CrossRef]

K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010).
[CrossRef] [PubMed]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Deligiannidis, S.

D'Oosterlinck, W.

K. Huybrechts, W. D'Oosterlinck, G. Morthier, and R. Baets, “Proposal for an All-Optical Flip-Flop Using a Single Distributed Feedback Laser Diode,” IEEE Photon. Technol. Lett. 20(1), 18–20 (2008).
[CrossRef]

Germani, L.

M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a Smart Card IC,” IEEE Trans. Comput. 52(4), 403–409 (2003).
[CrossRef]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Grassberger, P.

P. Grassberger and I. Procaccia, “Characterization of Strange Attractors,” Phys. Rev. Lett. 50(5), 346–349 (1983).
[CrossRef]

Harayama, T.

T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011).
[CrossRef]

K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010).
[CrossRef] [PubMed]

Hirano, K.

K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010).
[CrossRef] [PubMed]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Huybrechts, K.

K. Huybrechts, W. D'Oosterlinck, G. Morthier, and R. Baets, “Proposal for an All-Optical Flip-Flop Using a Single Distributed Feedback Laser Diode,” IEEE Photon. Technol. Lett. 20(1), 18–20 (2008).
[CrossRef]

K. Huybrechts, G. Morthier, and R. Baets, “Fast all-optical flip-flop based on a single distributed feedback laser diode,” Opt. Express 16(15), 11405–11410 (2008).
[CrossRef] [PubMed]

Inoue, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Kanter, I.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[CrossRef]

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[CrossRef] [PubMed]

Kocarev, L.

T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators - Part II: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48(3), 382–385 (2001).
[CrossRef]

T. Stojanovski and L. Kocarev, “Chaos-based random number generators - Part I: analysis [cryptography],” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48, 281–288 (2001).
[CrossRef]

Kurashige, T.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Li, P.

Y. C. Wang, P. Li, and J. Z. Zhang, “Fast random bit generation in optical domain with ultrawide bandwidth chaotic laser,” IEEE Photon. Technol. Lett. 22, 1680–1682 (2010).

P. Li, Y. C. Wang, and J. Z. Zhang, “All-optical fast random number generator,” Opt. Express 18(19), 20360–20369 (2010).
[CrossRef] [PubMed]

Li, X.

Lieberman, M. A.

G. M. Bernstein and M. A. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circ. Syst. 37(9), 1157–1164 (1990).
[CrossRef]

Lin, F.

D. Y. Tang, L. M. Zhao, and F. Lin, “Numerical studies of routes to chaos in passively mode-locked fiber soliton ring lasers with dispersion-managed cavity,” Europhys. Lett. 71(1), 56–62 (2005).
[CrossRef]

L. M. Zhao, D. Y. Tang, F. Lin, and B. Zhao, “Observation of period-doubling bifurcations in a femtosecond fiber soliton laser with dispersion management cavity,” Opt. Express 12(19), 4573–4578 (2004).
[CrossRef] [PubMed]

Luzzi, R.

M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a Smart Card IC,” IEEE Trans. Comput. 52(4), 403–409 (2003).
[CrossRef]

Morikatsu, S.

Morthier, G.

K. Huybrechts, G. Morthier, and R. Baets, “Fast all-optical flip-flop based on a single distributed feedback laser diode,” Opt. Express 16(15), 11405–11410 (2008).
[CrossRef] [PubMed]

K. Huybrechts, W. D'Oosterlinck, G. Morthier, and R. Baets, “Proposal for an All-Optical Flip-Flop Using a Single Distributed Feedback Laser Diode,” IEEE Photon. Technol. Lett. 20(1), 18–20 (2008).
[CrossRef]

Murphy, T. E.

Naito, S.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Okumura, H.

Oowada, I.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Ornstein, D. S.

D. S. Ornstein, “Ergodic theory, randomness, and “chaos”,” Science 243(4888), 182–187 (1989).
[CrossRef] [PubMed]

Petrie, C. S.

C. S. Petrie and J. A. Connelly, “A noise-based IC random number generator for applications in cryptography,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 47(5), 615–621 (2000).
[CrossRef]

Pihl, J.

T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators - Part II: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48(3), 382–385 (2001).
[CrossRef]

Pikasis, E.

Prichard, D.

D. Prichard and J. Theiler, “Generating surrogate data for time series with several simultaneously measured variables,” Phys. Rev. Lett. 73(7), 951–954 (1994).
[CrossRef] [PubMed]

Procaccia, I.

P. Grassberger and I. Procaccia, “Characterization of Strange Attractors,” Phys. Rev. Lett. 50(5), 346–349 (1983).
[CrossRef]

Qiao, Z. D.

L. Z. Yang, J. F. Zhu, Z. D. Qiao, X. Y. Yan, and Y. C. Wang, “Periodic intensity variations on the pulse-train of a passively mode-locked fiber ring laser,” Opt. Commun. 283(19), 3798–3802 (2010).
[CrossRef]

Reidler, I.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[CrossRef]

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[CrossRef] [PubMed]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Rosenbluh, M.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[CrossRef]

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[CrossRef] [PubMed]

Roy, R.

Salevan, J. C.

Shiki, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Someya, H.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Stojanovski, T.

T. Stojanovski and L. Kocarev, “Chaos-based random number generators - Part I: analysis [cryptography],” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48, 281–288 (2001).
[CrossRef]

T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators - Part II: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48(3), 382–385 (2001).
[CrossRef]

Sunada, S.

T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011).
[CrossRef]

Syvridis, D.

Tam, H. Y.

Tang, D. Y.

Theiler, J.

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K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010).
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I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
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[CrossRef]

Opt. Commun. (1)

L. Z. Yang, J. F. Zhu, Z. D. Qiao, X. Y. Yan, and Y. C. Wang, “Periodic intensity variations on the pulse-train of a passively mode-locked fiber ring laser,” Opt. Commun. 283(19), 3798–3802 (2010).
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T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011).
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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the all-optical RNG based on the pulse amplitude chaos in a mode-locked fiber ring laser (MLFRL). EDF, erbium-doped ðber; PC1 and PC2, two polarization controllers; PDI, polarization-dependent isolator; OC, optical coupler; WDM, wavelength-division-multiplexing coupler; Pump, pump light; 3 dB, 3-dB coupler; EDFA, erbium-doped fiber amplifier; Att., optical attenuator; FDL, ðber delay line; CW, continuous-wave light; DFB, distributed-feedback laser diode; BPF, optical bandpass ðlter; OSC, oscilloscope.

Fig. 2
Fig. 2

Quasi-periodic route to chaos with a fixed linear cavity phase delay bias of 1.6 π under different pump strength which are given in three-dimensional form (left column) and two-dimensional form (right column). (a) Period-1 state, G = 338 km−1; (b) Period-2 state, G = 342 km−1; (c) Multi-periodic state, G = 346 km−1; (d) Chaotic state, G = 348 km−1.

Fig. 3
Fig. 3

Characteristics of pulse amplitude chaos. (a) Auto-correlation curve of the chaotic pulse power; (b) First return map of the chaotic pulse power; (c) Stochastic histogram of the chaotic pulse power.

Fig. 4
Fig. 4

Result of Grassberger-Procaccia algorithem (GPA) analysis on chaotic data set shown in Fig. 3. Black squares: GPA analysis of original data. Red circles: GPA analysis of surrogate data.

Fig. 5
Fig. 5

Bistability curve: laser output power as a function of the power of the injected light [31]. There are two typical threshold values: Pth1 is 1.6 mW and. Pth2 is 1.7 mW.

Fig. 6
Fig. 6

Temporal waveforms: (a) Pulse amplitude chaos from the MLFRL. (b) Pulse amplitude chaos injected into the left-hand of the AOFF. (c) Delayed pulse amplitude chaos injected into the right-hand of the AOFF. (d) Random bit sequence.

Fig. 7
Fig. 7

Random bit patterns with 300 × 300 bits are shown in a two-dimensional plane. Bits “1” and “0” are converted to white and black dots, respectively, and placed from left to right (and from top to bottom).

Fig. 8
Fig. 8

The frequency of “0” in a random bit sequence (black squares) and the number of passed NIST tests (blue circles) as a function of the threshold bias. Here the threshold bias represents the difference between the threshold of the all-optical flip-flop (AOFF) which is fixed at 0.1 mW and the average power of the chaotic pulse trains.

Tables (2)

Tables Icon

Table 1 Typical results of NIST statistical tests. Using 1000 samples of 1 Mb data and significance level α = 0.01, for “Success”, the P-value (uniformity of p-values) should be larger than 0.0001 and the proportion should be greater than 0.9805608.

Tables Icon

Table 2 Typical results of Diehard statistical tests. Using 74 Mb data and significance level α = 0.01, for “Success”, the P-value (uniformity of p-values) should be larger than 0.0001. “KS” indicates that single P-value is obtained by the Kolmogorov-Smirnov (KS) test.

Equations (2)

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u Z =i Δβ 2 uδ u T i β 2 2 2 u T 2 + β 3 6 3 u T 3 +iγ( | u | 2 + 2 3 | v | 2 )u+ iγ 3 v 2 u + g 2 u+ g 2 Ω g 2 2 u T 2 ,
v Z =i Δβ 2 uδ v T i β 2 2 2 v T 2 + β 3 6 3 v T 3 +iγ( | v | 2 + 2 3 | u | 2 )v+ iγ 3 u 2 v + g 2 v+ g 2 Ω g 2 2 v T 2 ,

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