Page 1

Direct generation of all-optical random numbers

from optical pulse amplitude chaos

Pu Li, Yun-Cai Wang,* An-Bang Wang, Ling-Zhen Yang, Ming-Jiang Zhang, and Jian-

Zhong Zhang

Institute of Optoelectronic Engineering, College of Physics and Optoelectronics, Taiyuan University of Technology,

Taiyuan 030024, China

*wangyc@tyut.edu.cn

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 Optical Society of America

OCIS codes: (230.1150) All-optical devices; (200.4740) Optical processing; (140.4050) Mode-

locked lasers; (190.3100) Instabilities and chaos; (060.4510) Optical communications.

References and links

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

(2002).

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

(1951).

3. 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).

4. C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator

using amplified spontaneous emission,” Opt. Express 18(23), 23584–23597 (2010).

5. J. Walker, “HotBits: Genuine Random Numbers, Generated by Radioactive Decay,”

http://www.fourmilab.ch/hotbits/.

6. 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).

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

8. G. M. Bernstein and M. A. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans.

Circ. Syst. 37(9), 1157–1164 (1990).

9. 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).

10. 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).

11. 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).

12. 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).

13. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat.

Photonics 4(1), 58–61 (2010).

14. 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).

15. 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).

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4297

Page 2

16. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random

bit generator based on a chaotic photonic integrated circuit,” Opt. Express 18(18), 18763–18768 (2010).

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

20369 (2010).

18. 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).

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

(1999).

20. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert,

J. Dray, and S. Vo, “NIST Statistical Tests Suite,”

http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html.

21. G. Marsaglia, “DIEHARD: A battery of tests of randomness,” http://www.stat.fsu.edu/pub/diehard/.

22. 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).

23. 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).

24. 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).

25. 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).

26. 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).

27. 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).

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

(1983).

29. D. Prichard and J. Theiler, “Generating surrogate data for time series with several simultaneously measured

variables,” Phys. Rev. Lett. 73(7), 951–954 (1994).

30. 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).

31. 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).

32. K. Huybrechts, A. Ali, T. Tanemura, Y. Nakano, and G. Morthier, “Numerical and experimental study of the

switching times and energies of DFB-laser based All-optical flip-flops,” presented at the International

Conference on Photonics in Switching, Pisa, Italy, 15–19 Sept. 2009.

1. Introduction

Random number generators (RNGs) are essential components with a variety of applications

from commerce to science, such as lottery games, cryptography and Monte-Carlo

calculations. For the majority of these applications, random numbers are mostly generated by

means of classical computer algorithms. Although this kind of methods possess the advantage

of being fast and easy to implement, they are fully deterministic and exhibit a finite period,

thus called as pseudorandom number generators (PRNGs). In a security system, the adoption

of PRNGs can lead to catastrophic results [1]. John von Neumann once famously said,

“Anyone who considers arithmetical methods of producing random digits is, of course, in a

state of sin.” [2]

The other type of generators, called as physical random number generators or true random

number generators (TRNGs), generate nondeterministic random numbers from stochastic

physical phenomena, including stochastic noise [3,4], radioactive decay [5], frequency jitter

of electronic oscillator [6] chaotic circuits [7–10] and chaotic lasers [11–18]. Among them,

chaotic laser is the most attractive entropy source for high-speed true random number

generation in recent years, due to its high bandwidth and large amplitude. Since Uchida et al.

firstly realized random number generation by using chaotic laser in the experiment, there have

been a large number of TRNGs based on chaotic lasers reported which utilize optoelectronic

[11–16] or all-optical techniques [17,18]. All of them employ the semiconductor laser with

optical feedback to produce chaotic signal. This kind of chaos is a continuous chaotic

intensity signal and has an inherent periodicity associated with the external cavity. To achieve

high-quality random bit sequence generation in these TRNGs, they must perform the

following procedures:

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4298

Page 3

1) Sampling and quantizing the analog chaotic signal with an external triggered clock.

2) Post-processing raw random number sequence to eliminate the periodicity introduced

by the external feedback cavity. Two typical approaches are exclusive OR (XOR)

operation [11] and high-order derivatives algorithm [13].

However, in the transformation from a continuous analog signal to a discrete digital

signal, the sampling aperture jitter may greatly deteriorate the conversion accuracy and signal

to noise ratio (SNR) [19]. Moreover, the existence of the external clock and the post-process

procedure will significantly increase the complexity of RNG.

In this paper, we theoretically present a direct method for all-optical random number

generation using the discrete pulse amplitude chaos in a mode-locked fiber ring laser

(MLFRL) rather than analog chaotic intensity signals from semiconductor lasers. The

principal advantages of this method are listed as bellows. 1) The pulse amplitude chaos is a

kind of discrete signal consisting of a train of chaotic pulses with a fixed repetition frequency

but random intensities. Therefore, we can directly quantize them into random bit sequences

without requiring the sampling procedure and the external clock. 2) The pulse amplitude

chaos has no periodicity and possesses a highly symmetric distribution of amplitude. Thus, in

theory, we do not require post-processing such as XOR operation and high-order derivatives

algorithm and the generated random bit sequences therefore can successfully pass the

standard statistical tests for randomness [20,21]. 3) The proposed RNG does all signal

processing in the optical domain and thus can be compatible with the optical communication

networks directly with no need of any external modulators.

2. Principle and simulation

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.

Figure 1 is the schematic diagram of the proposed all-optical RNG which consists of a mode-

locked fiber ring laser (MLFRL) and a distributed-feedback laser diode (DFB). The pulse

amplitude chaos generated by the MLFRL is split into two identical chaotic pulse trains by a

3-dB coupler (3 dB). The power of the pulse trains can be adjusted by the erbium-doped fiber

amplifiers (EDFA) and attenuators (Att.). One of them is injected into the right-hand side of

the DFB via a length of fiber delay line (FDL), while the other is injected into the left-hand

side of the DFB, combined with a continuous-wave light (CW) by a wavelength-division-

multiplexing coupler (WDM). Here, the DFB acts as an all-optical ñip-ñop (AOFF), which

plays the role of quantizing the pulse amplitude chaos in the whole system. Finally, with a

circulator and an optical band-pass filter (BPF), we can separate the light of the DFB laser

from the injected light and visualize the random bit sequence on the oscilloscope (OSC).

Specific simulation procedures are described in the followings.

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4299

Page 4

2.1 Pulse amplitude chaos generation and its characteristics

As the stochastic source of the RNG, pulse amplitude chaos is generated by a MLFRL as

illustrated in Fig. 1. The laser is a ring laser configuration using the nonlinear polarization

rotation (NPR) technique. The cavity is composed of a polarization-dependent isolator (PDI)

and two polarization controllers (PC1 and PC2) in a single mode fiber (SMF) and an erbium-

doped fiber (EDF), which is pumped via a wavelength-division multiplexed coupler (WDM)

at 1480 nm and provides gain to the cavity. The mode-locked pulse stream is finally coupled

out through an optical coupler (OC).

Pulse amplitude chaos is an intrinsic feature of NPR MLFRLs, which has been

theoretically and experimentally demonstrated by Zhao et al. [22, 23] and our group [24],

respectively. Herein, we focus on analyzing the characteristics of the pulse amplitude chaos

and confirm that it can be an ideal stochastic source for random number generation. The pulse

amplitude chaos generation can be well described by the extended coupled complex nonlinear

Schrödinger equations [23–27]:

,

232

22

2

3

2

232

g

2

2

()

2263322

uuuuigg

Ω ∂

u

iuiiuvuv u u

ZT

TTT

ββ

βγ

δγ

∗

∂

∂

∆∂

∂

∂

∂

∂

∂

∂

=−−++++++

(1)

,

232

22

2

32

232

g

2

2

()

2263322

vvvvigg

Ω ∂

v

iuiivuv u vv

ZT

TTT

ββ

βγ

δγ

∗

∂

∂

∆∂

∂

∂

∂

∂

∂

∂

=−−++++++

(2)

In the above equations, u and v are the normalized envelopes of the optical pulses along

the two orthogonal polarized modes of the optical ðber. ∆β = 2π/LB is the wave-number

difference between the two modes, where LB is the beat length. δ = β1x-β1y is the linear group-

velocity difference between the two orthogonal polarization modes, where β1x and β1y are the

linear group-velocity related to the two orthogonal polarized modes. β2 and β3 are the group

velocity dispersion (GVD) parameter and the third-order dispersion coefficient, respectively. γ

represents the nonlinearity parameter of the ðbers. g is the saturable gain coefficient of the

ðbers and Ωg is the bandwidth of the laser gain. For single mode ðbers, g = 0. For EDF, g = G

· exp[-∫(|u|2 + |v|2)dt/Psat], where G is the small signal gain coefficient and Psat is the saturation

energy. In our simulation, the whole cavity length L is set as 10 m, which consists of a 2 m-

long EDF with β2 = 50 ps/nm/km and two sections of 4 m-long SMF with β2 = −30 ps/nm/km.

Other parameters are set as follows: γ = 4 W−1km−1, β3 = 0.1 ps2/nm/km, Ωg = 25 nm, LB =

L/2, Psat = 250 and the orientation of passive polarizer to the ðber fast axis θ = 0.125π. More

simulation details see Ref [24].

With the above parameter selection, this MLFRL can emit mode-locked pulses in an

appropriate linear cavity phase delay bias range which corresponds to the orientations of the

polarization controllers. With a fixed linear cavity phase delay bias but different pump power,

stable uniform pulse train can always be obtained. However, once the pulse powers exceed a

certain threshold value, the MLFRL will experience quasi-periodic route to chaos [22–24].

Figure 2 shows an example of quasi-periodic route to chaos. Here, the linear cavity phase

delay bias is fixed at 1.6 π. When pump power is relatively weak (G = 338 km–1), a stable

pulse train with uniform amplitude can be obtained [Fig. 2(a(I))]. With further increasing the

pump power to a certain value (G = 342 km–1), the MLFRL will be operated in period-2 state

and the intensity of the pulse alters between two different values [Fig. 2(b(I))]. Further

slightly increasing the pump power (G = 346 km–1), a multi-periodic state appears as shown in

Fig. 2(c(I)). Eventually, the MLFRL will snap into the chaos state as the pump power are

sufficient strong (G = 348 km–1) [Fig. 2(d(I))]. In order to display the quasi-periodic route to

chaos more directly, we extract the output pulse powers in each round when the laser works at

different state and show them in the right column of Fig. 2, where the blue dots represent the

values of output pulse powers in each round.

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4300

Page 5

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.

In our simulation, the MLFRL can actually generate pulse amplitude chaos in a pump

power range of 348 km−1 < G < 350 km−1 under the above-mentioned condition. We

arbitrarily select an operating point in the chaos region with G = 349 km−1 in order to analyze

the characteristics of pulse amplitude chaos. Figure 3 are the autocorrelation function curve,

first return map and histogram of the extracted chaotic pulse powers, respectively. No

apparent harmonic peak in the autocorrelation characteristics is found [Fig. 3(a)]. This

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4301

Page 6

indicates the correlation of the generated pulse amplitude chaos is statistically insignificant

and has no periodic components harmful to true random number generation. The first return

map of the chaotic pulse powers is shown in Fig. 3(b) which has no discernable structure. It

reveals that there is no correlation between one pulse at one round to the neighboring round

and the trajectory in phase space move in a disordered form. Finally, we consider the

distribution of the pulse amplitude chaos which has great effect on the level of complexity in

equalizing the ratio of “0” and “1” in the following quantization procedure (Section 2.2). As

Fig. 3(c) shown, the distribution of the obtained pulse amplitude chaos in our method is

highly symmetric. For such a distribution, one can easily perform an even division of the

chaotic pulses into “0” and “1” without bias, which is very useful to the realization of a

TRNG.

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.

Although an aperiodic oscilloscope trace and a spike-like autocorrelation curve are

necessary for chaos, they are not sufficient. Stochastic noise can also have these features. A

careful analysis is necessary to determine whether the final state of MLFRL is chaos. Our

analysis starts with the reconstruction of a pseudo phase space from the original one-

dimension chaotic data in the final state of MLFRL via the delay-coordinate technique. We

then use Grassberger-Procaccia algorithem (GPA) [28] to obtain the correlation dimensions

(CD2) and embedding dimensions (m) for the original data. A graph of CD2 as a function of m

based on 20 000 data points is shown in Fig. 4. If the final state of MLFRL is chaos, CD2 will

be expected to converge to a value, i.e., become independent of m at least for large m, and that

value is usually considered as the estimate for the correlation dimension. In contrast, a white

noise has m that monotonically increases with CD2. As Fig. 4 showing, CD2 clearly

converges, indicating that the final state of MLFRL is not white noise. However, the GPA

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4302

Page 7

analysis is sensitive to linear as well as nonlinear correlations. Thus a further check must be

performed to ensure that colored noise is not responsible for the aperiodicity. It has been

pointed out that a comparison with so-called “surrogate” data [29], which have the same

linear correlations as the original data, can provide such a check. The surrogate data are

produced by Fourier transforming the original data, randomizing the phases, and then

reversing the Fourier transform. A comparison of the GPA analysis of the surrogate data with

the original data [Fig. 4] clearly shows convergence in the original data which is absent in the

surrogate data. Thus, we confirm that the final state of MLFRL is indeed chaotic and its

correlation dimension is a fractional number around 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.

2.2 All-optical flip-flop and random bits generation

The quantization to pulse amplitude chaos is similar to that of Refs [18], which is realized

through an all-optical flip-flop (AOFF) proposed by Huybrechts et al. [30, 31]. The AOFF is

realized by means of the bistability existing in a single λ/4-shifted DFB laser (λ/4 DFB),

which can be described as below: A λ/4 DFB with antireflection-coated facets is biased above

threshold. When an external light outside the stop-band of the grating is injected into the laser,

the laser can operate at two different stable states: switching on and switching off.

Figure 5 is one of Huybrechts’ static experimental results under the injection of only

continuous-wave (CW) light [31]. From Fig. 5, we can clearly see that only when injection

light power is above Pth2, the output power of lasing light can jump down to a tiny level of

nearly 0 mW. While the injection light power is below Pth1, the output power of lasing light

will maintain a higher level around 1 mW. To obtain flip-flop operation, the DFB must

operate in the bistable regime through injecting a CW light as the holding beam. The

switching of the two states of the bistability can be controlled by the injected optical pulses

from the left and right facet of the DFB. The left-hand pulses switch the laser off by

disturbing the uniformity of the carrier distribution. The right-hand pulses switch the laser on

by restore the uniformity in the cavity again. To trigger the AOFF, the pulse power should be

higher above a certain threshold so as to induce the conversion of the output state of the DFB.

In our simulation about the AOFF, the power of CW light was set to be Pth1 = 1.6 mW so that

the threshold (i.e. ∆P = Pth2 – Pth1) equals to 0.1 mW, the average power of the chaotic pulse

trains. Thus, the quantizing to pulse amplitude chaos can be achieved, as illustrated in Fig. 6.

Figure 6(a) is a time series of pulse amplitude chaos generated by the MLFRL, which consists

of a chaotic pulse train with a fixed frequency of 20 MHz but randomly distributed pulse

powers. Figure 6(b) shows the chaotic pulse train after the 3-dB coupler, which is injected

into the DFB from its left facet and with a mean power about 0.1 mW. Figure 6(c) is the

identical pulse train delayed by 25 ns on the right side of the DFB. Figure 6(d) is an output

waveform of the AOFF (the DFB) through the BPF transmitting only the lasing light of the

DFB. The output is coded in the way as Fig. 6(d) shown. From it, one can see that each bit of

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4303

Page 8

“0” or “1” occupies a duration time of 50 ns and the extinction ratio between “0” and “1” is as

high as 30 dB.

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. 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.

3. Randomness verification

After the above processes, we got a train of random bits with a rate of 20 Mbps which is

determined by the repetition rate of the MLFRL. A rough verification of its statistical

randomness can be characterized by a bitmap image [Fig. 7] constructed from 300 × 300 bits

of the generated random sequences. As shown in Fig. 7, we can see clearly the bitmap image

exhibits no apparent pattern or bias.

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4304

Page 9

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).

Further, to better qualify the statistical randomness of the random bits, we used the

industry-standard statistical test suite of the National Institute of Standards and Technology

(NIST) [20] and the Diehard test suite [21]. The NIST test suite consists of 15 statistical tests

as shown in Table 1, and each test is performed using 1000 samples of 1 Mb data and

significance level α = 0.01. The passing criteria are that the proportion of the sequences

satisfying condition for the p-value, p > α, should be in the range of 0.99 ± 0.0094392, and the

P-value of the uniformity of the p-values should be larger than 0.0001. Diehard test suite

consist of 18 statistical tests as shown in Table 2, which are performed using 74 Mb data and

significant level α = 0.01. For “success”, the P-value (uniformity of the p-values) of each test

should be within [0.01, 0.99]. As shown in Table 1 and 2, the bit sequences generated in our

method can pass all tests of both the NIST and Diehard tests. These results confirm that the

generated random bits can be statistically regarded as truly independent random bits.

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.

STATISTICAL TEST

Frequency

Block frequency

Cumulative sums

Runs

Longest-run

Rank

FFT

Non-periodic templates

Overlapping templates

Universal

Approximate entropy

Random excursions

Random excursions variant

Serial

Linear Complexity

P-value

0.864510

0.104102

0.452681

0.119896

0.308143

0.699774

0.801956

0.030957

0.444908

0.339508

0.630192

0.125470

0.198021

0.736782

0.561793

Proportion

0.9950

0.9860

0.9940

0.9830

0.9940

0.9950

0.9860

0.9930

0.9890

0.9870

0.9950

0.9830

0.9900

0.9920

0.9910

Result

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4305

Page 10

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.

STATISTICAL TEST

Birthday spacing

Overlapping 5-permutation

Binary rank for 31 × 31 matrices

Binary rank for 32 × 32 matrices

Binary rank for 6 × 8 matrices

Bitstream

Overlapping-Pairs-Space-Occupancy

Overlapping-Quadruples-Space-Occupancy

DNA

Count –the-1’s on a stream of bytes

Count –the-1’s for specific bytes

Parking lot

Minimum distance

3D-spheres

Squeeze

Overlapping sums

Runs

Craps

P-value

0.546187

0.447600

0.564080

0.627815

0.876204

0.145736

0.087900

0.105129

0.383215

0.152096

0.120932

0.644797

0.958178

0.743550

0.623810

0.631732

0.423050

0.552007

Result

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

Success

KS

KS

KS

KS

KS

KS

KS

4. Discussions

4.1 Generation rate of RNG and its improvements

One critical factor for many applications of RNG is a high bit generation rate. The rate in our

RNG is primarily determined by the repetition rate of the pulse amplitude chaos

corresponding to the cavity length of the MLFRL. Huybrechts et al. [32] demonstrated that

the rising time of the AOFF can be down to 40 ps in experiment and even 10 ps in theory. It

indicates the bandwidth of AOFF can be larger than 30 GHz. Therefore, if the repetition rate

of the pulse amplitude chaos generated by the MLFRL is increased, our RNG has great

potential to work at a tremendous bit generation rate.

There are two ways to improve the repetition rate of the pulse amplitude chaos generated

by the MLFRL. The first one is the introduction of time-division multiplexing (TDM)

technology into the proposed RNG. Using a large number of independent MLFRLs which

works at different conditions with each other and multiplexing their output signal (pulse

amplitude chaos) before the quantization procedure, we can yield high repetition rate of

chaotic pulses and then achieve random bits generation at a rate of several Gbps or higher.

The second one is to shorten the cavity length of the MLFRL. For example, we can replace

the EDF with the small size SOA and substitute ordinary SMFs with high birefringence

optical fibers with ultra-short beat length.

4.2 Tolerance of RNG and its improvements

Another critical factor for the applications of RNG is the robustness or tolerance of the

system. Here, we discuss the tolerance of our RNG by analyzing the effect of threshold bias

on the randomness of the generated random numbers. Notice that here the threshold bias

represents the difference between the threshold of the all-optical flip-flop (AOFF) and the

average power of the chaotic pulse trains. In our system, the threshold (i.e. ∆P = Pth2 – Pth1) of

AOFF is fixed at 0.1 mW and the average power of the chaotic pulse trains can be adjusted by

the erbium-doped fiber amplifiers (EDFA) and attenuators (Att.) as shown in Fig. 1.

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4306

Page 11

Figure 8 shows the frequency of “0” in a random bit sequence and the number of passed

NIST tests as a function of the threshold bias. A value of “15” for the number of passed tests

means that all the tests are passed. The frequency of “0” can play the role of indicator for the

quality of the randomness. Figure 8 indicates that the frequency of “0” decreases almost

linearly with the increase of threshold bias and only the random sequences having a frequency

of “0” in the range from 49.87% to 50.13% can passed all the NIST tests. To get a high-

quality random numbers in this condition, the threshold bias should be within a range between

−0.001 and 0.001 mW.

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.

However, it should be noticed that the tolerance of threshold bias is so small, because the

threshold (i.e. ∆P = Pth2 – Pth1) of AOFF in our system is fixed at 0.1 mW. This causes the

average power of the chaotic pulses has to be attenuated to a level around 0.1 mW. If the

threshold of AOFF (i.e. the hysteresis curve width expressed as∆P = Pth2 – Pth1) is big enough,

the average power of chaotic pulses will be able to vary on a much larger scale and without

any loss of randomness and thus the tolerance of our system can be greatly improved. As a

matter of fact, the threshold of AOFF indeed can be easily enhanced by adjusting the bias

current of AOFF. This point has been successfully demonstrated by Huybrechts numerically

and experimentally [30,31]. So, there is still room for improvement in the robustness of our

RNG.

In addition, we want to point out that the post-processing procedure such as XOR

operation in the previous RNGs based on continuous analog chaotic light from the optical

feedback semiconductor laser is not only to improve the 1/0 ratio in random bit sequences and

then allow a much larger threshold bias, but, more importantly, to eliminate the inherent weak

periodicity introduced by the external feedback cavity [11–18]. For this kind of chaotic signal,

the random numbers generated based on it and without post-processing procedure cannot pass

industry-standard statistical tests (such as NIST and Diehard tests) even if the random

numbers has an even 1/0 ratio in practice. Different with them, our pulse amplitude chaos has

no periodicity [Fig. 3(a)] and processes a highly symmetric distribution [Fig. 3(c)]. Therefore,

our method does not require post-processing procedures in theory. In this sense, we believe

that our method also provides a clue in theory to the implementation of RNG with no need of

post-processing procedures.

5. Conclusions

One direct method for all-optical random bit generation using discrete pulse amplitude chaos

in a mode-locked fiber ring laser is theoretically demonstrated. Our simulation results show

that the pulse amplitude chaos consisting of a stream of pulses processes several excellent

characteristics favorable for high-quality true random bits generation, such as aperiodicity and

symmetric stationary distribution. Different from previously reported random number

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4307

Page 12

generators, our RNG do not require the sampling procedure but directly quantizing the chaotic

pulse train to random bit sequences. So it can bypass the possible aliasing problem caused by

sampling procedure and reduce hardware complexity. Free from post-processing, the

generated random bit sequences pass successfully the standard statistical tests for randomness.

In addition, the RNG in this paper is a conceptual protocol and its bit rate can be further

increased by using MLFRLs with higher repetition rate or multiplexing technology.

Acknowledgments

We acknowledge Koen Huybrechts from Ghent University for providing the data about the

all-optical flip-flop. We thank Han Zhang from Université libre de Bruxelles for helpful

discussions and comments. This work is supported partially by the Key Program of National

Natural Science Foundation of China (Grant 60927007 and Grant 61001114) and in part by

the open subject of the State Key Laboratory of Quantum Optics and Quantum Optics devices

of China (Grant 200903).

#158919 - $15.00 USD

(C) 2012 OSA

Received 28 Nov 2011; revised 15 Jan 2012; accepted 16 Jan 2012; published 7 Feb 2012

13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4308