Random bit generation using an optically
laser in chaos with oversampling
Xiao-Zhou Li and Sze-Chun Chan*
Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China
*Corresponding author: email@example.com
Received February 21, 2012; revised April 12, 2012; accepted April 20, 2012;
posted April 23, 2012 (Doc. ID 163393); published June 1, 2012
Random bit generation is experimentally demonstrated using a semiconductor laser driven into chaos by optical
injection. The laser is not subject to any feedback so that the chaotic waveform possesses very little autocorrelation.
Random bit generation is achieved at a sampling rate of 10 GHz even when only a fractional bandwidth of 1.5 GHz
within a much broader chaotic bandwidth is digitized. By retaining only 3 least significant bits per sample, an out-
put bit rate of 30 Gbps is attained. The approach requires no complicated postprocessing and has no stringent
requirement on the electronics bandwidth.© 2012 Optical Society of America
OCIS codes: 140.5960, 140.1540, 140.3520, 190.3100.
Random bit generation requires each output bit to be
associated with an unbiased probability for 0 and 1, in-
dependent of the rest of the bits. Applications of random
bit generators include stochastic modeling, encryption,
and secure communication, where a fast generation
speed often has positive impact on the performances.
The broad bandwidths offered by photonic devices
have recently been utilized for high-speed random bit
generation [1,2]. For instance, random bit generation
using quantum randomness in a pulsed laser was demon-
strated . Randomness of spontaneous emissions from
incoherent light sources wasalso utilized [4,5]. Pioneered
by Uchida et al., chaotic dynamics of semiconductor
lasers continue to be important for fast random bit
generation . In their work, chaotic intensities were
emitted from two lasers subject to optical feedback.
Upon detection by photodetectors, chaotic waveforms
with electronic bandwidths of about 3 GHz were
obtained. The waveforms were digitized at a sampling
rate of 1.7 GHz by analog-to-digital converters (ADCs)
of 1-bit resolution. The digitized signals were merged
using exclusive-or (XOR) so the resultant output bit rate
was 1.7 Gbps. Progress was made by further optically in-
jecting the chaotic waveforms into another laser, which
enhanced the bandwidths to attain increment of the out-
put bit rates . Simulations on all-optical generation
were also conducted [7,8]. Additionally, by incorporating
high-resolution ADCs, more than 1 bit can be generated
in each sampling period so that the effective output bit
rate can be significantly increased. Kanter et al. demon-
strated generation of 15 random bits per sampling period
. With an electronic detection bandwidth of 12 GHz
and a sampling rate of 20 GHz, an effective output rate
of 300 Gbps was achieved after much postprocessing.
High-order derivatives were computed to enhance small
fluctuations of the digitized signals in order to pass the
randomness tests. Simplifications by discarding the most
significant bits (MSBs) and retaining the least significant
bits (LSBs) were also reported .
Two features are common to the above approaches.
First, the chaotic waveforms were always generated
by optical feedback into the lasers, even if the waveforms
were further broadened by optical injection into or from
another laser. This often leads to residual autocorrelation
at the feedback round-trip time, which should be set in-
commensurate with the sampling period through careful
design [10,11]. Second, the chaotic waveforms were
always undersampled in which two times their electronic
bandwidths exceeded the respective sampling rates.
Undersampling violates the Nyquist criterion to cause
aliasing and flattening of the spectrum for random bit
generation. However, the chaotic waveforms, photode-
tectors, and front ends of the ADCs must have suffi-
ciently large bandwidths in order to support high
In this Letter, we experimentally demonstrate random
bit generation using an optically injected laser and over-
sampling. A chaotic waveform is generated by the in-
jected semiconductor laser. The waveform is digitized
by an 8-bit ADC with a low front-end bandwidth of only
1.5 GHz, but is oversampled at a sampling rate of 10 GHz.
By retaining only the 3 LSBs and performing XOR on
consecutive samples, random bits are generated at an
output bit rate of 30 Gbps. Since the laser is not subject
to any feedback, there is no round-trip time for the
sampling period to avoid. The use of oversampling also
relaxes the requirements on the front-end bandwidth of
the ADC. The output bits are shown to pass the standard
test suite for random numbers from the National Institute
of Standards and Technology (NIST).
Figure 1 shows our experimental setup for random
bit generation. The master and slave lasers are both
SL, slave laser; VA, variable attenuator; CIR, circulator; OI,
optical isolator; PD, photodetector; A, amplifier; PSA, power
Schematic of the experimental setup. ML, master laser;
June 1, 2012 / Vol. 37, No. 11 / OPTICS LETTERS2163
0146-9592/12/112163-03$15.00/0© 2012 Optical Society of America
ML920T43S-01) emitting at about 1.55 μm. The slave
laser is biased above threshold at 40 mA and is tempera-
ture stabilized at 26.50 °C. It emits about 10 mW with a
relaxation resonance frequency of about 11 GHz. The
master laser is biased at 128.5 mA and is also temperature
stabilized. Its continuous-wave emission passes through
a variable attenuator, a free-space circulator, and im-
pinges on the slave laser facet with about 0.3 mW. The
optical frequency detuning of the master laser from
the slave laser is about 12.5 GHz. Such optical injection
drives the slave laser into chaotic oscillation [12,13].
Chaotic oscillations are capable of amplifying the effects
of intrinsic laser noise . The emission from the
slave laser then passes the circulator and an isolator
so that about 0.5 mW enters a photodetector (Newport
AD-10ir) for optical-to-electrical conversion. The electri-
cal chaotic waveform is then amplified by 40 dB using a
microwave amplifier. For digitization, the electrical
chaotic waveform is monitored by an 8-bit ADC in an
oscilloscope (Agilent 90254A) for subsequent processing
in a computer. The front-end bandwidth of the ADC is
only 1.5 GHz, but the sampling rate is set at 10 GHz.
To ensure randomness, only the 3 LSBs are retained
for each sample. The bits of each sample are then com-
pared by an XOR to the respective bits of the previous
sample, which are stored in a buffer. The buffer is equiva-
lent to a delay of one sampling period. So the output from
the XOR is a bit stream at 30 Gbps.
The chaotic waveform is first characterized by con-
necting the output of the amplifier in Fig. 1 to a power
spectrum analyzer (Agilent N9010A) instead of the
ADC. The red curve in Fig. 2(a) shows the power spec-
trum of the chaotic waveform. The signal bandwidth is
about 10.27 GHz according to the convention of 80% con-
tainment of total power . The gray curve in Fig. 2(a)
shows the noise spectrum measured when the master
laser is turned off and the slave laser is free-running.
It is clear that the chaos spectrum is stronger than and
different from the noise spectrum. The corresponding
autocorrelation trace of the chaotic waveform is shown
in Fig. 2(b). The autocorrelation quickly diminishes as
the delay time increases from zero. Its magnitude is lower
than 0.02 when the delay is longer than 2 ns. By contrast,
chaotic lasers under optical feedback are often asso-
ciated with stronger autocorrelation peaks at multiples
of the feedback round-trip time, which cause restric-
tions on the sampling frequency [10,16]. Therefore, the
absence of any feedback round-trip time is a unique ad-
vantage of optical injection over optical feedback.
The chaotic waveform after the amplifier is input to the
ADC in Fig. 1. The peak-to-peak amplitude of the input
waveform is about 0.2 V. The digitized signal obtained
at position a immediately after the ADC in Fig. 1 is shown
in Fig. 3(i). Each sampled data point is digitized into
8 bits corresponding to 256 digitization levels, as the cor-
responding inset shows. The front-end of the ADC has a
low-pass cutoff frequency of only 1.5 GHz, but the ADC
performs oversampling at a sampling frequency of
10 GHz. This resulted in the rather smooth digitized sig-
nal, which cannot be directly used as random bits.
However, Fig. 3(ii) shows a very irregular signal at
position b of Fig. 1. It is because the 5 MSBs of each data
point are discarded and only the 3 LSBs are selected.
Selecting the LSBs is equivalent to a modulo operation,
or a folding action, that scrambles the data points .
The final output at position c of Fig. 1 is then obtained
by comparing consecutive data points through bitwise
XOR operations, as shown in Fig. 3(iii). The XOR opera-
tions effectively reduce any small statistical bias of the
bits . Overall, the output bits are generated at a
rate of 30 Gbps. They are verified to be random through
passing the NIST Special Publication 800-22 statistical
amplifier. (a) Power spectrum of chaos (red) and noise (gray).
(b) Magnitude of autocorrelation of chaos.
(Color online) Measurements at the output of the
(ii) position b, and (iii) position c of the experimental setup.
Table 1. Results of the NIST Special Publication
800-22 Statistical Tests for Random Bits
Statistical TestP-value ProportionResult
2164OPTICS LETTERS / Vol. 37, No. 11 / June 1, 2012
tests, which are standard tests for randomness. A total of
1000 sequences, each of size 1 Mbit, are collected for test-
ing. At significance level α ? 0.01, the success proportion
should be in the range of 0.99 ? 0.0094392. The compo-
site P-value should be larger than 0.0001 to ensure uni-
formity. The testing results are summarized in Table 1,
where the worst case is shown for tests producing multi-
ple P-values and proportions. Besides, when the master
running slave laser and the electronics is responsible
for generating the output bits. Because the correspond-
ing noise spectrum in Fig. 2(a) is very weak, the output
bits are found to fail the randomness tests even if 7 MSBs
are discarded in retaining only 1 LSB. This shows that
the chaotic waveforms are essential to random bit
generation in the experiment.
Random bits can be generated from the oversampled
signal because the signal bandwidth is significantly broa-
dened when the MSBs are discarded. Figure 4 shows
the spectra of the digitized signals at positions a, b,
and c of the setup in Fig. 1, which are Fourier transforms
of the digitized signals in Figs. 3(i), 3(ii), and 3(iii),
respectively. The full frequency-span of 5 GHz at half
the sampling frequency is presented. The red curve
shows the signal at position a. Limited by the front-
end bandwidth of the ADC, the signal drops quickly
at 1.5 GHz and is thus not suitable for random bit genera-
tion. However, the gray curve obtained at position b
shows a much broadened and nearly white spectrum.
This is because the MSBs are discarded in selecting
the LSBs, which is a very nonlinear operation that causes
significant frequency mixing. Experimentations show
that at least 5 MSBs have to be discarded in order to
ensure sufficient frequency mixing for passing the
randomness tests. The final output at position c has a
featureless white spectrum as shown by the black curve,
which is an essential indicator that the output bits are
In summary, random bit generation is demonstrated
using a chaotic laser driven by optical injection
alone. Because of the absence of optical feedback, the
autocorrelation of the chaotic waveform is free from
any problematic side peaks. Also, in spite of oversam-
pling, random bit generation is enabled by selecting only
the LSBs to effectively broaden the spectrum during
postprocessing. This relaxes the electronic bandwidth re-
quirements of the ADC front end. Utilizing only 1.5 GHz
of a much broader chaos spectrum, random bit genera-
tion at 30 Gbps is successfully demonstrated.
The work described in this paper was fully supported
by a grant from the Research Grants Council of Hong
Kong, China (Project No. CityU 111210).
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at position a (red), position b (gray), and position c (black) of
the experimental setup.
(Color online) Spectra of the digitized signals obtained
June 1, 2012 / Vol. 37, No. 11 / OPTICS LETTERS 2165