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\begin{document}

\title{Quantum random numbers from vacuum fluctuations}

\author{Yicheng Shi}
\affiliation{Department of Physics, National University of Singapore, 2
  Science Drive 3,  Singapore, 117542}
\affiliation{Center for Quantum Technologies, National University of
  Singapore, 3 Science Drive 2, Singapore, 117543}
\author{Brenda Chng}
\affiliation{Center for Quantum Technologies, National University of
  Singapore, 3 Science Drive 2, Singapore, 117543}
\author{Christian Kurtsiefer}
\affiliation{Department of Physics, National University of Singapore, 2
  Science Drive 3,  Singapore, 117542}
\affiliation{Center for Quantum Technologies, National University of
  Singapore, 3 Science Drive 2, Singapore, 117543}
\email[]{christian.kurtsiefer@gmail.com}
\date{\today}

\begin{abstract}
We implement a quantum random number generator based on a balanced
homodyne measurement of vacuum fluctuations of the electromagnetic field.
The digitized signal is directly processed with a fast randomness
extraction scheme based on a liner feedback shift register. The random bit
stream is continuously read in a computer at a rate of about 480 Mbit/s and
passes an extended test suite for random numbers.
\end{abstract}

% \pacs{}
\maketitle

\section{Introduction}
Various cryptographic schemes, classical or quantum, require high
quality and trusted random numbers for key generation and other aspects of the
protocols. A weakness in random number generation may jeopardize the security
of the entire system. In order to keep up with data rates in
modern communication schemes, these random numbers need to be generated at a
high rate~\cite{Gisin2002a}.
%\cite{Syed2011, Frauchiger2013, Abellan2014, Kanter2009, Fuerst2010,
%Sanguinetti2014, Gabriel2010}
Equally, large amounts of random numbers are at the core Monte Carlo
simulation methods~\cite{Metropolis:1987}.  
Algorithmically generated pseudo-random numbers are available at very high
rates, but are deterministic by definition and
may be unsuitable for cryptographic purposes, as they may contain backdoors in
the particular algorithm used to generate them. For
applications that require truly unpredictable random numbers, physical random
number generators have been used in the past~\cite{GALTON1890} and more
recently~\cite{INTEL}. These random 
number generators (RNG) involve measuring noisy physical processes and
conversion of the outcome into random numbers. Since it is either practically
(e.g. for thermal noise sources) or fundamentally (for certain quantum
processes) impossible to predict the outcome of such measurements, these 
physically generated random numbers are considered truly random.

Quantum random number generators (QRNG) belong to a class of physical random
number sources where the source of randomness is a fundamentally unpredictable
outcome of a quantum measurement. Early RNG of this class were based 
on observing the decay statistics of radioactive nuclei
\cite{gude1985,figotin2004random}. 
More recently, physical random number generators based on similar Poisson
statistics in optical photon detection were
implemented~\cite{Stipcevic:07,Fuerst2010}. Different schemes use the 
randomness of a single photon scattered by a beam splitter into either of two
output ports~\cite{Jennewein2000,Stefanov2000}.
Since the reflection/transmission of the photon is  intrinsically 
random due to the quantum nature of the process, the unpredictability of the
generated numbers is ensured~\cite{Frauchiger2013}.
Other implementations of QRNGs  measure the
vacuum fluctuations of the electromagnetic field~\cite{Gabriel2010,Syed2011}, or
the intensity and phase noise of different light
sources~\cite{Kanter2009,Abellan2014,Sanguinetti2014}. 


In this paper we report on a quantum random number generator based on
measuring vacuum fluctuations similar as in  \cite{Gabriel2010,Syed2011}, as
such measurements have a very high bandwidth compared to schemes based on
photon counting~\cite{Stipcevic:07,Fuerst2010}. Coupled with an efficient
randomness extractor, we obtain an unbiased, uncorrelated stream of random
bits at high speed.  

\section{Implementation}
Figure~\ref{fig:schematic} schematically shows the setup of our QRNG. A
continuous wave laser
(wavelength 780\,nm) is used as the local oscillator (LO) for the vacuum
fluctuations of the electromagnetic field entering the beam splitter at the
empty port. The output of the beam splitter is directed on two pin
photodiodes, and the photocurrent difference is processed further. 
\begin{figure}
\centerline{
%\includegraphics[width=0.8\columnwidth]{./plot/setup/setup1}
  \includegraphics[width=0.8\columnwidth]{figure1.eps}
}
\caption{Schematic of the quantum random number generator. A polarizing beam
  splitter (PBS) distributes the light of a 780\,nm laser diode equally on two
  fast photodiodes, generating photocurrents $i_1$ and
  $i_2$. The fluctuations in the photocurrent difference $i_1-i_2$ are
  amplified, digitized, and sent to a randomness extractor to generate
  unbiased true random numbers.}
\label{fig:schematic}
\end{figure}
This setup is known as a balanced homodyne
detector~\cite{jakeman1974,yuen1983} and maps the the electrical field in the
second mode entering the beam splitter to the photocurrent difference
$i_1-i_2$. Here, the second input port is empty, so the homodyne measurement
is probing the vacuum state of the electromangetic field. According to quantum
optics, this field fluctuates~\cite{Glauber:1963}, and these fluctuations are
the source of randomness here. As the vacuum field does not
require any external physical quantity to enter the system, it can not be
tampered with. Since the optical power impinging
on the two photodiodes is balanced, any power
fluctuation in the local oscillator will be simultaneously detected by the two
diodes, and therefore cacncel in the photocurrent
difference~\cite{yuen1983,schumaker1984}. In  
an alternative view, the laser beam can be seen as generating photocurrents
$i_1,i_2$ with a noise power proportional to the average optical power. The
shot noise currents from the two diodes will add up because they are
uncorrelated, while amplitude fluctuations in the laser intensity (referred to
as classical noise) represented by the average current of the photodiodes does
not affect the photocurrent difference.

The power between the two output ports is balanced by rotating the laser diode
in front of a polarizing beam splitter (PBS). The output light leaving 
the PBS is detected by a pair of 
reversely biased silicon pin photodiodes (Hamamatsu S5972) connected in
series to perform the current subtraction. The balancing of the photocurrents
is monitored by observing the voltage drop across a resistor $R_D$ providing a
DC path for the current difference from the common node to ground. The
fluctuations above 20\,MHz are amplified with the
transimpedance amplifier chain shown in Fig.~\ref{fig:amplifier} with a
calculated effective transimpedance of $R_{\rm eff}\approx540\,$k$\Omega$. 

\begin{figure}
%\centerline{\includegraphics[width=0.9\columnwidth]{plot/setup/gainstage}}
\centerline{\includegraphics[width=0.9\columnwidth]{figure2.eps}}
\caption{Gain stage. A transimpedance amplifier (Analog devices AD8015)
  converts the photocurrent difference $i_1-i_2$ into a low impedance voltage
  signal, which is amplified by two consecutive gain blocks (Mini Circuits
  MAR6). The effective transimpedance is $R_{\rm
    eff}=590\,\rm{k}\Omega$.\label{fig:amplifier}}
\end{figure}


\begin{figure}
%\centerline{\includegraphics[width=0.9\columnwidth]{./plot/spectrum/spectrum2}}
\centerline{\includegraphics[width=0.9\columnwidth]{figure3.eps}}

\caption{Noise levels measured after amplification into a bandwidth
  $B=1\,$kHz. Between 20 and 120\,MHz, the total noise is measured from the
  photocurrent difference 
  $i_1-i_2$ with a balanced optical power impinging on both photodiodes comes
  close to the theoretical shot noise level of -52\,dBm given by
  (\ref{eq:shot_noise}). The current $i_1$ of a single photodiode reveals
  colored classical amplitude noise. The electronic
  noise is measured without any optical input.}
\label{fig:spectrum}
\end{figure}

To ensure that the fluctuations at the output of the amplifier chain are
dominated by 
quantum fluctuations of the vacuum field, the spectral power density at the
output of the amplifier chain is measured (see
Fig.~\ref{fig:spectrum}). With an optical power of 3.1\,mW 
received by each photodiode, a noise power of $P=-53.5\,$dBm (at 75\,MHz) in
a bandwidth of $B=1\,$kHz was measured. This is about 1.5\,dB lower
than the theoretically expected shot noise value (dashed trace) of 
\begin{equation}\label{eq:shot_noise}
P=\frac{4eIB{R_{\rm eff}}^2}{Z}\approx-52 \, \rm{dBm}\,,
\end{equation}
with an average photocurrent $I$ in each diode, the electron charge $e$ and a
load impedance $Z=50\,\Omega$.
The difference is compatible with uncertainties in determining the
transimpedance of the amplifier chain.
 to the theoretical value (\ref{eq:shot_noise}).
The measured total noise after the amplifier has a relatively flat power density in the
range of 20 to 120\,MHz, while the high pass filters in the amplifier chain
suppress low frequency fluctuations. The high end of the
pass band is defined by the cutoff frequency of the amplifier. To illustrate
the effectiveness of removing classical noise in the photocurrents, the
spectral power density of a photocurrent generated from a single diode is also
shown. Strong spectral peaks at various radio frequencies appear
that enter  the system probably via the laser diode current. For
completeness, the spectral power density of the electronic noise
of the amplifier chain is recorded without any light input, and found to be
at least 10\,dB below the total noise level, i.e., the total noise is
dominated by quantum fluctuations.


The amplified total noise signal is digitized into signed 16 bit wide words
$x_i$ at a sampling rate of 60\,MHz. The sampling rate is set to be lower than
the cut-off frequency of the noise signal in order to avoid temporal
correlation between samples. As shown in
Fig.~\ref{fig:autocorrelation}, the normalized autocorrelation
\begin{equation}
A(d)=\langle x_i\,x_{i+d}\rangle_n/\langle x_i^2\rangle_n
\end{equation}
computed over $n=10^6$ samples falls into the expected confidence
interval which indicates no significant correlation between samples.
\begin{figure}
% \centerline{\includegraphics[width=0.9\columnwidth]{./plot/autocorrelation/autocorrelation}}
\centerline{\includegraphics[width=0.9\columnwidth]{figure4.eps}}
  \caption{Autocorrelation of the total noise signal sampled at 60\,MHz,
    computed over $10^6$ samples (solid line), compared with the $2\sigma$
    confidence level (dashed line).} 
\label{fig:autocorrelation}
\end{figure}

\section{Entropy estimation}
The total noise we measured at the output of the detector consists of both
quantum noise and the electronic noise of the detector.  To determine how much
randomness we can safely extract from the system in the sense that it
originates from a quantum process, it is necessary to quantitatively estimate
the entropy contributed by the quantum noise.

To estimate the entropy of the quantum noise $H(X_q)$, we follow the approach
in \cite{Ma2012, Sanguinetti2014}: the measured total noise signal $X_t$ is
assumed to be the sum of the quantum noise signal $X_q$ and the
electronic noise signal $X_e$, such that $X_t=X_q+X_e$, where $X_q$ and $X_e$
are two random variables independent to each other. Furthermore, all three
variables $X_q$, $X_e$ and $X_t$ are assumed to have discrete distributions over
the digitization interval from $-2^{15}$ to $2^{15}-1$.
Since the origin of electronic noise is uncertain, we take the
worst case scenario that the adversary gains full knowledge of the electronic
noise, i.e., is able to predict the exact outcome of variable $X_e$
at any moment. In this case, the accessible amount of randomness in the acquired
total noise signal is quantified by the conditional entropy $H(X_t|X_e)$,
i.e. the amount of entropy left in the total signal, given full knowledge of
the electronic noise $X_e$. As the variables are assumed to be additive and
independent, the conditional entropy is calculated as
$H(X_t|X_e)=H(X_q+X_e|X_e)=H(X_q|X_e)=H(X_q)$.  


The variance of the total noise, $\sigma_t^2$, is given by the sum of the
variances $\sigma_q^2$ for the quantum noise, and $\sigma_e^2$ of the
electronic noise.  In an ensemble of $10^9$ samples, we find $\sigma_t=4504.41$
and $\sigma_e=1481.8$, which is directly measured by blocking all light (see
Fig.~\ref{fig:distribution}).
Note that for the total noise, the observed distribution is slightly skewed
compared to a Gaussian distribution [solid line in
Fig.~\ref{fig:distribution}(a)]. We believe this is due to a distortion in the
digitizer.
Assuming the quantum noise $X_q$ has a Gaussian
distribution~\cite{Glauber:1963}, we would assign
$\sigma_q^2=\sigma_t^2-\sigma_e^2\approx 4253.7^2$
To estimate the entropy
for a Gaussian distribution, we use the Shannon entropy
\begin{equation}\label{eq:shannon}
H_q=\sum_{x=1}^{2^{16}} -p_{q}(x)\log_2 p_{q}(x)\,,
\end{equation}
where $p_q(x)$is the probability
distribution of the quantum noise with variance
$\sigma_q^2$. For $\sigma_q\gg1$, ${H_q}$ can be well approximated by 
\begin{equation} \label{eq:differential_entropy}
H'(X_q)=\int\limits_{-\infty}^{+\infty}\!\!-{f(x)}\log_2{f(x)}\,\mathrm{d}x=\log_2(\sqrt{2{\pi}e}\,\sigma_q)\,,
\end{equation}
where $f(x)$ is a Gaussian probability density function with variance
$\sigma_q^2$, and $e$ the base of the natural logarithm~\footnote{One can show
  that
  $|H(X_q)-H'(X_q)|<\log_2{(\sqrt{2\pi}\sigma_q)}/(\sqrt{2\pi}\sigma_q)\approx 0.0013$\,bit  for $\sigma_q=4108$.}.
This yields 14.1 bits of entropy per 16-bit sample.


\begin{figure}
% \centerline{\includegraphics[width=0.9\columnwidth]{./plot/distribution/distribution}}
\centerline{\includegraphics[width=0.9\columnwidth]{figure5.eps}}
\caption{Probability distribution of the measured total output noise with
  variance ${{\sigma}_t}^2$ (a), electronic noise with variance
  ${{\sigma}_e}^2$ (b), and the estimated quantum noise with variance
  ${{\sigma}_q}^2$ (c). The filled areas in (a), (b) show the actual
  measurements over $10^9$ samples, the solid lines approximate the Gaussian distributions.}
\label{fig:distribution}
%\end{subfigure}
\end{figure}

We note that this numerical estimation of entropy only serves as an upper
bound of randomness, i.e. the maximum possible amount of entropy one can
extract from the source of randomness under the assumption of a Gaussian
distribution of the independent random variables $X_q$ and $X_e$. In
\cite{Syed2011, Haw2014}, entropy is estimated under the more strict
assumption that the electronic noise is not only known to a third party, but
also could be tampered with.

\section{Randomness extraction}
In many applications, random numbers are required to be not only
unpredictable, but also uniformly distributed. As such, the raw data at the
amplifier output cannot be directly used since they are non-uniformly
distributed. Randomness extraction is the essential process required to
convert our biased raw data into a uniformly distributed binary stream at the
final output~\cite{miklos:86}. 

Various implementations of randomness extractors have been reported, such as
Trevisan's extractor and Toeplitz-hashing extractor in~\cite{Ma2012},
random-matrix multiplication in~\cite{Sanguinetti2014}, or the family of
secure hashing algorithms (SHA) in~\cite{Gabriel2010}. 

In this work, we use a randomness extractor based on Linear
Feedback Shift Register (LFSR). The LFSRs are well known for quickly
generating long pseudo-random streams with little computational resources and
are in widespread use in communication applications for
spectrum whitening and other purposes~\cite{Krawczyk1994,Barkan2007}.


\begin{figure}
%\centerline{\includegraphics[width=0.9\columnwidth]{./plot/extractor/extractor1}}
\centerline{\includegraphics[width=0.9\columnwidth]{figure6.eps}}
\caption{Schematic of a randomness extractor based on a 63 bit long LFSR
  structure.}
\label{fig:extractor}
\end{figure}

Our randomness extractor is schematically shown in
Fig.~\ref{fig:extractor}. The main body of the extractor is a n-bit shift
register initialized with $n$ random bits. The extractor has a serial input
that receives the raw bit stream from the digitizer sampling the amplified
photodetector difference signal. At each clock cycle, one of the serialzied
input bits is XORed with selected bits of the shift register (also known as
the LFSR taps). The result is fed back into the left most cell, and all other
bits are shifted one cell to the right. At the same time, the result of 
the XOR operation builds up the output stream.
The taps of the shift register are chosen to generate a pseudo-random sequence
of maximal length ${2^n}-1$ in the case that the input receives an stream of
zeroes~\cite{Golomb:1981}.

At each clock cycle, the extractor receives one bit input and sends one bit out. Only a portion of output bits are kept as output random numbers while the rest are discarded, such that the amount of output bits does not exceed the entropy bound.


\begin{figure}
% \centerline{\includegraphics[width=0.9\columnwidth]{./plot/histogram/multiplot}}
\centerline{\includegraphics[width=0.9\columnwidth]{figure7.eps}}
\caption{Distribution of random data before (blue) and after (red) the randomness extractor, shown in time domain (left) and histogram (right).}
\label{fig:histogram}
\end{figure}

In our demonstration, we use a 63-bit shift register, in which cell 62 and 63
(rightmost two cells) are selected as taps. For every 16 output bits, 4 bits
are discarded in order to conservatively honor the entropy bound of 14.1 bits
per sample. The effect of the extractor on data distribution is visualized in
Fig.~\ref{fig:histogram}.
A merit of such an extractor is that it is of low circuit complexity, and can
be easily implemented either in high speed or low power technology. Therefore,
the extraction process does not limit the  random number generation rate, as
there is only one clock cycle necessary to convert a single bit. Adequate
circuit topology allows to further parallelize the processing of a complete 16
bit word form the converter into a 12 bit word of random bits in a single
clock step.



\section{Performance}

To evaluate the quality of the extracted random numbers, we apply two suites
of randomness tests: the statistical test suite from
NIST~\cite{NISTtestsuite}, and the ``Die-harder'' randomness test
battery~\cite{dieharder}. The output of our RNG passed both tests consistently
when evaluated over a sample of 50 gigabytes. We also tried compressing our random number data file with several commercial compression software (Gzip, winRAR, LZMA) and so far none of them can generate compressed files smaller than the original file size.


Our current setup can theoretically reach generation rate of 720\,Mbit/s with
the digitizer unit sampling at 60\,MHz and randomness extraction of 12 bits
out of 16. In practice, due to limitation of data transmission rate (USB2.0),
we can process random bits at a rate of 480\,Mbit/s. Apart from using
different interfaces, the practically
available rate random number generation can be greatly improved by extending
the bandwidth of the photodiodes, amplifiers, and digitizer devices,
while maintaining the relatively simple randomness extraction mechanism. Given
a fast enough data transfer to a host, the random number generation rate of
this scheme would probably be limited by the digitizer bandwidth.


\section{Conclusion}
In conclusion, we demonstrated a random number generation scheme by measuring
the vacuum fluctuations of the electromagnetic field. By estimating the amount
of usable entropy from quantum noise and using an efficient randomness
extractor, we are able to generate uniformly distributed random numbers at a
high rate from a fundamentally unpredictable quantum measurement.

We acknowledge support by the National Research Foundation (partly under grant
No. NRF-CRP12-2013-03) \& Ministry of Education in Singapore, partly through
the Academic Research Fund MOE2012-T3-1-009.


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  {\bibfnamefont {A.}~\bibnamefont {Gordon}}, \bibinfo {author} {\bibfnamefont
  {J.}~\bibnamefont {Quinn}}, \ and\ \bibinfo {author} {\bibfnamefont
  {N.}~\bibnamefont {Stavrakas}},\ }\href
  {http://www.google.com/patents/US6745217} {\enquote {\bibinfo {title} {Random
  number generator based on the spontaneous alpha-decay},}\ } (\bibinfo {year}
  {2004}),\ \bibinfo {note} {uS Patent 6,745,217}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Stipcevic}\ and\ \citenamefont
  {Rogina}(2007)}]{Stipcevic:07}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
  {Stipcevic}}\ and\ \bibinfo {author} {\bibfnamefont {B.~Medved}\ \bibnamefont
  {Rogina}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Quantum random
  number generator based on photonic emission in semiconductors},}\ }\href
  {\doibase 10.1063/1.2720728} {\bibfield  {journal} {\bibinfo  {journal} {Rev.
  Sci. Instrum.}\ }\textbf {\bibinfo {volume} {78}},\ \bibinfo {pages} {045104}
  (\bibinfo {year} {2007})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {F\"{u}rst}\ \emph {et~al.}(2010)\citenamefont
  {F\"{u}rst}, \citenamefont {Weier}, \citenamefont {Nauerth}, \citenamefont
  {Marangon}, \citenamefont {Kurtsiefer},\ and\ \citenamefont
  {Weinfurter}}]{Fuerst2010}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Martin}\ \bibnamefont
  {F\"{u}rst}}, \bibinfo {author} {\bibfnamefont {Henning}\ \bibnamefont
  {Weier}}, \bibinfo {author} {\bibfnamefont {Sebastian}\ \bibnamefont
  {Nauerth}}, \bibinfo {author} {\bibfnamefont {Davide~G.}\ \bibnamefont
  {Marangon}}, \bibinfo {author} {\bibfnamefont {Christian}\ \bibnamefont
  {Kurtsiefer}}, \ and\ \bibinfo {author} {\bibfnamefont {Harald}\ \bibnamefont
  {Weinfurter}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {High speed
  optical quantum random number generation},}\ }\href {\doibase
  10.1364/oe.18.013029} {\bibfield  {journal} {\bibinfo  {journal} {Optics
  Express}\ }\textbf {\bibinfo {volume} {18}},\ \bibinfo {pages} {13029}
  (\bibinfo {year} {2010})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Jennewein}\ \emph {et~al.}(2000)\citenamefont
  {Jennewein}, \citenamefont {Achleitner}, \citenamefont {Weihs}, \citenamefont
  {Weinfurter},\ and\ \citenamefont {Zeilinger}}]{Jennewein2000}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Thomas}\ \bibnamefont
  {Jennewein}}, \bibinfo {author} {\bibfnamefont {Ulrich}\ \bibnamefont
  {Achleitner}}, \bibinfo {author} {\bibfnamefont {Gregor}\ \bibnamefont
  {Weihs}}, \bibinfo {author} {\bibfnamefont {Harald}\ \bibnamefont
  {Weinfurter}}, \ and\ \bibinfo {author} {\bibfnamefont {Anton}\ \bibnamefont
  {Zeilinger}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {A fast and
  compact quantum random number generator},}\ }\href {\doibase
  10.1063/1.1150518} {\bibfield  {journal} {\bibinfo  {journal} {Review of
  Scientific Instruments}\ }\textbf {\bibinfo {volume} {71}},\ \bibinfo {pages}
  {1675} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Stefanov}\ \emph {et~al.}(2000)\citenamefont
  {Stefanov}, \citenamefont {Gisin}, \citenamefont {Guinnard}, \citenamefont
  {Guinnard},\ and\ \citenamefont {Zbinden}}]{Stefanov2000}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
  {Stefanov}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Gisin}},
  \bibinfo {author} {\bibfnamefont {O.}~\bibnamefont {Guinnard}}, \bibinfo
  {author} {\bibfnamefont {L.}~\bibnamefont {Guinnard}}, \ and\ \bibinfo
  {author} {\bibfnamefont {H.}~\bibnamefont {Zbinden}},\ }\bibfield  {title}
  {\enquote {\bibinfo {title} {Optical quantum random number generator},}\
  }\href {\doibase 10.1080/09500340008233380} {\bibfield  {journal} {\bibinfo
  {journal} {Journal of Modern Optics}\ }\textbf {\bibinfo {volume} {47}},\
  \bibinfo {pages} {595--598} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Frauchiger}\ and\ \citenamefont
  {Renner}(2013)}]{Frauchiger2013}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Daniela}\ \bibnamefont
  {Frauchiger}}\ and\ \bibinfo {author} {\bibfnamefont {Renato}\ \bibnamefont
  {Renner}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Truly random
  number generation: an example},}\ }\href {\doibase 10.1117/12.2032183}
  {\bibfield  {journal} {\bibinfo  {journal} {Emerging Technologies in Security
  and Defence; and Quantum Security II; and Unmanned Sensor Systems X}\ }
  (\bibinfo {year} {2013}),\ 10.1117/12.2032183}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Gabriel}\ \emph {et~al.}(2010)\citenamefont
  {Gabriel}, \citenamefont {Wittmann}, \citenamefont {Sych}, \citenamefont
  {Dong}, \citenamefont {Mauerer}, \citenamefont {Andersen}, \citenamefont
  {Marquardt},\ and\ \citenamefont {Leuchs}}]{Gabriel2010}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Christian}\
  \bibnamefont {Gabriel}}, \bibinfo {author} {\bibfnamefont {Christoffer}\
  \bibnamefont {Wittmann}}, \bibinfo {author} {\bibfnamefont {Denis}\
  \bibnamefont {Sych}}, \bibinfo {author} {\bibfnamefont {Ruifang}\
  \bibnamefont {Dong}}, \bibinfo {author} {\bibfnamefont {Wolfgang}\
  \bibnamefont {Mauerer}}, \bibinfo {author} {\bibfnamefont {Ulrik~L.}\
  \bibnamefont {Andersen}}, \bibinfo {author} {\bibfnamefont {Christoph}\
  \bibnamefont {Marquardt}}, \ and\ \bibinfo {author} {\bibfnamefont {Gerd}\
  \bibnamefont {Leuchs}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {A
  generator for unique quantum random numbers based on vacuum states},}\ }\href
  {\doibase 10.1038/nphoton.2010.197} {\bibfield  {journal} {\bibinfo
  {journal} {Nature Photon}\ }\textbf {\bibinfo {volume} {4}},\ \bibinfo
  {pages} {711–715} (\bibinfo {year} {2010})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Symul}\ \emph {et~al.}(2011)\citenamefont {Symul},
  \citenamefont {Assad},\ and\ \citenamefont {Lam}}]{Syed2011}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
  {Symul}}, \bibinfo {author} {\bibfnamefont {S.~M.}\ \bibnamefont {Assad}}, \
  and\ \bibinfo {author} {\bibfnamefont {P.~K.}\ \bibnamefont {Lam}},\
  }\bibfield  {title} {\enquote {\bibinfo {title} {Real time demonstration of
  high bitrate quantum random number generation with coherent laser light},}\
  }\href {\doibase http://dx.doi.org/10.1063/1.3597793} {\bibfield  {journal}
  {\bibinfo  {journal} {Applied Physics Letters}\ }\textbf {\bibinfo {volume}
  {98}},\ \bibinfo {eid} {231103} (\bibinfo {year} {2011})}\BibitemShut
  {NoStop}%
\bibitem [{\citenamefont {Kanter}\ \emph {et~al.}(2009)\citenamefont {Kanter},
  \citenamefont {Aviad}, \citenamefont {Reidler}, \citenamefont {Cohen},\ and\
  \citenamefont {Rosenbluh}}]{Kanter2009}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Ido}\ \bibnamefont
  {Kanter}}, \bibinfo {author} {\bibfnamefont {Yaara}\ \bibnamefont {Aviad}},
  \bibinfo {author} {\bibfnamefont {Igor}\ \bibnamefont {Reidler}}, \bibinfo
  {author} {\bibfnamefont {Elad}\ \bibnamefont {Cohen}}, \ and\ \bibinfo
  {author} {\bibfnamefont {Michael}\ \bibnamefont {Rosenbluh}},\ }\bibfield
  {title} {\enquote {\bibinfo {title} {An optical ultrafast random bit
  generator},}\ }\href {\doibase 10.1038/nphoton.2009.235} {\bibfield
  {journal} {\bibinfo  {journal} {Nature Photon}\ }\textbf {\bibinfo {volume}
  {4}},\ \bibinfo {pages} {58–61} (\bibinfo {year} {2009})}\BibitemShut
  {NoStop}%
\bibitem [{\citenamefont {Abellán}\ \emph {et~al.}(2014)\citenamefont
  {Abellán}, \citenamefont {Amaya}, \citenamefont {Jofre}, \citenamefont
  {Curty}, \citenamefont {Acín}, \citenamefont {Capmany}, \citenamefont
  {Pruneri},\ and\ \citenamefont {Mitchell}}]{Abellan2014}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
  {Abellán}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Amaya}},
  \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Jofre}}, \bibinfo
  {author} {\bibfnamefont {M.}~\bibnamefont {Curty}}, \bibinfo {author}
  {\bibfnamefont {A.}~\bibnamefont {Acín}}, \bibinfo {author} {\bibfnamefont
  {J.}~\bibnamefont {Capmany}}, \bibinfo {author} {\bibfnamefont
  {V.}~\bibnamefont {Pruneri}}, \ and\ \bibinfo {author} {\bibfnamefont
  {M.~W.}\ \bibnamefont {Mitchell}},\ }\bibfield  {title} {\enquote {\bibinfo
  {title} {Ultra-fast quantum randomness generation by accelerated phase
  diffusion in a pulsed laser diode},}\ }\href {\doibase 10.1364/oe.22.001645}
  {\bibfield  {journal} {\bibinfo  {journal} {Optics Express}\ }\textbf
  {\bibinfo {volume} {22}},\ \bibinfo {pages} {1645} (\bibinfo {year}
  {2014})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Sanguinetti}\ \emph {et~al.}(2014)\citenamefont
  {Sanguinetti}, \citenamefont {Martin}, \citenamefont {Zbinden},\ and\
  \citenamefont {Gisin}}]{Sanguinetti2014}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Bruno}\ \bibnamefont
  {Sanguinetti}}, \bibinfo {author} {\bibfnamefont {Anthony}\ \bibnamefont
  {Martin}}, \bibinfo {author} {\bibfnamefont {Hugo}\ \bibnamefont {Zbinden}},
  \ and\ \bibinfo {author} {\bibfnamefont {Nicolas}\ \bibnamefont {Gisin}},\
  }\bibfield  {title} {\enquote {\bibinfo {title} {Quantum random number
  generation on a mobile phone},}\ }\href@noop {} {\bibfield  {journal}
  {\bibinfo  {journal} {Phys. Rev. X}\ }\textbf {\bibinfo {volume} {4}},\
  \bibinfo {pages} {031056} (\bibinfo {year} {2014})},\ \Eprint
  {http://arxiv.org/abs/1405.0435} {1405.0435} \BibitemShut {NoStop}%
\bibitem [{\citenamefont {Jakeman}\ \emph {et~al.}(1975)\citenamefont
  {Jakeman}, \citenamefont {Oliver},\ and\ \citenamefont {Pike}}]{jakeman1974}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {E}~\bibnamefont
  {Jakeman}}, \bibinfo {author} {\bibfnamefont {C.J.}\ \bibnamefont {Oliver}},
  \ and\ \bibinfo {author} {\bibfnamefont {E.R.}\ \bibnamefont {Pike}},\
  }\bibfield  {title} {\enquote {\bibinfo {title} {Optical homodyne
  detection},}\ }\href {\doibase 10.1080/00018737500101421} {\bibfield
  {journal} {\bibinfo  {journal} {Advances in Physics}\ }\textbf {\bibinfo
  {volume} {24}},\ \bibinfo {pages} {349} (\bibinfo {year} {1975})}\BibitemShut
  {NoStop}%
\bibitem [{\citenamefont {Yuen}\ and\ \citenamefont {Chan}(1983)}]{yuen1983}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Horace~P}\
  \bibnamefont {Yuen}}\ and\ \bibinfo {author} {\bibfnamefont {Vincent~W.S.}\
  \bibnamefont {Chan}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Nosie
  in homodyne and heterodyne detection},}\ }\href {\doibase
  10.1364/OL.8.000177} {\bibfield  {journal} {\bibinfo  {journal} {Optics
  Letters}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages} {177--179}
  (\bibinfo {year} {1983})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Glauber}(1963)}]{Glauber:1963}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Roy~J.}\ \bibnamefont
  {Glauber}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Coherent and
  incoherent states of the radiation field},}\ }\href {\doibase
  10.1103/PhysRev.131.2766} {\bibfield  {journal} {\bibinfo  {journal} {Phys.
  Rev.}\ }\textbf {\bibinfo {volume} {131}},\ \bibinfo {pages} {2766--2788}
  (\bibinfo {year} {1963})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Schumaker}(1984)}]{schumaker1984}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Bonny~L}\ \bibnamefont
  {Schumaker}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Noise in
  homodyne detection},}\ }\href {\doibase 10.1364/OL.9.000189} {\bibfield
  {journal} {\bibinfo  {journal} {Optics Lettes}\ }\textbf {\bibinfo {volume}
  {9}},\ \bibinfo {pages} {189} (\bibinfo {year} {1984})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Ma}\ \emph {et~al.}(2013)\citenamefont {Ma},
  \citenamefont {Xu}, \citenamefont {Xu}, \citenamefont {Tan}, \citenamefont
  {Qi},\ and\ \citenamefont {Lo}}]{Ma2012}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Xiongfeng}\
  \bibnamefont {Ma}}, \bibinfo {author} {\bibfnamefont {Feihu}\ \bibnamefont
  {Xu}}, \bibinfo {author} {\bibfnamefont {He}~\bibnamefont {Xu}}, \bibinfo
  {author} {\bibfnamefont {Xiaoqing}\ \bibnamefont {Tan}}, \bibinfo {author}
  {\bibfnamefont {Bing}\ \bibnamefont {Qi}}, \ and\ \bibinfo {author}
  {\bibfnamefont {Hoi-Kwong}\ \bibnamefont {Lo}},\ }\bibfield  {title}
  {\enquote {\bibinfo {title} {Postprocessing for quantum random-number
  generators: Entropy evaluation and randomness extraction},}\ }\href {\doibase
  10.1103/PhysRevA.87.062327} {\bibfield  {journal} {\bibinfo  {journal} {Phys.
  Rev. A}\ }\textbf {\bibinfo {volume} {87}},\ \bibinfo {pages} {062327}
  (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
\bibitem [{Note1()}]{Note1}%
  \BibitemOpen
  \bibinfo {note} {One can show that $|H(X_q)-H'(X_q)|<\mathop {\mathgroup
  \symoperators log}\nolimits _2{(\protect \sqrt {2\pi }\sigma _q)}/(\protect
  \sqrt {2\pi }\sigma _q)\approx 0.0013$\protect \,bit for $\sigma
  _q=4108$.}\BibitemShut {Stop}%
\bibitem [{\citenamefont {Haw}\ \emph {et~al.}(2015)\citenamefont {Haw},
  \citenamefont {Assad}, \citenamefont {Lance}, \citenamefont {Ng},
  \citenamefont {Sharma}, \citenamefont {Lam},\ and\ \citenamefont
  {Symul}}]{Haw2014}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.~Y.}\ \bibnamefont
  {Haw}}, \bibinfo {author} {\bibfnamefont {S.~M.}\ \bibnamefont {Assad}},
  \bibinfo {author} {\bibfnamefont {A.~M.}\ \bibnamefont {Lance}}, \bibinfo
  {author} {\bibfnamefont {N.~H.~Y.}\ \bibnamefont {Ng}}, \bibinfo {author}
  {\bibfnamefont {V.}~\bibnamefont {Sharma}}, \bibinfo {author} {\bibfnamefont
  {P.~K.}\ \bibnamefont {Lam}}, \ and\ \bibinfo {author} {\bibfnamefont
  {T.}~\bibnamefont {Symul}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
  {Maximization of extractable randomness in a quantum random-number
  generator},}\ }\href {\doibase 10.1103/PhysRevApplied.3.054004} {\bibfield
  {journal} {\bibinfo  {journal} {Phys. Rev. Applied}\ }\textbf {\bibinfo
  {volume} {3}},\ \bibinfo {pages} {054004} (\bibinfo {year}
  {2015})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Santha}\ and\ \citenamefont
  {Vazirani}(1986)}]{miklos:86}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Miklos}\ \bibnamefont
  {Santha}}\ and\ \bibinfo {author} {\bibfnamefont {Umesh~V.}\ \bibnamefont
  {Vazirani}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Generating
  quasi-random sequences from semi-random sources},}\ }\href {\doibase
  10.1016/0022-0000(86)90044-9} {\bibfield  {journal} {\bibinfo  {journal} {J.
  Comput. Syst. Sci.}\ }\textbf {\bibinfo {volume} {33}},\ \bibinfo {pages}
  {75} (\bibinfo {year} {1986})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Krawczyk}(1994)}]{Krawczyk1994}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Hugo}\ \bibnamefont
  {Krawczyk}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Lfsr-based
  hashing and authentication},}\ }\href {\doibase 10.1007/3-540-48658-5_15}
  {\bibfield  {journal} {\bibinfo  {journal} {Advances in Cryptology — CRYPTO
  ’94}\ ,\ \bibinfo {pages} {129–139}} (\bibinfo {year}
  {1994})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Barkan}\ \emph {et~al.}(2007)\citenamefont {Barkan},
  \citenamefont {Biham},\ and\ \citenamefont {Keller}}]{Barkan2007}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Elad}\ \bibnamefont
  {Barkan}}, \bibinfo {author} {\bibfnamefont {Eli}\ \bibnamefont {Biham}}, \
  and\ \bibinfo {author} {\bibfnamefont {Nathan}\ \bibnamefont {Keller}},\
  }\bibfield  {title} {\enquote {\bibinfo {title} {Instant ciphertext-only
  cryptanalysis of {GSM} encrypted communication},}\ }\href {\doibase
  10.1007/s00145-007-9001-y} {\bibfield  {journal} {\bibinfo  {journal} {J
  Cryptol}\ }\textbf {\bibinfo {volume} {21}},\ \bibinfo {pages} {392–429}
  (\bibinfo {year} {2007})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Golomb}(1981)}]{Golomb:1981}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Solomon~W.}\
  \bibnamefont {Golomb}},\ }\href {http://dl.acm.org/citation.cfm?id=578271}
  {\emph {\bibinfo {title} {Shift Register Sequences}}}\ (\bibinfo  {publisher}
  {Aegean Park Press, Laguna Hills, CA, USA},\ \bibinfo {year}
  {1981})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Rukhin}\ \emph {et~al.}(2010)\citenamefont {Rukhin},
  \citenamefont {Soto}, \citenamefont {Nechvatal}, \citenamefont {Smid},
  \citenamefont {Barker}, \citenamefont {Leigh}, \citenamefont {Levenson},
  \citenamefont {Vangel}, \citenamefont {Banks}, \citenamefont {Heckert},
  \citenamefont {Dray},\ and\ \citenamefont {Vo}}]{NISTtestsuite}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Andrew}\ \bibnamefont
  {Rukhin}}, \bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont {Soto}},
  \bibinfo {author} {\bibfnamefont {James}\ \bibnamefont {Nechvatal}}, \bibinfo
  {author} {\bibfnamefont {Miles}\ \bibnamefont {Smid}}, \bibinfo {author}
  {\bibfnamefont {Elaine}\ \bibnamefont {Barker}}, \bibinfo {author}
  {\bibfnamefont {Stefan}\ \bibnamefont {Leigh}}, \bibinfo {author}
  {\bibfnamefont {Mark}\ \bibnamefont {Levenson}}, \bibinfo {author}
  {\bibfnamefont {Mark}\ \bibnamefont {Vangel}}, \bibinfo {author}
  {\bibfnamefont {David}\ \bibnamefont {Banks}}, \bibinfo {author}
  {\bibfnamefont {Alan}\ \bibnamefont {Heckert}}, \bibinfo {author}
  {\bibfnamefont {James}\ \bibnamefont {Dray}}, \ and\ \bibinfo {author}
  {\bibfnamefont {San}\ \bibnamefont {Vo}},\ }\href@noop {} {\emph {\bibinfo
  {title} {A Statistical Test Suite for Random and Pseudorandom Number
  Generators for Cryptographic Applications}}},\ \bibinfo {organization}
  {National Institute of Standards and Technology} (\bibinfo {year}
  {2010})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Robert G. Brown~(rgb)}(2004)}]{dieharder}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {David~Bauer}\
  \bibnamefont {Robert G. Brown~(rgb)}, \bibfnamefont {Dirk~Eddelbuettel}},\
  }\href@noop {} {\enquote {\bibinfo {title} {Dieharder: A random number test
  suite},}\ } (\bibinfo {year} {2004})\BibitemShut {NoStop}%
\end{thebibliography}%


\end{document}
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