## 2 Delay-Free Phase Noise QRNG Scheme
### Post-Processing Method  
The raw data, measured from the phase noise of laser diode, is approximately a Gaussian variable on the probability distribution function (PDF) [53; 54]. However, random sequences of general QRNG schemes and applications should be a uniform distribution, hence post-processing is essential. Another function of post-processing is eliminating the unexpected randomness from the environment, which may be utilized by the adversary Eve, specifically the classical noise. Therefore, generally speaking,a traditional post-processing method includes two phases: entropy estimation and randomness extraction. The entropy estimation phase calculate the upper bound of randomness, which can be extracted from the raw data as a discretized random variable \(X_{dis}\), based on the observable parameters. Then, one can adopt various extractors to distill the randomness calculated before in the randomness extraction phase. There exist two efficient methods in QRNG post-processing, based on different devices in implementation and application requirements, namely the \(m\)-least significant bit (\(m\)-LSB) method and universal (Toeplitz) hashing method. The former one, \(m\)-LSB method, belongs to the deterministic extractor. The \(m\)-LSB method is extremely simple to implement both on hardware and software, and could run at very high sampling and generation rate. One just truncates the raw data, takes the last \(m\)-bit random numbers and outputs the final sequence (logical exclusive OR (XOR) operation is also optional if necessary). The reason for taking the LSB instead of its counterpart, the most significant bit (MSB), is that the LSB has a better distribution and lower auto-correlation coefficient after post-processing, and thus is more difficult to be predicted by the adversary. This method is quite effective if the implementation is trusted and has a relatively high quantum-to-classical noise ratio (QCNR). However, in untrusted device scenarios, one can still extract several bits with high sampling resolution. We adopt a \(m\)-LSB method by treating our implementation as a trusted-device scenario and according to the analysis in [51]. It is secure for \(m\)-LSB to truncate four bits out of a 16-bit discretized signal in noise-free cases, and secure to truncate five (seven) bits out of 16 bits with the deviation of classical noise \(\sigma_{E}\) three (four) times larger than quantum noise \(\sigma_{Q}\). In fact, quantum noise is dominate in our implementation, thus truncate a moderate number of six bits from raw data (including the unoccupied three most significant bits) to form the extracted sequence is adequately conservative for our scheme. Therefore, the generation rate of adopting \(m\)-LSB method is \(100\times 6=600\) Mb/s. Universal hashing method is another post-processing method often chosen in QRNG schemes, which belongs to the seeded extractor, indicating that this method should consume some short random seed to generate the universal hashing functions. Among these functions, the Toeplitz matrix is an outstanding solution for its low complexity in computation and implementation. For a binary Toeplitz matrix utilized for QRNG post-processing, the size of the original matrix is \(M\times N\), where \(N\) is the size of raw data, and \(M\) is the size of extracted sequence. The ratio \(M/N\) is a crucial parameter which is closely related to the min-entropy \(H_{\infty}(X)\) calculated in the entropy estimation phase:  
Figure 2: The waveform at photo-detectors in time domain, recorded and shown by oscilloscope (DSAX-91604A, Agilent). Two signals below are the input signal at the frequency mixer, and the signal above with smaller scale is the output signal of mixer. Pictured in (**a**) is the stable waveform with appropriate center wavelength selection. The waveform is very stable during the test, and sine shape phase noise signal is distilled by the frequency mixer and band-pass filter.