## 3 Test Results  
The test is divided into two parts, including \(3\sigma\) criterion and widely used test batteries such as DIEHARD or NIST-STS. Uniformity are also included in the batteries. Firstly, data are randomly chosen to perform the \(3\sigma\) test to compare the difference between raw data and extracted numbers. For raw data, the sample size equals the sampling depth of 50 M points/samples with 12-bit resolution in our implementation. While for extracted sequence, the sample size is 420 Mbits, which is consecutive data in 0.7 s after post-processing. The results are shown in Figures 3 and 4. Apparently post-processing methods impressively reduce the low-order auto-correlation coefficient, due to the limited bandwidth of devices may cause some correlation in the adjacent samples of raw data, particularly in oversampled scenarios, where sampling rate and auto-correlation trade-off should be carefully dealed with in practical devices. Figure 3: Auto-correlation coefficient \(a_{k}\) of raw data with different self-delay value \(k\), ranging from 1 to 1000 samples (\(k=0\) always leads to \(a_{k}=1\), thus makes no sense). The dashed line indicates reference calculated by the \(3\sigma\) criterion. Apparently there exists a relatively high correlation among adjacent 10 samples, that occasionally goes beyond the \(3\sigma\) reference threshold, due to the limited bandwidth of devices. In order to evaluate randomness, we also adopt widely-accepted randomness test batteries, namely DIEHARD and NIST-STS test, both of which are hypothesis tests of a statistically based randomness test, with a couple of \(p\)-values indicating whether to accept or reject the hypothesis in each sub-test. Generally speaking, if all the final \(p\)-values located between \([0.01,0.99]\) (with default significance level \(\alpha=0.01\)), the whole test is considered successful. Random sequences extracted from either the \(m\)-LSB or Toeplitz hashing method should pass both test batteries, and we choose one typical sequence from the Toeplitz hashing method as an example: the result of DIEHARD test is shown in Table 1, and result of NIST-STS test is shown in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Statistical Test** & \(p\)**-Value** & **Result** \\ \hline Birthday spacings & 0.122824[KS] & success \\ Overlapping permutations & 0.430620 & success \\ Ranks of 31 \(\times\) 31 matrices & 0.605645 & success \\ Ranks of 32 \(\times\) 32 matrices & 0.427548 & success \\ Ranks of 6 \(\times\) 8 matrices & 0.260611[KS] & success \\ Monkey tests on 20-bit words & 0.136669[KS] & success \\ Monkey test OPSP & 0.43930[KS] & success \\ Monkey test OQSO & 0.68062[KS] & success \\ Monkey test DNA & 0.61412[KS] & success \\ Count 1’s in stream of bytes & 0.662425 & success \\ Count 1’s in specific bytes & 0.561794[KS] & success \\ Parking lot test & 0.312073[KS] & success \\ Minimum distance test & 0.377192[KS] & success \\ Random spheres test & 0.440218[KS] & success \\ Squeeze test & 0.019830[KS] & success \\ Overlapping sums test & 0.053688[KS] & success \\ Run test(up) & 0.314213 & success \\ Run test(down) & 0.492526 & success \\ Craps test No. of wins & 0.420491 & success \\ Craps test throw/game & 0.965724 & success \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The result of the DIEHARD test after post-processing. DIEHARD includes 18 terms and 20 randomness test results, each with a \(p\)-value. For the sub-tests with multiple \(p\)-values, a Kolmogorov-Smirnov (KS) test is performed to obtain the final \(p\)-value. It is considered successful for certain sub-tests if the final \(p\)-value is between \([0.01,0.99]\) (significance level \(\alpha=0.01\)). The length of test sequence is 420 Mbits. Figure 4: Auto-correlation coefficient \(a_{k}\) of extracted sequences. Pictured in (**a**) is auto-correlation with different self-delay value \(k\), ranging from 1 to 300 bits. Pictured in (**b**) is a first-order auto-correlation coefficient \(a_{1}\) with different sequence length, ranging from 1 M bit to 420 M bits.