# MRNG: Accessing Cosmic Radiation as an Entropy Source for a Non-Deterministic Random Number Generator

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- We show that random numbers can be extracted from a yet unused entropy source for randomness, namely, UHECR.
- We describe four methods that, in combination, extract randomness out of UHECR, as discussed in Section 4.2.2.
- Our proposed Muon Random Number Generator (MRNG) prototype (named in honor of the research on muons collected among UHECR) does not need any external services or devices in order to extract randomness from UHECR. Our MRNG prototype works on any Android smartphone with an API level of 14 or higher, including Android 4.0, which was released in 2011. This is stated in more detail in Section 4.2.
- Most importantly, we have proved that our extracted random sequence, out of the UHECR entropy source, is truly random when tested against NIST SP.800-22 statistical test suite. This is stated within Section 6.
- Furthermore, we may have accidentally discovered a new splash-like representation of (presumably) UHECR, as discussed and shown in Section 7.1.

## 2. Threat Model

## 3. Background

#### 3.1. Random Number Generators

#### 3.2. Relevant Existing Random Number Generators

#### 3.2.1. RNG Proposed by Park

#### 3.2.2. RNG Proposed by Zhang

#### 3.2.3. RNG Proposed by Leschiutta

#### 3.2.4. RNG Proposed by Chen

#### 3.2.5. RNG Proposed by Krhovják

#### 3.2.6. RNG Proposed by Reezwana

#### 3.3. Random Number Test Suites

- (NIST Test 2.1) The Frequency (Monobit) Test;
| $n\ge $ 100 |

- (NIST Test 2.2) Frequency Test within a Block;
| $n\ge $ 100 |

- (NIST Test 2.3) The Runs Test;
| $n\ge $ 100 |

- (NIST Test 2.4) Tests for the Longest-Run-of-Ones in a Block;
| $n\ge $ 128 |

- (NIST Test 2.12) The Approximate Entropy Test; 10.0 cm
| $n\ge $ 100 |

- (NIST Test 2.13) The Cumulative Sums (Cusum) Test.
| $n\ge $ 100 |

#### 3.4. Cosmic Radiation

^{−1}, which is $99.4$ % of the speed of light ($0.994\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}c$); they have a mean lifetime of $2.197$ μs and a half-life of $1.523$ μs [41,42]. It can be assumed that there is one muon hit within an area of 1 cm${}^{2}$ every minute. The detection rate for muons doubles with every 1500 m increase in altitude. Muons are interesting for several reasons. They not only prove Einstein’s special relativity theories, namely time dilation and length contraction [43,44], but can also interact with semiconductor material, despite having weak interference properties [45,46,47].

#### 3.4.1. Single Event Effect

#### SEU Incidents

#### 3.4.2. Smartphone-Based Cosmic Ray Detectors

## 4. Experiment

#### 4.1. Experiment Setup

#### 4.2. Experiment Implementation

#### 4.2.1. Application

#### 4.2.2. Algorithm

- P1 Time.

- P2 Position.

- P3 Color.

- P4 Outlier.

#### 4.3. Experiment Execution

## 5. Evaluation

**Table 1.**Excerpt showing 6 of the hits visible in Figure 3 alongside the extracted random sequences P1, P2, P3, and P4; the length of each sequence is indicated in the P#L column, and the overall length of the sequence from mode MRNG-P124 is shown in the P124L column.

Timestamp | P124L | P#L | P1–P4 |
---|---|---|---|

1647655594901 | 52 | 17 2 34 33 | P1: 10111001010110101P2: 11P3: 1101001010000101011001011010010001P4: 110100101000010101100101101000001 |

1647670326687 | 46 | 15 2 29 29 | P1: 110100000111111P2: 11P3: 11000011100101111010001111110P4: 11000011100101111010001111110 |

1647889001366 | 49 | 11 2 44 36 | P1: 10101010110P2: 00P3: 11100001101000111000101100001111100101100011P4: 100011010011001011000011111001011001 |

1647992216262 | 37 | 14 2 27 21 | P1: 11111110000110P2: 00P3: 111110011110101110110110111P4: 110011110101110111011 |

1648193196369 | 34 | 17 2 17 15 | P1: 10111100001110001P2: 10P3: 00100110111000010P4: 001001011000010 |

1648302783786 | 28 | 17 2 11 9 | P1: 10100011101001010P2: 01P3: 11111111110P4: 111111110 |

#### Detection

^{−6}; this means that among 1,000,000 detections, it is likely that, on average, 3 detections share the same centered position on the image. In contradiction to this, Table 2 counts 3551 detections out of 3830 that share the same position. However, such detections are characterized as so-called hot pixels by the inventors of the forked CREDO application. Further, Homola et al. [10], Bar et al. [63] discuss the possibility that detections not exceeding 2 illuminated pixels are most likely not caused by cosmic radiation. Removing detections as stated above gives us a final set of 414 hits collected throughout 8 devices. This suggests that a hit caused by cosmic radiation can be expected every $62.7778$ min over all 8 devices, on average.

## 6. Results

## 7. Discussion

#### 7.1. Splash-like Particle Representation

#### 7.2. Muon Random Number Generator (MRNG)

#### 7.3. Reproducibility

#### 7.4. Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Miscellaneous

**Table A1.**Showing the sequence concatenations along with the extracted random bits. The ‘Pass’ column states if a sequence has successfully passed the NIST SP.800-22 statistical test suite.

Filename | Bits | Pass |
---|---|---|

MRNG-P1234.txt | 157.843 | NO |

MRNG-P123.txt | 153.087 | NO |

MRNG-P124.txt | 12.052 | YES |

MRNG-RP1234.txt | 389.890 | NO |

MRNG-RP123.txt | 361.987 | NO |

MRNG-RP124.txt | 126.363 | YES |

## Appendix B. Code Snippets

Listing A1. Displays P4 outlier detection algorithm. |

for x in range (crop_bitmap.width): |

for y in range (crop_bitmap.height): |

red, green, blue, alpha = crop_bitmap.getpixel ((y,x)) |

if prev_red > 0 or prev_green > 0 or prev_blue > 0: |

loc_dist_red = abs (prev_red - red) |

loc_dist_green = abs (prev_green - green) |

loc_dist_blue = abs (prev_blue - blue) |

if red > epsilon or green > epsilon or blue > epsilon: |

if (loc_dist_red > avg_dist_red ∗ outlier_multiplier or |

loc_dist_green > avg_dist_green ∗ outlier_multiplier or |

loc_dist_blue > avg_dist_blue ∗ outlier_multiplier): |

mrng_raw_p4_outlier=(mrng_raw_p4_outlier + str (red)+";" |

+str (green) + ";" + str (blue) + ";" + str (alpha)+";" |

+str (x) + ";" + str (y) + ";\n") |

mrng_p4_outlier = (mrng_p4_outlier |

+ str (((red % 2) + (green % 2) + (blue % 2)) % 2)) |

prev_red = red |

prev_green = green |

prev_blue = blue |

## Appendix C. ‘Hits’ Supplemental Material

**Figure A1.**The first row represents data obtained from the NOAA Space Weather Prediction Center: ‘OVATION Aurora Model Forecast North’ [67,68,69], ‘Solar Wind Predicted at Earth Geospace Timeline Lates 24 h’ [70]—depicted in inverted colors, ‘Aurora Hemispheric Power Tabular Values—Ovation Aurora Short Term Forecast’ [73], ‘Estimated Planetary K index (3 h data)’ [71]. The second row shows the image of the hit, the P4 representation of the used pixel (colored in black), and the textual data of the extracted random sequence.

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**Figure 1.**The first row represents fetched data from the NOAA Space Weather Prediction Center: ‘OVATION Aurora Model Forecast North’ [67,68,69], ‘Solar Wind Predicted at Earth Geospace Timeline Lates 24 h’ [70]—depicted in inverted colors, ‘Aurora Hemispheric Power Tabular Values—Ovation Aurora Short Term Forecast’ [73], ‘Estimated Planetary K index (3 h data)’ [71], whereas the second row shows the image of the hit, the P4 representation of the used pixel colored in black, and the textual data of the extracted random sequence.

**Figure 3.**The hits of (presumably) UHECR and muons, respectively, cropped into a 32 × 32 frame, where the first row represents spots, the second represents tracks, and the third and last represent worms. The number beneath each 32 × 32 frame represents the UNIX timestamp when the hit is detected.

**Figure 4.**Comparison of the pixels taken into consideration for the random sequence from the visual representation of UHECR (first row) by feature P3 (second row) or feature P4 (last row).

**Figure 5.**The results of the outcome from the Borel normality criterion [34] test. It analyzes the mean frequencies of bit patterns in 414 MRNG-P124 sequences (12 052 bits), with error bars representing the standard error. The red lines represent 12.5%, 25%, and 50%.

**Figure 6.**The results of the outcome from the Borel normality criterion [34] test. The illustration analyzes the mean frequencies of bit patterns in 5567 MRNG-RP124 sequences (126 363 bits), with error bars representing the standard error. The red lines represent 12.5%, 25%, and 50%.

**Figure 7.**The figure displays the splash-like representation of (presumably) cosmic radiation found during the analysis of the dataset that has not, until now, been described or mentioned.

**Table 2.**Displays the number of double hits “cntDH” and programmatically detected potential pixel errors “cntPE”. The device identifiers (deviceID) were masked with asterisks as a full deviceID adds no value to this table but has a very small chance of being used in a malicious way.

deviceID | Width × Height | cntDH | cntPE |
---|---|---|---|

sm_a320fl-***14bd | 640 × 480 | 2503 | 2157 |

sm_a320fl-***b4db | 640 × 480 | 15 | 44 |

sm_a320fl-***1419 | 640 × 480 | 1033 | 16 |

sm_a505fn-***YVCM | 1920 × 1080 | 1 | 752 |

mi_a1-***0804 | 1280 × 640 | 1 | 2 |

Parameter | Value |
---|---|

Length of a bitstream | 128 |

Number of bit streams | 94 |

Applied statistical tests | 1–5;11 |

Input file format | ASCII |

Block frequency test—block length (M) | 8 |

Approximate entropy test—block length (m) | 2 |

**Table 4.**NIST statistical test suite results for the random bit sequence

**“MRNG-P1234”**from our MRNG prototype. Tests that failed are marked with ‘*’ by the test suite.

p-Value | Proportion | Statistical Test | Pass |
---|---|---|---|

0.000000 * | 26/94 * | Frequency | NO |

0.000000 * | 49/94 * | BlockFrequency | NO |

0.000000 * | 28/94 * | CumulativeSums | NO |

0.000000 * | 28/94 * | CumulativeSums | NO |

0.000000 * | 50/94 * | Runs | NO |

0.000000 * | 38/94 * | LongestRun | NO |

0.000000 * | 31/94 * | ApproximateEntropy | NO |

**Table 5.**NIST statistical test suite results for the random bit sequence

**“MRNG-P123”**from our MRNG prototype. Tests that failed are marked with ‘*’ by the test suite.

p-Value | Proportion | Statistical Test | Pass |
---|---|---|---|

0.000000 * | 27/94 * | Frequency | NO |

0.000000 * | 59/94 * | BlockFrequency | NO |

0.000000 * | 30/94 * | CumulativeSums | NO |

0.000000 * | 30/94 * | CumulativeSums | NO |

0.000000 * | 53/94 * | Runs | NO |

0.000000 * | 44/94 * | LongestRun | NO |

0.000000 * | 30/94 * | ApproximateEntropy | NO |

**Table 6.**NIST statistical test suite results for the random bit sequence

**“MRNG-P124”**from our MRNG prototype.

p-Value | Proportion | Statistical Test | Pass |
---|---|---|---|

0.013153 | 92/94 | Frequency | YES |

0.000677 | 94/94 | BlockFrequency | YES |

0.189397 | 93/94 | CumulativeSums | YES |

0.804337 | 93/94 | CumulativeSums | YES |

0.100508 | 93/94 | Runs | YES |

0.332797 | 92/94 | LongestRun | YES |

0.879806 | 94/94 | ApproximateEntropy | YES |

**Table 7.**NIST statistical test suite results for the random bit sequence

**“MRNG-RP1234”**from our MRNG prototype. Tests that failed are marked with ‘*’ by the test suite.

p-Value | Proportion | Statistical Test | Pass |
---|---|---|---|

0.000000 * | 26/94 * | Frequency | NO |

0.000000 * | 49/94 * | BlockFrequency | NO |

0.000000 * | 28/94 * | CumulativeSums | NO |

0.000000 * | 28/94 * | CumulativeSums | NO |

0.000000 * | 50/94 * | Runs | NO |

0.000000 * | 56/94 * | LongestRun | NO |

0.000000 * | 43/94 * | ApproximateEntropy | NO |

**Table 8.**NIST statistical test suite results for the random bit sequence

**“MRNG-RP123”**from our MRNG prototype. Tests that failed are marked with ‘*’ by the test suite.

p-Value | Proportion | Statistical Test | Pass |
---|---|---|---|

0.000000 * | 35/94 * | Frequency | NO |

0.000000 * | 52/94 * | BlockFrequency | NO |

0.000000 * | 33/94 * | CumulativeSums | NO |

0.000000 * | 34/94 * | CumulativeSums | NO |

0.000000 * | 55/94 * | Runs | NO |

0.000000 * | 49/94 * | LongestRun | NO |

0.000000 * | 37/94 * | ApproximateEntropy | NO |

**Table 9.**NIST statistical test suite results for the random bit sequence

**“MRNG-RP124”**from our MRNG prototype.

p-Value | Proportion | Statistical Test | Pass |
---|---|---|---|

0.000283 | 93/94 | Frequency | YES |

0.824517 | 93/94 | BlockFrequency | YES |

0.019334 | 93/94 | CumulativeSums | YES |

0.130453 | 92/94 | CumulativeSums | YES |

0.332797 | 93/94 | Runs | YES |

0.490050 | 93/94 | LongestRun | YES |

0.949602 | 94/94 | ApproximateEntropy | YES |

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## Share and Cite

**MDPI and ACS Style**

Kutschera, S.; Slany, W.; Ratschiller, P.; Gursch, S.; Dagenborg, H.
MRNG: Accessing Cosmic Radiation as an Entropy Source for a Non-Deterministic Random Number Generator. *Entropy* **2023**, *25*, 854.
https://doi.org/10.3390/e25060854

**AMA Style**

Kutschera S, Slany W, Ratschiller P, Gursch S, Dagenborg H.
MRNG: Accessing Cosmic Radiation as an Entropy Source for a Non-Deterministic Random Number Generator. *Entropy*. 2023; 25(6):854.
https://doi.org/10.3390/e25060854

**Chicago/Turabian Style**

Kutschera, Stefan, Wolfgang Slany, Patrick Ratschiller, Sarina Gursch, and Håvard Dagenborg.
2023. "MRNG: Accessing Cosmic Radiation as an Entropy Source for a Non-Deterministic Random Number Generator" *Entropy* 25, no. 6: 854.
https://doi.org/10.3390/e25060854