# Attacks and Defenses for Single-Stage Residue Number System PRNGs

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. System Description

#### 2.2. Assumptions and Goals

#### 2.3. Mathematical Tools

#### Measures of Randomness

#### 2.4. RNS-Based PRNG Attacks

## 3. Results

#### 3.1. Toy Example

#### 3.1.1. Toy Example: Implementing the Attack

#### 3.1.2. Analysis of Toy Example Attack

#### 3.2. Implementing PRNG on IOT-Caliber Example

#### 3.3. Extending to Even Larger Examples

#### 3.4. RNS-Based PRNG Defenses and Perturbations

#### 3.4.1. Noise

#### 3.4.2. Code Hopping

#### 3.4.3. Time Hopping

#### 3.5. Modified RNS Attack to Account for Noise

Algorithm 1: Modified RNS Attack for a Single Error |

#### 3.6. Analysis of RNS-Based PRNG Defenses

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**A visualization of permutations $\sigma \left(3\right),\sigma \left(5\right)$, and $\sigma \left(11\right)$ as gears. At each time increment, the gears turn clockwise by one tooth. After $M=165$ iterations, the initial value (2,0,1) realigns.

**Figure 3.**A one−dimensional ($L=1$) frequency analysis is visualized as one “hallway” in the corner-turning process.

**Figure 4.**The two-dimensional ($L=2$) analysis represents turning a corner. We are lead into a hallway that distributes the search space ${\Phi}^{-1}\left(0\right)$ into smaller batches.

**Figure 5.**Three-dimensional (L = 3) visual of the corner-turning process, assuming the turns $0,{2}^{k}-1$.

**Figure 8.**Expected two-dimensional transitional frequency data of toy example outputs, shown through a surface plot histogram.

**Figure 9.**A two−dimensional frequency analysis of toy example output $\overrightarrow{Z}$ component values.

**Figure 10.**A visualization of the sliding window and respective current state used in the large example.

**Figure 12.**A communication system block diagram of the transmit-receive devices generating an RNS sequence and applying/removing noise to the outputs.

**Figure 13.**Fishbone diagram displaying a visual of the possible output branching when a single $\pm 1$ error occurs.

**Figure 14.**An example of code hopping with three different sets of parameters, where $\alpha ,\beta $, and $\gamma $ represent the top clockwise orderings ${\alpha}_{1},{\alpha}_{2},\dots {\alpha}_{9}$, etc.

**Figure 15.**An example of time hopping, where $\alpha ,\beta ,$ and $\gamma $ represent the original counterclockwise orderings ${\alpha}_{1},{\alpha}_{2},\dots {\alpha}_{9},$ etc. The original permutations are not modified during a time hop, but each permutation is essentially rotated, as shown by the gears.

**Table 1.**Descriptions of the mathematical computations in Figure 1.

Label | Description |
---|---|

${X}_{l}$ | ${X}_{i}\left[l\right]={X}_{i}[l\mathrm{mod}{p}_{i}]={\sigma}_{l}\left({p}_{i}\right)$ |

mCRT | ${X}_{l}\left[i\right]=Y\left[l\right]\xb7{\left(\frac{M}{{p}_{i}}\right)}^{-1}\mathrm{mod}{p}_{i}$ |

$Y\left[l\right]$ | $Y\left[l\right]={\sum}_{i=0}^{n-1}{X}_{l}\left[i\right]\frac{M}{{p}_{i}}\mathrm{mod}M$ |

$Z\left[l\right]$ | $Z\left[l\right]=\varphi \left(Y\left[l\right]\right)=Y\left[l\right]\mathrm{mod}{2}^{k}$ |

$\varphi $ | Surjective mapping ${\mathbb{Z}}_{M}\to {\mathbb{Z}}_{{2}^{k}}$ |

l | ${\overrightarrow{\mathit{X}}}_{\mathit{l}}$ | CRT Output | $\mathit{Y}\left[\mathit{l}\right]$ | $\mathit{Z}\left[\mathit{l}\right]$ |
---|---|---|---|---|

0 | $<1,0,2>$ | 145 | 85 | 5 |

1 | $<0,1,9>$ | 141 | 3 | 3 |

2 | $<2,3,1>$ | 23 | 59 | 3 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

163 | $<0,2,5>$ | 27 | 141 | 5 |

164 | $<2,4,8>$ | 74 | 32 | 0 |

n | $\mathbf{Max}\mathcal{P}$ | ${\mathit{\zeta}}_{\mathbf{mem}}$ | ${\mathit{\nu}}_{\mathbf{mem}}$ | ${\mathit{\mu}}_{\mathbf{mem}}$ | ${\mathit{\mu}}_{\mathbf{time},\mathit{r}=100}$ |
---|---|---|---|---|---|

3 | 11 | 132 | 1216 | 0.1 | ≈20.2 |

4 | 479 | 25,893 | $4.9\times {10}^{11}$ | $5.3\times {10}^{-8}$ | ≈$1.29\times {10}^{-5}$ |

16 | 1021 | 242,038 | $5.7\times {10}^{49}$ | $4.2\times {10}^{-45}$ | ≈$2.73\times {10}^{-33}$ |

24 | 2003 | 126,229 | $7.66\times {10}^{61}$ | $1.65\times {10}^{-57}$ | ≈$1.26\times {10}^{-53}$ |

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**MDPI and ACS Style**

Vennos, A.; George, K.; Michaels, A.
Attacks and Defenses for Single-Stage Residue Number System PRNGs. *IoT* **2021**, *2*, 375-400.
https://doi.org/10.3390/iot2030020

**AMA Style**

Vennos A, George K, Michaels A.
Attacks and Defenses for Single-Stage Residue Number System PRNGs. *IoT*. 2021; 2(3):375-400.
https://doi.org/10.3390/iot2030020

**Chicago/Turabian Style**

Vennos, Amy, Kiernan George, and Alan Michaels.
2021. "Attacks and Defenses for Single-Stage Residue Number System PRNGs" *IoT* 2, no. 3: 375-400.
https://doi.org/10.3390/iot2030020