# Design and Test of an Integrated Random Number Generator with All-Digital Entropy Source

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Design of the Random Number Generator

#### 2.1. RNG Engine

#### 2.2. Entropy and Design of the All-Digital Entropy Source

#### 2.3. Design of the Deterministic Random Bit Generator Module

- CTR-DRBG, which relies on the CTR mode of operation of block ciphers;
- HMAC-DRBG, which relies on the HMAC scheme of hash algorithms.

- $4\xb7{f}_{clk}\phantom{\rule{4pt}{0ex}}\frac{\mathrm{bit}/\mathrm{s}}{\mathrm{area}}$, for the SHA2-256 case;
- $3.2\xb7{f}_{clk}\phantom{\rule{4pt}{0ex}}\frac{\mathrm{bit}/\mathrm{s}}{\mathrm{area}}$, for the SHA2-512 case.

#### 2.4. Synthesis Design

## 3. Results

#### 3.1. Results on FPGA and 7 nm Standard Cell Technologies

^{2}.

#### 3.2. RNG Assessment

- The batteries of tests PractRand [26], employed in [27,28], and TestU01 [29], whose usage is reported in [27,28,30,31,32,33]: both essentially represent an enhancement of Diehard(er) suite, because they include some improvements such as the possibility of setting the parameters of some of the offered statical tests (feature not offered by Diehard(er));

#### 3.2.1. Entropy Evaluation

- $7.888$ bits of entropy per byte, corresponding to $0.986$ bit of entropy per bit, estimated with the NIST EA suite;
- $7.999$ bits of entropy per byte, corresponding to $0.999$ bit of entropy per bit, estimated with the BSI suite.

#### 3.2.2. Randomness Tests

- The p-value of each sequence is calculated, discarding the sequences for which p-value < $\alpha $;
- The ratio between the number of sequences that passed the test (i.e., the one for p-value $\ge \alpha $) and the total number of tested sequences (i.e., k) is computed, and it is labeled as PRoportion (PR);
- The p-value of sequences that passed the test are distributed in the range $[0,\phantom{\rule{3.33333pt}{0ex}}1)$ by splitting it into 10 equal sub-intervals, and the uniformity of the distribution of p-value is calculated: basing on the chi-square (chi-squared or ${\chi}^{2}$) function, the uniformity of distribution is determined by computing a figure that can be considered as a p-value of p-value (PoP);

- For each test, PR lies in the confidence interval defined as $(1-\alpha )\pm 3\sqrt{\frac{\alpha (1-\alpha )}{k}}$;
- For each test, PoP ≥ 0.0001 (i.e., the p-values of sequences that passed the test are uniformly distributed).

## 4. Comparison to the State of the Art

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Internal architecture of entropy source module of RNG engine (

**a**), and of FiGaRO stages composing it (

**b**).

**Figure 5.**Typical outcome of the synthesis of a GaRO. This schematic has been extracted by using the Synopsys Design Compiler tool.

**Figure 6.**Outcome of GaRO synthesis with proper constraints. This schematic has been extracted by using the Synopsys Design Compiler tool and refers to the synthesis of the same RTL code used also for the synthesis of the circuit represented in Figure 5 but including dedicated synthesis constraints.

**Figure 7.**Graph of NIST STS PR metric results for tested DRBG sequences. The dark golden points represent the PR values for each test, or sub-test, and the blue dashed lines represent the boundaries of the confidence interval(s): it the PR value lies outside the confidence interval(s), the test is failed: three (six) failing tests are tolerated for the widest (narrowest) confidence interval.

**Figure 8.**Graph NIST STS PoP metric results for tested DRBG sequences. The dark golden points represent the PoP values for each test, and the light blue dashed line traces the threshold for the pass–fail criterion: if the PoP value is greater than (or equal to) threshold, then the test is passed; otherwise, it is failed. The vertical axis uses the logarithmic scale.

**Figure 9.**Histograms of p-values distributions from NIST STS tests. The three-dimensional histograms of the distribution of p-values are reported only for single-experiment tests of NIST STS and, according to the results of PoP metric (Figure 8), their uniformity can be noted.

Design Strategy | Physical Phenomena Generating Entropy | Main Characteristics |
---|---|---|

TERO | Latches oscillatory metastability | Low throughputs, large dependence on placement of logic cells |

Meta-RO | Analogue metastability of inverter gates | PLL required, dependence on placement of logic cells |

FiRO | Jitter and metastability | Good independence from placing |

GaRO | Jitter and metastability | Good independence from placing |

FiGaRO | Jitter and metastability | Independence from placing, higher entropy and robustness respect to single FiRO and GaRO |

B | A | $\mathit{C}=\mathit{B}\xb7\mathit{A}$ | $\mathit{Y}=\overline{\mathit{C}}=\overline{\mathit{B}\xb7\mathit{A}}$ |
---|---|---|---|

0 | 0 | 0 | 1 |

0 | 1 | 0 | 1 |

1 | 0 | 0 | 1 |

1 | 1 | 1 | 0 |

**Table 3.**Implementation results on FPGA VU37P. The percentage data between round brackets refer to the relative utilization of the corresponding entity with respect to the total of resources offered by the FPGA device.

Entity | Frequency [MHz] | CLB (162,960) | CLB LUTs (1,303,680) | CLB Registers (2,607,360) |
---|---|---|---|---|

RNG engine | 260 | 2151 | 9842 | 7121 |

Entropy Source | 260 | 384 | 1567 | 2137 |

DRBG | 260 | 1528 | 7327 | 3685 |

**Table 4.**Post-synthesis results for the RNG engine on 7 nm technology. Frequency data are expressed in GHz, while area data are expressed in kGE.

Entity | Frequency [GHz] | Area [kGE] |
---|---|---|

RNG engine | 4.325 | 127.16 |

Entropy Source | 4.325 | 69.29 |

DRBG | 4.325 | 46.51 |

True Situation | Conclusion | |
---|---|---|

Data Are Random (Accept H0) | Data Are Not Random (Accept Ha) | |

Data are random (H0 is true) | No error | Type I error |

Data are not random (Ha is true) | Type II error | No error |

**Table 6.**Comparison between FPGA implementations of TRNG. Data reported in this table have been extracted from the results of this works and from [39]. In case multiple citations are present, it indicates that the analyzed TRNG implementation was built merging the contributions from each of the works documented in the corresponding citation.

Implementation | FPGA | Bit Rate [Mbit/s] | Entropy per Bit (from BSI Suite) | Entropy Rate [Mbit/s] |
---|---|---|---|---|

This work | Virtex Ultrascale+ VU37P | 2080 | 0.999 | 2077.92 |

[54] | Spartan 6 | 0.0042 | 0.999 | 0.004 |

Cyclone V | 0.0027 | 0.990 | 0.003 | |

SmartFusion 2 | 0.014 | 0.980 | 0.013 | |

[55] | Spartan 6 | 0.54 | 0.999 | 0.539 |

Cyclone V | 1.44 | 0.999 | 1.438 | |

SmartFusion 2 | 0.328 | 0.999 | 0.327 | |

[56,57,58] | Spartan 6 | 2.57 | 0.999 | 2.567 |

Cyclone V | 2.2 | 0.999 | 2.197 | |

SmartFusion 2 | 3.62 | 0.999 | 3.616 | |

[59,60] | Spartan 6 | 0.44 | 0.981 | 0.431 |

Cyclone V | 0.6 | 0.986 | 0.592 | |

SmartFusion 2 | 0.37 | 0.921 | 0.340 | |

[11,61] | Spartan 6 | 0.625 | 0.999 | 0.624 |

Cyclone V | 1 | 0.987 | 0.985 | |

SmartFusion 2 | 1 | 0.999 | 0.999 | |

[62,63] | Spartan 6 | 154 | 0.998 | 154.121 |

Cyclone V | 245 | 0.999 | 244.755 | |

SmartFusion 2 | 188 | 0.999 | 188.522 |

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**MDPI and ACS Style**

Crocetti, L.; Di Matteo, S.; Nannipieri, P.; Fanucci, L.; Saponara, S. Design and Test of an Integrated Random Number Generator with All-Digital Entropy Source. *Entropy* **2022**, *24*, 139.
https://doi.org/10.3390/e24020139

**AMA Style**

Crocetti L, Di Matteo S, Nannipieri P, Fanucci L, Saponara S. Design and Test of an Integrated Random Number Generator with All-Digital Entropy Source. *Entropy*. 2022; 24(2):139.
https://doi.org/10.3390/e24020139

**Chicago/Turabian Style**

Crocetti, Luca, Stefano Di Matteo, Pietro Nannipieri, Luca Fanucci, and Sergio Saponara. 2022. "Design and Test of an Integrated Random Number Generator with All-Digital Entropy Source" *Entropy* 24, no. 2: 139.
https://doi.org/10.3390/e24020139