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The advent of quantum computers and algorithms challenges the semantic security of symmetric and asymmetric cryptosystems. Thus, the implementation of new cryptographic primitives is essential. They must follow the breakthroughs and properties of quantum calculators which make vulnerable existing cryptosystems. In this paper, we propose a random number generation model based on evaluation of the thermal noise power of the volume elements of an electronic system with a volume of 58.83 cm
^{3}. We prove through the sampling of the temperature of each volume element that it is difficult for an attacker to carry out an exploit. In 12 seconds, we generate for 7 volume elements, a stream of randomly generated keys of 187 digits that will be transmitted from source to destination through the properties of quantum cryptography.

The emergence of quantum computers exposes classical cryptosystems. These cryptosystems whose semantic security is based on difficult mathematical problems and algorithmic complexity become vulnerable. Shor’s [

In this work, we propose a random number generation model using the thermal noise theory. This model is described as a sequence of concatenation of the integer and decimal parts of the thermal power of each volume element of an electronic system. The power is evaluated by sampling the temperature in non-equilibrium state according to Fourrier’s law [

We devote the first and second sections respectively to an exhaustive study of TRNG using the properties of quantum physics, and the description of the proposed mechanism. In third section, we carry out the experiments and performances analysis.

In this section, we make an exhaustive study of the random numbers generators based on the properties of quantum physics.

A True Random Number Generator (TRNG) is a device able to produce a sequence of numbers for which there is no known deterministic link paradoxically to the pseudo-random number generator [

In computer science, a hardware random number generator is a device that generates random numbers from a physical phenomenon rather than use of a computer program [

However, they have the limits that require researchers to move towards other innovative primitives. For illustration purposes, research topics are oriented towards nano-devices, inverters, oxide distribution and random telegraph noise. Although these methods are efficient for producing true random numbers, their implementation proves to be complex for 14 nm processors and its derivatives [

Noise refers to all harmful signals that overlap with the useful signal at any point in a measurement chain or transmission system. The useful signal represents the information, while noise is a hindrance to understanding the information conveyed by the signal. In electronics, it presents interesting properties due to its randomness. According to Johnson-Nyquist work [

- when we evaluate the noise across resistor [

ν ¯ b 2 = 4 K T R Δ F ; (1)

with:

ν ¯ b 2 : Voltage variance across the resistor,

K: Boltzmann constant, K = 1.3806 × 10 − 23 J ⋅ K − 1 ,

T: resistor absolute temperature expressed in kelvin,

R: resistance expressed in Ohms,

Δ F : bandwidth expressed in Hertz.

This application enables to predict the minimum noise in electronic system and its detection limit:

- when we evaluate the power of thermal noise [

η 0 = K T Δ F ; (2)

with:

K: Boltzmann constant, K = 1.3806 × 10 − 23 J ⋅ K − 1 ,

T: conductor temperature expressed in Kelvin,

Δ F : bandwidth in Hertz,

η 0 : thermal noise power, expressed in Watt.

Thermal noise is inevitable and unpredictable in electronic systems and has quite important characteristics when Shannon theory is associated it [

Classification | Technology | Advantages | Limits | |
---|---|---|---|---|

AAmplify Noise | Analog | Simple structure | High energy consumption | |

Oscillator | Couple Oscillator | Digital | Easy integration | Vulnerable to frequency attacks |

Ring Oscillator | Digital | Good portability | Hermetic | |

FIRO/GARO | Digital | More sensitive to jitter | Vulnerable to feedback connections leading to arbitrary output | |

Metastability | Digital | Easy integration | Sensitive to physical phenomena and vulnerable to symmetry of metastability | |

Chaos | Continuous Time | Analog | High rate | High energy consumption |

Discret Time | Digital | High rate | Finite computable precision with a pseudo-random output |

Scott A. Wilber [

lim t → + ∞ g ( t ) = 0 ; (3)

an attacker who studies behavior of the system, could compute the entropy accurate values. Therefore, they are many theories and implementation for true random numbers generation [

In this section, we present logical structure of the proposed true random number generation mechanism. Also we perform the tests.

Let’s consider an embedded system in non-equilibrium state. Its density is given by:

ρ = m v ; (4)

with:

ρ : density expressed in kg∙m^{−3},

m: mass expressed in kg,

v: volume expressed in m^{3}.

According to Fourier’s law [

F = I × S × G r a d T ; (5)

with:

F: heatflow in Watts,

S: plane area expressed in m^{2},

I: thermal conductivity expressed in W∙m^{−1}∙K^{−1},

G r a d T : temperature gradient expressed in K∙m^{−1}.

Let’s consider:

a volume element of embedded system defined by:

ν = ∭ Σ d x d y d z ; (6)

Δ T : the measured temperature according to time (t) and space ( ν ). We evaluate it considering two parameters:

- time(t): it is sampling period of temperature;

- volume element ( ν ): it is the volume element considered during temperature evaluation. The evaluation of thermal noise power in relation to its volume element is defined by:

P ν = K Δ T Δ F ; (7)

with:

K: Boltzmann constant, K = 1.38 × 10 − 23 J ⋅ K − 1 ,

Δ T : volume element temperature expressed in Kelvin,

Δ F : bandwidth expressed in Hertz,

P ν : thermal noise, expressed in Watt.

Let’s consider:

P e ν and P d ν respectively as the integer part and the decimal part of the thermal noise power.

TRNG as the concatenation of P e ν and P d ν ( P e ν | | P d ν ) such as :

T R N G i = P e ν i | | P d ν i ; (8)

where T R N G i : the sequence of random numbers generated and i ∈ [ 0 ; + ∞ [ the clock step of each temperature evaluation. Also, we describe through an algorithm, the proposed mechanism for true random numbers generation.

We evaluate the robustness of the proposed mechanism through the notion of entropy derived from Shannon [

- the numbers are generated following the measured temperature ( Δ T i ) within each volume element ( ν i ) of the proposed device;

- the measured value determines the power ( P ν i ) of the thermal noise.

Let’s note respectively: X, Y, Z the random variables associated to the sources ( P ν i ), ( Δ T i ), ( ν i ) and H(X), H(Y), H(Z), their entropies.

Let’s consider the determination of the thermal noise power of a volume element as a source of information. Its probability and entropy follow respectively the relation:

P ( X = x i ) = P ( Y | Z ) ; (9)

H ( X = x i ) = − ∑ x P ( X = x i ) log ( P ( X = x i ) ) = − ∑ x P ( Y | Z ) log ( P ( Y | Z ) ) (10)

(By identification following to (10));

with: P ( Y | Z ) = P ( Y ∩ Z ) P ( Z ) .

For an infinity of volume elements (z), z → + ∞ :

1) P ( Z = z i ) → 0 (equiprobability);

2) P ( Y ∩ Z ) → 0 (nonequiprobable due to the source ( Y = y i );

3) P ( X = x i ) = P ( Y | Z ) → 0 .

From 1), 2) and 3), we have:

H ( X = x i ) = − ∑ x P ( X = x i ) log ( P ( X = x i ) ) = − ∑ x P ( Y | Z ) log ( P ( Y | Z ) ) → 0 bit . (11)

Thus, an attacker has none information to determine the thermal power of each volume element.

We conclude that the proposed mechanism is efficient.

We use an Arduino Uno ATMega 328p [

We mention that the function which characterizes each volume element of the solid ( Σ ) is defined by:

ν = ∭ Σ d x d y d z ; (12)

We define by framing in black (

For each volume element of the electronic system, we deploy a temperature sensor type LM 35. Then, we determine the power for each volume element according to the temperature values measured.

We summarize through

Index | Volume (cm^{3}) | Constant J·K^{−1} | Temperature (K) | Frequency (Hz) | Time (s) | Power (w) |
---|---|---|---|---|---|---|

ν 1 | 1.35 | 1.3806 × 10^{−23} | 305.25 | 16 × 10^{3} | 0 | 67,428,504 × 10^{−24} |

ν 2 | 1.46 | 1.3806 × 10^{−23} | 303.55 | 16 × 10^{3} | 2 | 670,529,808× 10^{−25} |

ν 3 | 1.08 | 1.3806 × 10^{−23} | 305.85 | 16 × 10^{3} | 4 | 675,610,416× 10^{−25} |

ν 4 | 1.98 | 1.3806 × 10^{−23} | 297.85 | 16 × 10^{3} | 6 | 657,938,736× 10^{−25} |

ν 5 | 22.2 | 1.3806 × 10^{−23} | 304.75 | 16 × 10^{3} | 8 | 67,318,056 × 10^{−24} |

ν 6 | 1.2 | 1.3806 × 10^{−23} | 296.45 | 16 × 10^{3} | 10 | 654,846,192 × 10^{−25} |

ν 7 | 5.4 | 1.3806 × 10^{−23} | 296.95 | 16 × 10^{3} | 12 | 655,950,672× 10^{−25} |

Clock step | Time (s) | Volume (cm^{3}) | Power (w) | Integer part | Decimal part |
---|---|---|---|---|---|

1 | 0 | 1.35 | 67,428,504 × 10^{−24} | 0 | 67,428,504 × 10^{−24} |

2 | 2 | 1.46 | 670,529,808 × 10^{−25} | 0 | 670,529,808 × 10^{−25} |

3 | 4 | 1.08 | 675,610,416 × 10^{−25} | 0 | 675,610,416 × 10^{−25} |

4 | 6 | 1.98 | 657,938,736 × 10^{−25} | 0 | 657,938,736 × 10^{−25} |

5 | 8 | 22.2 | 67,318,056 × 10^{−24} | 0 | 67,318,056 × 10^{−24} |

6 | 10 | 1.2 | 654,846,192 × 10^{−25} | 0 | 654,846,192 × 10^{−25} |

7 | 12 | 5.4 | 655,950,672 × 10^{−25} | 0 | 655,950,672 × 10^{−25} |

We devote this section to the analysis of the results obtained during the tests. Thus, Figures 2-4 represent graphs relating to the achieved results during the experiments. It is constant to note that the thermal noise power varies for each volume element at

We generate a number by concatenation of the integer and decimal parts of the thermal noise power obtained per volume element ignoring the decimal point. A sequence of generated numbers is equivalent to a sequence of concatenation of integer and decimal parts of the power of each volume element according to its assignment index j. So:

for j ∈ [ 1 ; 7 ] ⇒ ν j ∈ { ν 1 , ν 2 , ν 3 , ν 4 , ν 5 , ν 6 , ν 7 }

⇒ P ν j ∈ { P ν 1 , P ν 2 , P ν 3 , P ν 4 , P ν 5 , P ν 6 , P ν 7 }

⇒ T R N G = { P e ν 1 | | P d ν 1 | | P e ν 2 | | P d ν 2 | | P e ν 3 | | P d ν 3 | | P e ν 4 | | P d ν 4 | | P e ν 5 | | P d ν 5 | | P e ν 6 | | P d ν 6 | | P e ν 7 | | P d ν 7 }

For 7 volume elements, we get a sequence of random numbers of 187 digits distributed as follows:

Let’s note:

n ν i : number of digits for each volume element,

n P e ν i : number of digits enumerated for the integer part of each volume ν i ,

n P d ν i : number of digits enumerated for the decimal part of each volume element ν i . The results are represented in

Therefore, for z volume elements, z ∈ [ 0 ; + ∞ [ , it is very difficult for an attacker to determine exactly the different temperatures within each volume element and:

T R N G z = P e ν 1 | | P d ν 1 | | P e ν 2 | | P d ν 2 | | P e ν 3 | | P d ν 3 | | P e ν 4 | | P d ν 4 | | P e ν 5 | | P d ν 5 | | P e ν 6 | | P d ν 6 | | P e ν 7 | | P d ν 7 | | ⋯ | | P e ν z − 4 | | P d ν z − 4 | | ⋯ | | P e ν z − 1 | | P d ν z − 1 | | P e ν z | | P d ν z .

ν i | n p e ν i | n p d ν i | n ν i |
---|---|---|---|

ν 1 | 1 | 25 | 26 |

ν 2 | 1 | 26 | 27 |

ν 3 | 1 | 26 | 27 |

ν 4 | 1 | 26 | 27 |

ν 5 | 1 | 25 | 26 |

ν 6 | 1 | 26 | 27 |

ν 7 | 1 | 26 | 27 |

Total number of digits | 187 |

The obtained TRN is converted into binary and recovered as a keystream. This keystream will be transmitted from the transmitter to the receiver through quantum cryptography properties. We will associate it on-time pad cryptographic method to secure the transmitted data.

In this paper, we have proposed a mechanism for true random number generation which can resist to an attacker with quantum computers. This mechanism uses the fundamentals of thermal noise theory which is a random phenomenon. For tests and experiments, we used an ATMega microcontroller as a solid space that generates volume elements. We sample the temperature of these volume elements to determine the power of thermal noise for each volume element. Thus, we have obtained for 7 volume elements, a series of random numbers of 187 digits which conversion into binary represents the cryptographic key. Our analysis shows that it is not possible for an attacker to determine the generated sequence numbers for infinity of volume elements. In future work, we will propose a quantum cryptography mechanism to exchange the generated keystream and associate it the One-Time Pad cryptographic method.

The authors would like to thank the reviewers for their constructive comments. Many thanks to the Doctoral School of University of Burundi, IMSP-UAC of Benin and Antenne Afrique des Grands lacs de l’AUF-Burundi for their support.

The authors declare no conflicts of interest regarding the publication of this paper.

Ndagijimana, P., Nahayo, F., Assogba, M.K., Ametepe, A.F.-X. and Shabani, J. (2020) Towards Post-Quan- tum Cryptography Using Thermal Noise Theory and True Random Numbers Generation. Journal of Information Security, 11, 149-160. https://doi.org/10.4236/jis.2020.113010