Intrinsic Entropy for True Random Number Generation

Abstract

Random numbers play a central role in a myriad of applications today ranging from statistical sampling and analysis, to computer simulation, graphics, cryptography, and hardware security. Most of these applications require a very high degree of randomness in the generated numbers, which warrants that the entropy source generating the randomness be pure and unpredictable. But this randomness is limited by the physical processes governing the entropy source of the specific True Random Number Generator (TRNG). In this chapter, we study how the intrinsic metastability-induced entropy in emerging devices can be a powerful tool to meet the requirements of a low power, energy-efficient, and robust TRNG for future security applications.

Appendices

Appendix 1

The dynamics and performance of the majority of spin-based devices are modeled using the stochastic Landau–Lifshitz–Gilbert–Slonczewski (s-LLGS) equation. This equation describes the temporal evolution of the magnetization vector, M, of a monodomain nanomagnet under the effects of magnetic fields, STT, and thermal noise [Slo96, RS08, SZ02]. Mathematically, the s-LLGS equation is given as

$$\displaystyle \begin{aligned} \frac{d\boldsymbol{M}}{dt} = -\gamma\mu_0(\boldsymbol{M}\times \boldsymbol{H}_{\text{eff}}) + \frac{\alpha}{M_{\text{s}}} \Big(\boldsymbol{M}\times\frac{d\boldsymbol{M}}{dt}\Big) - \frac{\boldsymbol{M} \times (\boldsymbol{M} \times \boldsymbol{I_{\text{s}}})}{qN_{\text{s}}M_{\text{s}}}, {} \end{aligned} $$

(5.1)

where H _eff is the effective magnetic field experienced by the nanomagnet, α is the dimensionless Gilbert damping constant, I _s is the applied spin current, q is the elementary charge, and other quantities are as defined previously.

The first term on the right hand side (RHS) in Eq. (5.1) is the conservative precessional torque that governs the precession of the magnetization vector around the effective field acting on the nanomagnet. This effective field comprises the magnetocrystalline anisotropy field, the shape anisotropy field, and the external applied field. A Langevin field $\boldsymbol {h}_r = h_x \hat {\boldsymbol {x}} + h_y \hat {\boldsymbol {y}} + h_z \hat {\boldsymbol {z}}$, representing Gaussian white noise, is added into the effective field in the s-LLGS equation to model thermal noise. The second term on the RHS in (5.1) is the Gilbert damping torque, which is responsible for damping the precessions of the magnetization vector and eventually relaxing it to one of its stable states [dAq05]. The final term on the RHS in Eq. (5.1) is the Slonczewski spin torque arising from the deposition of spin angular momentum by the itinerant electrons of the spin-polarized current. For simplicity of analysis, Eq. (5.1) is often transformed into its dimensionless form, expressed as

$$\displaystyle \begin{aligned} \frac{d\boldsymbol{m}}{dt} = - \boldsymbol{m} \times \boldsymbol{h}_{\text{eff}} + \alpha \left( \boldsymbol{m} \times \frac{d\boldsymbol{m}}{dt} \right) - \boldsymbol{m} \times ( \boldsymbol{m} \times \boldsymbol{i}_{\text{s}}) , \end{aligned} $$

(5.2)

where we have the normalized quantities $\boldsymbol {m} = \frac {\boldsymbol {M}}{M_{\text{s}}}$, $\boldsymbol {h}_{\text{eff}} = \frac {\boldsymbol {H}_{\text{eff}}}{M_{\text{s}}}$, and $\boldsymbol {i_{\text{s}}} = \frac {\boldsymbol {I}_{\text{s}}}{I}$. Here, the scaling factor, I, for spin current is defined as I = qγμ ₀ M _s N _s. The time scale is normalized using the factor (γμ ₀ M _s)⁻¹. The advantages of the normalized equation Eq. (5.2) over Eq. (5.1) are: (a) it is easier to deal with normalized quantities in terms of numerical complexity, and (b) normalized entities are mathematically well behaved under the application of a numerical scheme. The explicit form of Eq. (5.2) obtained by decoupling d m∕dt is given as

$$\displaystyle \begin{aligned} \frac{d\boldsymbol{m}}{dt} = &-\frac{1}{1+\alpha^2} [ \boldsymbol{m} \times \boldsymbol{h}_{\text{eff}} + (\boldsymbol{m} \times (\boldsymbol{m} \times \boldsymbol{i}_{\text{s}}))\\ &+ \alpha \left( \boldsymbol{m} \times( \boldsymbol{m} \times \boldsymbol{h}_{\text{eff}}))- \alpha (\boldsymbol{m} \times \boldsymbol{i}_{\text{s}} \right)]. \end{aligned} $$

(5.3)

To model the thermal field in s-LLGS, it in expressed in terms of the Wiener process as H _T(t)dt = νd W(t) [Aqu+ 06], where W(t) is the Wiener process, and $\nu =\sqrt {\frac {2\alpha K_{\text{b}}T}{\mu _0 M_{\text{s}}^2 V }}$ [Sun06, MNY12]. Here, K _b T is the thermal energy. The statistical properties of this thermal field discussed by Brown and Kubo are given as [Bro63, KH70]

• (1) The mean thermal field: 〈H _T,i(t)〉 = 0,

• (2) The correlation between the components of H _T(t) defined over a time interval τ,

  $$\displaystyle \begin{aligned} \langle \boldsymbol{H}_{\text{T},i}(t)\boldsymbol{H}_{\text{T},j}(t+\tau) \rangle = \frac{2K_{\text{b}}T\alpha}{\gamma \mu_0^2 M_{\text{s}} V}\delta_{ij}\delta(\tau), \end{aligned} $$

  (5.4)

  where δ _ij is the Kronecker delta function. To simulate the thermal effects numerically, the model is discretized in time

  $$\displaystyle \begin{aligned} \boldsymbol{H}_{\text{T}}(t) \Delta t = \nu \Delta \boldsymbol{W}(t), \end{aligned} $$

  (5.5)

  where ΔW(t) = W(t +  Δt) −W(t). The normalized standard deviation of the thermal field is given by

  $$\displaystyle \begin{aligned} \sigma = \sqrt{\frac{2\alpha K_{\text{b}}T}{\mu_0 M_{\text{s}}^2 V}} \sqrt{ \frac{\Delta t' }{\gamma \mu_0 M_{\text{s}}} }, \end{aligned} $$

  (5.6)

  where Δt is the time step of the numerical method used and t′ = (γμ ₀ M _s)t.

We then have

$$\displaystyle \begin{aligned} \boldsymbol{h}_{\text{T}}(t) \Delta t' = \sigma \xi_t, \end{aligned} $$

(5.7)

where the normalized thermal field h _T = H _T∕M _s, and $\xi _t \sim \mathcal {N}(0,1)$ is a standard Gaussian vector.

Now, the total normalized effective field is given as

$$\displaystyle \begin{aligned} \boldsymbol{h}_{\text{eff}} &= \frac{\boldsymbol{H}_{\text{eff}}}{M_{\text{s}}} = \frac{-1}{\mu_0 M_{\text{s}} V}\nabla_{\boldsymbol{M}}E_{\text{total}}(\boldsymbol{M}) + \boldsymbol{h}_{\text{T}}\\ &= \boldsymbol{h}_{\text{app}} + \frac{H_{\text{k}}}{M_{\text{s}}}(\hat{\boldsymbol{n}} \cdot \boldsymbol{m}) \hat{\boldsymbol{n}} - \sum_i N_i\boldsymbol{m}_i +\boldsymbol{h}_{\text{T}}, \end{aligned} $$

(5.8)

where $\boldsymbol {h}_{\text{app}} = \frac {\boldsymbol {H}_{\text{app}}}{M_{\text{s}}}, \boldsymbol {m}_i = \frac {\boldsymbol {M}_i}{M_{\text{s}}},$ and ∇_M is the gradient with respect to the magnetization M.

Appendix 2

A brief description of the various tests encompassed in the NIST SP 800-22 statistical test suite [Ruk+ 01] is given below.

• 1. Frequency (Monobit) Test: Evaluates proportion of ones and zeroes in the entire sequence.

• 2. Frequency Test within a Block: Divides entire sequence into n-bit blocks and then evaluates proportion of ones within each n-bit block.

• 3. Runs Test: Evaluates the number of uninterrupted runs of identical bits in the sequence.

• 4. Test for the Longest-Run-of-Ones in a Block: Determines the longest uninterrupted sequence of ones in n-bit blocks.

• 5. Binary Matrix Rank Test: Constructs disjoint sub-matrices of the entire sequence and then evaluates their rank.

• 6. Discrete Fourier Transform (DFT) Test: Detects periodic features and peaks in the DFT spectrum of the sequence.

• 7. Non-overlapping Template Matching Test: Matches the sequence against m-bit target string templates in a non-overlapping fashion and returns the number of such occurrences.

• 8. Overlapping Template Matching Test: Matches the sequence against m-bit target string templates in an overlapping fashion and returns the number of such occurrences.

• 9. Maurer’s Universal Statistical Test: Identifies similar patterns in the sequence and then evaluates the number of bits between such matching patterns.

• 10. Linear Complexity Test: Calculates the length of a linear feedback shift register for the sequence.

• 11. Serial Test: Determines the frequency of all possible overlapping n-bit patterns in the sequence.

• 12. Approximate Entropy Test: Determines the frequency of all possible overlapping n-bit and (n+1)-bit patterns in the sequence and compares them against statistics for an ideal random sequence.

• 13. Cumulative Sums Test: Converts the sequence of ones and zeroes to (1,−1) and then calculates the maximum excursion (from 0) of a random walk defined by the cumulative sum of the new sequence.

• 14. Random Excursions Test: Evaluates the number of states having K visits in the cumulative sum random walk defined in the previous test.

• 15. Random Excursions Variant Test: Evaluates the number of visits to various states in the cumulative sum random walk defined in the previous tests.