Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Evaluation and monitoring of free running oscillators serving as source of randomness Elie Noumon Allini • Maciej Sk´ orski • Oto Petura • Florent Bernard • Marek Laban • Viktor Fischer CHES 2018, Amsterdam, September 2018 Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Introduction 2 Jittery clock – commonly used source of randomness in digital devices Clock jitter caused by several noise sources ◮ White noise (thermal noise, ...) → Best source of randomness, non manipulable ֒ ◮ Autocorrelated noise (low frequency noises, e.g. flicker noise) → Entropy rate (unpredictability measure) difficult to quantify ֒ ◮ Data dependent noise → Dangerous (manipulable), must be avoided ֒ Jitter monitoring Continuous embedded monitoring is preferable Jitter – usually quantified using the variance var( X ) = E ( X 2 ) − [ E ( X )] 2 (1) Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Introduction 3 Free running oscillators – sources of the jittery clocks V 1 V 2 V 3 s F 1 F 1 C 1 C 1 F 2 F 2 C 2 C 2 F L F L C L C L s R 1 R 1 R 2 R 2 R L R L V 1 V 2 V 3 s ena Ring oscillators (RO) Self-timed ring (STR) based on Müller gates Randomness extraction methods from jittery clocks Sampling m -bit flip-flop counter 1-bit raw l -bit raw s 1 random random signal D ena FRO 1 signal ( l < m ) Q Q s 1 clk clk FRO 1 Counter of k periods Counter of k periods s 2 s 2 FRO 2 clk FRO 2 clk Sampler based randomness extraction Counter based randomness extraction Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Introduction 4 Objectives Analyze the use of variance for entropy estimation Use high order Markov model to estimate entropy coming from auto-correlated noises Compare performance of ROs and STRs as sources of randomness Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Variance and Allan variance 1 High order Markov model for entropy rate estimation from 2 autocorrelated signals Experimental results 3 Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Characterization of random fluctuations of the clock frequency 6 Power spectral density (PSD) Defined as: S y ( f ) = h α f α (2) ◮ y – dimensionless fractional frequency ( y = ( ν − ν 0 ) /ν 0 ) ◮ α – constant characterizing the noise process ◮ h α – intensity of this noise Characterizes random fluctuations of the clock frequency Type of the noise process α α α − 2 Random Walk Frequency (RWF) − 1 Flicker Noise Frequency (FF) 0 White Noise Frequency (WF) or Random Walk Phase (RWP) 1 Flicker Noise Phase (FP) 2 White Noise Phase (WP) Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Variance of the frequency fluctuations 7 Main assumption y is an infinite zero-mean stationary process ◮ characterized by its variance computed from a window of length τ Variance can be computed using the power spectral density Corollary of the Wiener-Khinchin theorem Variance of y computed from the power spectral density S y ( f ): � + ∞ σ 2 S y ( f ) × | H τ ( f ) | 2 df , y ( τ ) = (3) 0 whenever it exists. ◮ H τ ( f ) is the transfer function of the variance operator: → Fourier transform of the impulse response function h τ ֒ → Depends on the type of variance computed ֒ Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Computation of the statistical variance from the PSD 8 Time domain Frequency domain h τ ( t ) � 2 � sin( πτ f ) 1 /τ | H τ ( f ) | 2 = (4) πτ f t τ Variance of the jitter computed for α ∈ [ − 2; 2] from time window τ � f h 2 h α f α − 2 sin 2 ( πτ f ) df . σ 2 � y ( τ ) = (5) ( πτ ) 2 0 α = − 2 Problem: if α � − 1, the integral does not converge as f tends to 0 ◮ The use of the statistical variance can cause entropy overestimation Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Allan variance and its computation from the PSD 9 Time domain Frequency domain h τ ( t ) 1 / 2 τ � 2 � sin( πτ f ) − τ | H τ ( f ) | 2 sin 2 ( πτ f ) (6) = t τ πτ f − 1 / 2 τ Allan Variance of the jitter computed for α ∈ [ − 2; 2] from window τ � f h 2 2 h α f α − 2 sin 4 ( πτ f ) df σ 2 � y ( τ ) = (7) ( πτ ) 2 0 α = − 2 Convergence ensured for α > − 3 as f tends to 0: ◮ Allan variance is accurate, even in presence of low frequency noises Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Allan variance estimation from a limited data set 10 An average fractional frequency can be used Average frequency deviation y k over a time interval of length τ ◮ Corresponds to the fluctuations while counting the number of periods of the jittery signal over τ Estimate of the Allan variance: M − 1 1 � 2 . σ 2 � � y ( τ ) = y i +1 − y i (8) 2( M − 1) i =1 ֒ → M : total number of y k ’s. For α = 0 , σ 2 y ( τ ) is an unbiased estimator of the variance even for a finite M Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Experimental results 11 12 Allan variance 1e+07 Allan variance Statistical variance Statistical variance 1e+06 10 100000 8 10000 variance variance 1000 6 100 4 10 1 2 0.1 0 0.01 64 256 1024 4096 16384 65536 262144 1.04858e+06 100 1000 10000 100000 1e+06 1e+07 1e+08 M k Variance dependence on the Variance dependence on the number of samples M jitter accumulcation period k ◮ Allan variance stable ◮ Allan variance always below statistical variance ◮ Statistical variance ◮ Statistical variance causes increases with M entropy rate overestimation Similar results for both types of free running oscillators studied Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Hardware implementations 12 Statistical variance Allan variance 12.0 Reg Reg cnt 24.12 ena 12.0 24.0 Reg ena cnt c c ena rst c 24.24 y i+ 1 y i 16.0 3.13 Var (cnt) c V(cnt) Reg 4.0 8.0 8.0 Reg Reg Reg Reg ena ena 12.12 24.24 c ena rst c Reg ena ena c c c c rst ena c cnt_rdy ena cnt_rdy ena var_rdy M counter M counter c c 3 adders/subtractors, 2 multipliers 1 adder/subtractor, 1 multiplier Comparison with the state-of-the-art methods Method Area f max f max f max Power ALM/Regs DSPs [MHz] [mW] Haddad et al. 119/160 2 178.3 6-7 Fischer and Lubicz 169/200 4 187.7 7-8 Proposed method, Eq. (8) 49/117 1 238.5 4-5 Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
Variance and Allan variance High order Markov model for entropy rate estimation Experimental results Variance and Allan variance 1 High order Markov model for entropy rate estimation from 2 autocorrelated signals Experimental results 3 Noumon Allini, Sk´ orski, Petura , Bernard, Laban, Fischer Free running oscillators as sources of randomness
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