addabbo2013b

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Physically Unclonable Functions Derived

From Cellular Neural Networks

Tommaso Addabbo, Ada Fort, Luca Pancioni,

Mauro Di Marco, Valerio Vignoli

Department of Information Engineering and Mathematical Sciences

University of Siena, 53100 Italy

(e-mail: [email protected]

)

RESEARCH MANUSCRIPT

PLEASE REFER TO THE PUBLISHED PAPER1

Abstract

We propose the design of Physically Unclonable Functions (PUFs)exploiting the nonlinear behavior of Cellular Neural Networks (CNNs).Our work derives from some theoretical results achieved within the theoryof CNNs, adapted to a simpler case. The theoretical analysis discussedin this work has a general validity, whereas the presented basic hardware

solution (i.e., the PUF electronic implementation) has to be understood asa reference demonstrating circuit to be further optimized and developedfor a profitable use of the proposed approach. Theoretical results havebeen validated experimentally.

1 Introduction

Silicon Physical Unclonable Functions (PUFs) are innovative circuit primitiveswhose digital output strictly depends on the manufacture random physical varia-tions introduced during the fabrication process of integrated circuits (ICs) [13].These circuits have been recently proposed for cryptographic key generation andstorage in secure applications, since PUFs can be used to obtain unique digital

signatures different among nominally identical chips [49].PUFs are typically used to implement the so-called challenge-response chip

authentication, i.e., a PUF can be though as a digital function :{0, 1}n {0, 1}k, where the output k-bit words are unambiguously identified by boththe n challenge bits and the unclonable, unpredictable (but repeatable) systembehavior, different from chip to chip.

Most implementations of silicon PUFs can be classified as either delay-basedcircuits or bi-stable circuits, depending on the physical phenomenon exploited togenerate the output bits [110]. In delay-based PUFs two matched digital delaylines are compared to generate one bit: due to the random process variations

1IEEE Transaction on Circuits and Systems I, 2013

1

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Research manuscript. Please refer to the published paper 2

the delays are slightly different and the output bit is determined by the faster

delay-line. In bi-stable PUFs, a circuit like a SRAM or a sense amplifier isproperly activated in an unstable state, being its final equilibrium dependenton the bias introduced by the process variations.

These mentioned solutions involve low-complexity hardware digital circuits,but their behavior typically suffers from circuit aging and environmental per-turbations (i.e., noise, power supply and temperature variations), producingunstable outcomes that introduce the need of correction techniques based onerror control coding (ECC) or fuzzy extractors, with a significant increase ofthe area and power consumption [810].

In this work we investigate whether nonlinear dynamical systems can beexploited to design reliable silicon PUFs. Starting from the seminal work ofChua [11] a number of researchers have successfully proposed to use nonlineardynamics for ICT applications, with a special reference to, e.g., analog parallel

computing [12], signal processing and cryptography [1322], wireless telecom-munication systems [23] and circuit and instrumentation testing [24,25].

In this paper we show that simple systems inspired by Cellular Neural Net-works (CNNs) may be considered as good candidates to design 1-bit PUF coremodules (i.e., n = k = 1), as an alternative to traditional solutions. The au-thors believe that CNNs may represent a promising and interesting resourceto the design of PUFs. Indeed, both the theory and the deterministic equa-tions ruling these nonlinear systems may offer new engineering tools to designreliable analog PUFs. As it is shown in this paper, in CNN-based PUFs thefinal equilibrium is determined by a set of parameters that can result in strongbiasing toward one stable state, and this fact can be used against the effects ofaging and environmental perturbations. Moreover, adopting an approach sim-

ilar to [26, 27], the process variations can be modeled modifying deterministicequations, obtaining a reliable theoretical tool to investigate the PUF behavior,even from a statistical point of view. Finally, the pervading theory of nonlineardynamics has influenced a number of research groups working on circuits andsystems, and the authors believe that the idea presented in this work can beexplored from different sides.

This paper is organized as in the following. In Sec. 2 we present the two-neurons CNN exploited to design the proposed 1-bit PUF core module. Inthe same section we analyze the nonlinear system, discussing its dynamicalbehavior using some well-known results achieved within the theory of CNNs[2832], adapted to our specific case. Furthermore, we propose a basic electronicimplementation of the CNN, that is afterwards used to explain the workingmechanism of the PUF. The presented hardware solution has to be understoodas a demonstrating circuit, whereas for a profitable use of the proposed approachthe hardware complexity can be drastically reduced adopting a transistor leveldesign point of view [3338], not discussed in this work. In Sec. 3 we discuss howthe circuit non-idealities can define a reliable unclonable function characterizingthe physical device implementing the circuit itself. Accordingly, we investigatethe dynamics of the perturbed CNN system, presenting the statistical analysis ofthe effects induced by the non-idealities on the PUF result. Finally, we discussthe experimental results in Sec. 4. Conclusions and references close the paper.

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x y

x y

u

kg(x)xkg(y)

kg(x)

kg(y)y

Figure 1: A graphical representation of the simple two-neurons CNN defined in(1).

2 The CNN systemLet us consider the following two-neurons CNN (see Fig. 1)

x=

1

x + k (g(x) g(y)) + u(t)

,

y= 1

y+ k (g(y) g(x)) + u(t)

,

(1)

wherek >1, is a time-constant modifying the dynamics speed, the function g :R R is the well-known standard piecewise linear (three-segments) activationfunction defined as

g() =

1, if 1,

(2)

and the function u : R {0, 1} is a digital control signal activating the excita-tion (the role ofu is investigated in the next Sections).

We split the analysis of the above system distinguishing the case in whichthe excitationis active (u= 1) or not (u= 0). In what follows we assume thegeneric pointv R2 as a column vector, and we denote with v its transpose.

2.1 The non-excited case

When the excitation is turned off (u = 0), the system has three equilibrium

points: two stable points located atp1 = (2k, 2k)

and p2 = (2k, 2k)

, andone unstable pointp0located at the origin. As highlighted by the velocity vectorfield shown in Fig. 2, the pointp0is a saddle type equilibrium point whereas theattraction basins ofp1and p2are the half-planes 1 = {(x, y) R2 :x > y} and2 ={(x, y) R2 : x < y}, respectively. Furthermore, any initial conditionsbelonging to the line x= y trigger trajectories asymptotically approaching theunstable pointp0 located at the origin.

We are interested in discussing the dynamical behavior of the non-excitedsystem from a probabilistic point of view. Accordingly, we assume the systeminitial condition as a random variable spatially distributed in R2 according to anarbitrary probability density function (pdf) Lebesgue-integrable with bounded

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8 4 0 4 88

4

0

4

8

x

y

s

i

n

ofat

t

r

c

t

i

o

n

or

d

e

r

basin

ofattr

actio

nborde

r

p

p0

p

2

p2

p

p1

qq

Figure 2: The velocity vector field of non-excited system (1) for k = 2, andthe trajectory triggered by the initial condition q= (6.5, 4.75). The quiver andcontour plots highlight the symmetry of the velocity field with respect to theline x = y.

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8 4 0 4 8

x

8

4

0

4

8

y

qq

p

t

pinit

Figure 3: The velocity vector field of the excited system (1) for k = 2 and= 3.75. The trajectory is triggered by the initial condition q= (1.75, 6.12).The quiver and contour plots highlight the symmetry of the velocity field withrespect to the line x = y.

variation2. Since the subset of the phase space for which x= y has zero measure,any initial condition in R2 is attracted towards either p1 orp2 with probability1. Furthermore, due to the symmetry of the velocity field, if the initial conditionis distributed symmetrically with respect to the line x = y, resulting trajectorieshave probability 0.5 to be attracted towards p1 (or p2).

As it is made clearer in the next Sections, the dynamical behavior of thenon-excited system rules the probabilistic working mechanism that sets the the-oretical basis of the PUF implementation proposed in this paper.

2.2 The excited case

When the excitation is turned on (u = 1), if > 2k

1 the system has an

unique exponentially stable equilibrium point located at pinit = (, ), insidethe region of positive saturation for the activation functions. As highlighted inFig. 3, the excited system preserves its symmetry with respect to the line x = y.

The global exponential stability of the unique attractive sink can be used toset the state of the excited system (1) arbitrarily close to the point pinit, andthis dynamical behavior can be exploited to initialize the CNN by means of thecontrol signalu. Since the initialization phase is critical for the correct behaviorof the proposed PUF implementation, it is important to investigate the rate of

2The set of pdfs that are Lebesgue-integrable with bounded variation provides an exhaus-tive set of probabilistic models that can be used to analyze cases of practical interest, e.g.,when implementing system (1) by means of an electronic circuit.

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convergence of the excited system to estimate how close to the point pinit will

be located the CNN state after it has been excited for a while. To this aim,we rewrite the excited system (1) in the canonical matrix form widely used inneural networks literature, i.e.,

v= D v+ A (v) + E, (3)

where v = (x, y) is the CNN state, A R22 is the interconnection matrix,E R2 is the input vector, : R2 R2 is the output activation function andD R22 is a diagonal matrix, with

D=

1

0

0 1

, A=

k kk k

, (4)

and(v) =

g(x), g(y)

, E=

,

. (5)

Accordingly, we can provide conservative estimate of the converging rate of (3)using the following theoretical result proved by S. Hu and J. Wang.

Theorem 1 ( [30], Th. 4) Let be locally Lipschitz continuous. If the matrixAis Lyapunov Diagonally Semi-Stable (LDSS) then the neural network (3) isglobal exponentially stable, and its exponential convergence rate is not smallerthan dmin

2 , where

dmin = mini=1,...,n

di, (6)

beingd1, . . . , dn the diagonal entries ofD.

In our case, the activation function is globally Lipschitz continuous withLipschitz constant 1. Moreover, the interconnection matrixA is LDSS sincethere exists the matrix

P =

1 00 1

(7)

such that the matrixP A + AP is semi-definite positive, which is the conditionforA to be LDSS [39]. As a consequence, the excited system (1) withu = 1is global exponentially stable with convergence rate not smaller than 1

2.

The previous result plays a key role in the working mechanism of the PUFproposed in this paper, and it will be discussed more in detail in Sec. 3.

2.3 Overall dynamical behavior

The previous analysis has highlighted some important properties of the system(1), that are independent of the value ofu. In detail:

the velocity field is symmetric with respect to the line x= y; any initial conditions with x > ytrigger trajectories contained in the half-

plane 1 = {(x, y) R2 :x > y} and, symmetrically, the same behavioroccurs on the mirroring half-plane 2 = {(x, y) R2 :x < y};

any initial conditions belonging to the line x= y trigger trajectories con-strained on the line x = y.

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To better evaluate the overall dynamical behavior of the system (1), let us

define two time instants 0< t0 < t1 such that

u(t) =

1, ift0 t < t1,0, otherwise.

(8)

In other words, we are assuming the control signalu as aninitialization impulsestarting att0 and having pulse-width t1 t0.

According to the above discussion, if the initial condition q = (x0, y0) hasx0 > y0 the system state is first confined in 1, being attracted towards p1during the time-lag [0, t0). In t0 the excitation is turned on and the systemstate, whatever is its value around p1 (i.e., in 1), starts to be attracted by thesink pinit. As a result, during the time-lag [t0, t1) the system state approachesexponentially the point (, ) from the side of 1. Finally, as far as the signal

u turns back off the excitation att1, the system state is again attracted towardthe equilibrium point p1, whatever is its value around pinit (i.e., in 1). As aresult, in the ideal case the excitation does not cause the trajectory to cross theline x = y , and the whole dynamics belongs to the same half-plane. Thus, it isclear that in the ideal case the initialization phase of the CNN has no practicalusefulness, since the final equilibrium state approached by the CNN is set bythe initial condition qat t= 0 (see Figs. 4 and 5). On the other hand, as it ismade clearer in the next Section, a proper initialization process for a non-idealelectronic implementation of the system (1) becomes necessary in order to makethe final equilibrium point not dependent on the start-up initial condition q.

2.4 Equivalent circuit model

The dynamical system (1) can be implemented exploiting the physical differen-tial law linking the current and the voltage across a capacitor, as depicted inFig. 6. According to Kirchoffs current law it results

CdVxdt

= VxRs

+ Ix+ I0(t),

CdVydt

= VyRs

+ Iy+ I0(t).

(9)

If we now define

Ix=

Iy = k

Rsg(Vx) g(Vy) (10)

and

I0(t) =u(t)

Rs, (11)

system (1) is obtained rearranging (9), i.e.,

dVxdt

= 1

RsC

Vx+ k (g(Vx) g(Vy)) + RsI0(t),dVydt

= 1

RsC

Vy+ k (g(Vy) g(Vx)) + RsI0(t).(12)

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8 4 0 4 88

4

0

4

8

x

y

p

2

p2

p

p0

p

p1

qq

p

t

pinit

Figure 4: The trajectory triggered by the initial conditionq= (3.37, 1.78)for the system (1) with k = 2 and = 3.75. According to the control signal (8)the excitation is turned on at t = 8and turned off at t = 16.

time (x )0 5 10 15 20 25 30

6

4

2

0

2

4

6

8

x y u

t0 t1

Figure 5: The evolution of the state (x(t), y(t)) for the system (1) (initialcondition q = (3.37, 1.78), k = 2 and = 3.75). According to the controlsignal (8) the excitation is turned on at t = 8and turned off at t = 16.

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Vx Vy

+

+

Rs Ix IyI0(t)

I0(t)C

NEURON X NEURON Y

Rs

C

Figure 6: The basic circuit implementing the system (1) used to set the initialcondition of the CNN.

It is interesting to note that if Vx = Vy the two currents Ix and Iy are equalto 0A and the voltage across each capacitor in Fig. 6 exponentially increasesor decreases with time-constant RsC, depending on the control signal u. This

physical behavior describes the dynamics of the system (1) when the CNN statebelongs to the line x = y.

2.5 Demonstration circuit implementation

The circuit model of Fig. 6 can be implemented by means of the circuit shownin Fig. 7. The operational amplifiers OA1X and OA1Y are used to realize twotransconductance amplifiers implementing the two voltage-dependent currentsources Ix+ I0(t) and Iy +I0(t) that feed the impedances Rs//C for the twoneurons, respectively. Since the transconductance amplifiers operate as invert-ing circuits, their driving input voltages are properly collected with oppositesign from the remaining parts of the circuit.

The output of the activation functions are obtained exploiting the four oper-ational amplifiers OA2X, OA2Y, OA3X and OA3Y, obtaining saturation levelsequal to V z

Rf, being Vz the clamping voltage across the series of the two

oppositely oriented Zener diodes. In order to properly set the system parametervalues, the resistance ofRfhas to be set such that

Rs= V zRs

Rf=k. (13)

Furthermore, the input levels that fix the final shape of the activation func-tions are determined by the resistors R1, R2 and R3, that have to be chosensuch that

R2

R1

= 1 +R3

R1

= Vz

1V

. (14)

The previous result can be obtained noting that the current terms kRs

g(Vx)

and kRs

g(Vx) in (10) must be equal to the saturation level k/Rs for Vx, Vy equalto 1V. If G is the gain of the amplifiers provided by OA2X, OA2Y, OA3X,OA3Y, the current feeding OA1X or OA1Y for Vx, Vy = 1V is

G 1Rf

= k

Rs=

VzRf

. (15)

Finally, the digital signal Vinit and the bipolar transistor Q allow to alter-nately activate the CNN excitation. When the transistor Q is turned on, the

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Vcc

VeeVee

Vinit

Vx

Vcc

Vee

Vcc

Vee

+

R1

Rs

C

OA1X

R1

Rf

Rinit

Rf

Rb

R3

R2

R0

R0

Vcc

Vee

Vcc

Vee

R1

R1

R3

R2

R0

R0

NEURON X NEURON Y

OA2Y

OA3Y

OA2X

OA3X Q

Figure7:A

basiccircuitim

plementin

gsystem

(9),usingoperationalam

plifiers

and

Zenerdiodes.

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current sourceI0 of Fig. 6 is active, being

I0|Vee|Rinit

. (16)

Accordingly, the resistance ofR init can be set such to obtain

I0Rs=|Vee|Rs

Rinit=. (17)

On the other hand, when the transistor Q is turned off the exciting currentsourceI0 of Fig. 6 is inactive.

The experimental results based on the circuit in Fig. 7 are evaluated anddiscussed more in detail in Sec. 4. As mentioned in the Introduction, thepresented hardware solution has to be understood as a demonstrating circuit,

whereas for a profitable use of the proposed approach the hardware complexitycan be drastically reduced adopting a transistor level design point of view [3338], not discussed in this work.

3 Definition of the 1-bit PUF core module

As discussed in the Introduction, the output of a PUF must characterize thephysical device implementing the PUF itself. Accordingly, we are interested ininvestigating how the implemented system dynamics is affected by the circuitnon-idealities.

Referring to the circuit of Fig. 7, taking into account the component tol-erances, the implemented system can be described by the following system of

differential equations

x= 1

x

x + k (g1(x) g2(y)) + u(t)x,y =

1

y

y+ k (g3(y) g4(x)) +u(t)y,(18)

where for i = 1, . . . , 4 the functions gi are defined introducing a set of pertur-bation parametersGi, i, i, i R, with Gi, i> 0, such that

gi() =Gig(i + i) + i=

=

Gi+ i, if < Li,

Gi(i + i) + i, ifLi Hi,Gi+ i, if > Hi,

(19)

and

Li=1 i

i, Hi=

1 ii

. (20)

In (18), x and y take into account of the differences between the excitingcurrentsI0 in the two neurons. The termsx and y are related to different timeconstantsRsC, whereas the activation functions gi are modified by the offsetsand gain errors introduced by the sub-circuits using operational amplifiers, alsoconsidering small deviations in the Zener clamping voltages.

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The perturbed terms in (18)-(19) must be seen as realizations of random

variables having mean values equal to the design nominal values of the param-eters in (1), i.e., we can write

x= (1 + X), y =(1 + Y),

x=(1 + X ), y =(1 + Y), (21)

and, fori = 1, . . . , 4,

Gi= 1 + Gi , i= 1 + i ,

i = i , i= i . (22)

where, according to above probabilistic approach, the perturbations with respectto the nominal design values are random variables having zero mean value andstandard deviation which is an expression of the relative parameter variations.

3.1 Analysis of the perturbed system

In the following we assume to deal with small perturbations, such to preserve theoverall qualitative behavior of the original dynamical system (1). In other words,we assume the magnitude of the perturbations to be small enough to preservethe presence of two stable equilibrium points p1 and p2 for the perturbed not-excited CNN (u= 0), and the presence of a global attractor pinit located insidethe positive saturation regions of the activation functions g1, g2, g3, g4 for theexcited CNN (u= 1). This is a reasonable assumption when referring to actualcircuits implementing CNNs [3338].

As far as we address the effects of the perturbations on the equilibrium

points, we remark that the time-constantsxand ydo not affect their positions,and we can write

p0= (x0, y0),

p1= (k(2 + x1), k(2 + y1)),p2= (k(2 + x2), k(2 + y2)),pinit = (+ xinit, + yinit)

,

(23)

where the random variables have zero mean value and are function of theperturbations introduced in (21)-(22). For example, it is easy to prove that

xinit = k(G1 G2+ 1 2) + x ,

yinit

= k(G3 G4 + 3 4) + y .(24)

We omit to provide the explicit expression of the position perturbations forthe other equilibrium points, since they are not relevant to the scope of thispaper.

To analyze the working mechanism of the PUF based on the perturbedsystem (18), it is necessary to evaluate how the border of the attraction basinsis affected by the random perturbations. Indeed, as highlighted in Fig.8, aftera proper initialization process performed by means of the excitations x and yand the control signal (8), the final CNN equilibrium depends on which basinof attraction pinit belongs to, and it does not depend on the initial condition q,as it occurs for the ideal system (Fig. 4).

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8 4 0 4 88

4

0

4

8

basin

ofattr

actionb

orde

r

x

y

p

2

p2

p0

p

p1

qq

p

t

pinit

positive saturation

region for

g1,g2,g3,g4

A

I

II

III

IV

V

B C D E

~ ~

~

~

Figure 8: The trajectory triggered by the initial conditionq= (3.63, 5.13)for the system (18) with nominal values k = 2,= 3.75 and affected by randomperturbations (for the relative parameter deviations 0.1). According to (8)the excitation is turned-on at t0 = 8 and turned-off at t1 = 16. Twenty-five linearity regions are highlighted in the plane background, being the regionsidentified by the reference coordinates {A , . . . , E }{I , . . . , V } set at the upperand right sides of the plot.

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Accordingly, as far as the first equation of (18) is addressed, we note that

the threshold input levels L and H in (20) split the plane xy into nine disjointregions in which we have nine different linear expressions, depending on theactivation functions g1, g2. The same result occurs for the second equation of(18), in which other nine different linearity regions are identified by the thresholdinput levels ofg3, g4. Consequently, as highlighted in Fig. 8, the overall analysisof the perturbed system (18) can be obtained analyzing up to a maximum oftwenty-five different linear dynamical systems, being the number of systemsto be analyzed dependent on the intersection of the two sets of nine regionspreviously discussed.

As a result, for the generic case (18) the analytic expression of the curvedescribing the border of the basin of attraction can not be written in an usefulform, even if for any specific case it can be obtained by simple integrationoperations, as discussed in the following.

Property 1 The curve describing the basin of attraction border is continuousand in any linearity region it can be described using trajectories of the non-excited system (18).

Proof. Let us first prove that in any linearity region the curve can be de-scribed using trajectories of the non-excited dynamical system (18); its conti-nuity derives consequently. Accordingly, let us assume ab absurdo that thereexists a tra jectory intersecting the basin of attraction border. In such case wewould have different points belonging to the same trajectory and belonging todifferent attraction basins, that is impossible. As a result, in any region thebasin of attraction border must be tangent to the velocity field.

On the basis of the previous property, we can provide a detailed expressionfor the curve describing the basin of attraction border in the positive saturationregion forg1, g2, g3, g4 containing the point pinit (regionE-I in Fig. 8).

Property 2 In the positive saturation region forg1, g2, g3, g4 the curve : R R2 describing the basin of attraction border has the exponential form

x() =Kxe x + k(G1 G2+ 1 2),

y() =Kye y + k(G3 G4+ 3 4).

(25)

Proof.The proof comes directly from the integration of the dynamical system(1) in the global positive saturation region (E-Iin Fig. 8), in which the system

is reduced to the form

x= 1

x

x + k(G1 G2+ 1 2),y =

1

y

y+ k(G3 G4+ 3 4).(26)

It is interesting noting that from (25) if x = y the curve is a straightline, regardless of the perturbations . The values of Kx and Ky can not bedetermined a priori as a function of the perturbation parameters, since they

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depend on the system dynamics in the other linearity regions. Nevertheless, for

the scopes of this paper it is sufficient to emphasize that after the initializationprocedure the final equilibrium point for the non-excited CNN depends on theposition of the equilibrium point pinit with respect to a curve having form (25)(Fig. 8).

3.2 Working mechanism of the 1-bit PUF core module

The working mechanism of the 1-bit PUF core module proposed in this paperis described by the following steps:

1. the circuit of Fig. 7 is powered on with the excitation active for a while(u = 1). As previously discussed the excited CNN state is attractedtowards the global equilibrium point pinit regardless of the CNN power-up

state condition;

2. after the initialization procedure has stabilized the CNN state around pinit,the control signaluis turned off. As a result, the PUF digital output is setto 0 if the CNN state converges to the equilibrium p1; conversely, the PUFdigital output is set to 1 if the CNN state converges to the equilibriump2. As shown in Fig. 8, the final CNN equilibrium depends on which sideof the attraction basin border contains the point pinit, i.e., it depends onthe circuit non-idealities.

On the basis of the above core module, a simple PUF :{0, 1} {0, 1}can be obtained by implementing two instances of the same circuit of Fig. 7 ona same chip, choosing one of the two outputs depending on the input challenge

bit. More complex PUFs can be obtained by optimizing and developing thisidea, but this issue goes beyond the scopes of this paper.Instead, it is convenient to remark that the initialization process (step 1)

plays a key role in the correct working of the proposed PUF, since it assures thefinal result to be not dependent on the power-up initial condition (the pointqatt= 0 in Fig. 8). Accordingly, the initialization process must hold the excitation for a time-interval t long enough to assure the CNN state to be sufficientlyclose to the initialization point pinit. In order to theoretically quantify suchinitialization time-interval, we use an approach similar to that one presented inthe proof of [30], Theorem 4. In detail, taking into account the dynamics of theCNN (1) and considering its electronic implementation by means of the circuitof Fig. 7, it can be proved that (we omit the proof)

v(t) pinit 2(Vcc+ ) + 8k2Vcc e t2 . (27)The physical dimensional consistency in the above inequality is assured by somemultiplicative constants that were not reported, since in our case they have uni-tary magnitude. Nevertheless, in practical cases the time-constantis typicallysmall enough to make the term 8k

2Vcc negligible, and a reasonable approxi-

mation of (27) is given by the inequality

v(t) pinit 4Vcc e t2wc , (28)

where we have taken into account that < Vcc and defining wc = max{x, y}.

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4 3 2 1 0 1 2 3 40

400

800

1200

#

trials

Figure 9: The distribution obtained for the quantity defined in (31) by means

of a Monte Carlo simulation of the system (18), with nominal values k = 2, = 3.75 and based on 20,000 trials. The perturbation terms were randomlychanged according to a Gaussian distribution having zero mean value and stan-dard deviation equal to 0.1 (expressed in relative parameter deviation).

The ineq. (28) can be inverted to give a theoretical minimum length of theinitialization time t. For example, if we denote withvthe standard deviationof the electronic noise present in the PUF circuit, from a theoretical point ofview it is convenient to set

t 2wc ln v4Vcc

. (29)

Ifv is in the order of 104

Vand if 4Vcc is in the order of 10V, the minimumtheoretical length of t satisfying (29) is 20wc.

3.3 Statistical analysis

The Euclidean distance of the point pinit given in (23)-(24) from the curve (25)can be expressed implicitly by means of a transcendental equation, as a functionof both the nominal parameter values and the perturbation terms in (21)-(22).Accordingly, the only way to statistically evaluate the positioning of the pointpinit with respect to the attraction basin border is by means of numerical MonteCarlo analysis.

To this aim, the authors performed 20,000 simulations of (18) with nominalvaluesk = 2,= 3.75Vand in which the perturbation terms in (21)-(22) were

randomly generated according to a Gaussian distribution having zero mean valueand standard deviation equal to 0.1 (remember that are relative deviations).For each simulation we computed the Euclidean distance of the point pinit fromthe attraction basin border (25) as

min = mintR

|pinit (t)| , (30)

and from the result we derived a new quantity , dependent on which equilib-rium is reached by the CNN after the excitation, i.e.,

=

min, if limt

x(t), y(t)

= p1,

min, if limt

x(t), y(t)

= p2.(31)

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200ms

2V/div

220ms 240ms 260ms 280ms 300ms

Vx Vy VinitVinit

Vx, Vy

t1t0

Figure 10: A SPICE simulation of the circuit of Fig. 7, assuming for the usedcomponents a 0.5% tolerance of their nominal values (almost ideal case).

2V/div

0ms 20ms 40ms 60ms 80ms 100ms

Vinit

t1t0

Vx Vy

Vx, Vy

Vinit

Figure 11: The dynamical behavior of the implemented circuit, acquired withan oscilloscope (to be compared with the simulation of Fig. 10).

As a result, from the Monte Carlo simulation we obtained the Gaussian-shaped distribution reported in Fig. 9. The histogram indicates that theproposed PUF structure is not polarized towards any of the two equilibriumpoints, also exhibiting a reasonably sensitivity to the random perturbations.Indeed, random fluctuations of the parameter values having 0.1 of standarddeviation produce fluctuations of , normalized to the nominal magnitude|pinit| =

2 = 3.75

2, with a standard deviation approximately equal to

0.2.

4 Experimental results

Two prototypes of the circuit in Fig. 7 have been assembled referring to thenominal design system (1) with k = 2 and = 3.75V. Since the clampingvoltage determined by the Zener diodes in our experiment was Vz 7.6V, using(13) we could set Rf 10k and Rs 2.63k. The feedback networks of

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the operational amplifiers OA2X, OA2Y, OA3X and OA3Y were determined

using (14) and setting R1 1k, R2 7.6k and R3 6.6k. Finally, bychoosing Vcc Vee 12V, using (17) we obtained Rinit 8.42k. Withsuch component values, using a 2.2Fstandard value for the capacitor C theresulting time-constant is about 5.8ms. The resistor R0 was set equal to220 Rf.

In order to evaluate the sensitivity of the PUF to the process variations, wesubstituted the resistors with trimmers and we adopted operational amplifierswith offset compensation (TL081) to adjust the perturbations for most of theparameters in (18). In Fig. 10 a SPICE simulation of the circuit of Fig. 7is reported, assuming for the used components a 0.5% maximum tolerance oftheir values (almost ideal case). The Fig. 11 shows the same transient of Fig.10, acquired with an oscilloscope and one circuit prototype in which we set byboth measurements and fine tuning the component values to closely match the

nominal design reference (k= 2 and = 3.75V).Furthermore, by means of accurate impedance measurements we selected

several capacitors with capacity close to 2.2F(i.e., with differences lower than1%). As a result, depending on the used components, the oscilloscope revealeddifferent final equilibrium points for the CNN, as expected (Fig. 12).

We have also evaluated the effectiveness of the theoretical bound (28). Theexperimental verification showed that in our case (with v 5mV) the mini-mum length for the initialization time twas over-estimated only by a factor 2,that represents a fairly good result for a theoretical bounding approach, whenconsidering nonlinear dynamical systems.

4.1 PUF outcome stability

The robustness of the circuit of Fig. 7 against some environmental perturbationswas evaluated by means of both accurate Monte Carlo SPICE simulations andtargeted experimental measurements. Thanks to the rejection capabilities ofthe operational amplifiers, supply voltage variations with relative amplitudesup to 10% did not cause any error (0% in 100 trials, for 100 PUFs with a 0.5%tolerance of their component values). Referring to the same set-up, temperaturevariations of 50 Celsius degrees caused an average error lower than 1% in 100trials, for 100 PUFs. Similar results were obtained forcing noisy perturbationsof the CNN state with a noise rms up to 10mV. These results can be ascribed tothe strong bias determined by the process variations toward one of the two stablestate. Referring to Fig. 8, temperature variations change both the position ofthe point pinit and the shape of the attraction basin border. In most cases

these changes in the plane{x, y}did not have disadvantageous directions. Thesame happened for the noise perturbations acting during the transient afterswithing-off the excitation.

Even if the above analysis indicates that the proposed PUF circuit exhibitsrobustness against environmental perturbations, from a complexity point of viewit can not compete with delay-based or bi-stable PUF circuits. The authors arecurrently working on the design of a transistor-level custom implementation ofthe circuit, to investigate if the quality of the above results can be confirmedeven for a silicon PUF solution that would be competitive with the traditionalones.

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(Vx,Vy) @ t1

0V

0V

2V

2V

4V

4V

4V 4V

2V

2V

(Vx,Vy) @ t1

Vx

Vy

0V

0V

2V

2V

4V

4V

4V 4V

2V

2V

Vx

Vy

(b)

(a)

Figure 12: The final equilibrium reached by the CNN depending on the circuitnon-idealities. The nominal positioning of the equilibrium points are p1 =(4V, 4V) and p2 = (4V, 4V), whereaspinit = (3.75V, 3.75V).

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5 Conclusions

In this paper we have proposed to exploit a two-neurons CNN to design a basic1-bit PUF core module. The analysis in Sec. 3.1 presents original resultsderived from the theory of CNNs, adapted to a simpler case (i.e., Th. 4 in [30]and the estimation of a theoretical minimum length for the CNN initializationtime-interval).

We have proposed a basic electronic implementation of the CNN to explainthe working mechanism of the PUF. The circuit has been evaluated experi-mentally, and results confirm that this type of structures may present goodrobustness against environmental perturbations.

The profitability of the proposed approach has to be further investigatedreferring to a transistor level design of the circuit, that would be competitivein terms of circuit complexity with the digital traditional PUFs proposed in

literature.

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