Molecular MUX-Based Physical Unclonable Functions
MMolecular MUX-Based Physical Unclonable Functions
Lulu Ge, and Keshab K. Parhi,
Fellow, IEEE
Dept. of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USAEmail: { ge000567, parhi } @umn.edu Abstract —Physical unclonable functions (PUFs) are small cir-cuits that are widely used as hardware security primitives forauthentication. These circuits can generate unique signaturesbecause of the inherent randomness in manufacturing and pro-cess variations. This paper introduces molecular PUFs based onmultiplexer (MUX) PUFs using dual-rail representation. It may benoted that molecular PUFs have not been presented before. Eachmolecular multiplexer is synthesized using 16 molecular reactions.The intrinsic variations of the rate constants of the molecularreactions are assumed to provide inherent randomness necessaryfor uniqueness of PUFs. Based on Gaussian distribution of therate constants of the reactions, this paper simulates intra-chip andinter-chip variations of linear molecular MUX PUFs containing , , and stages. These variations are, respectively, used tocompute reliability and uniqueness. It is shown that, for the rateconstants used in this paper, although -state molecular MUXPUFs are not useful as PUFs, PUFs containing 16 or higher stagesare useful as molecular PUFs. Like electronic PUFs, increasingthe number of stages increases uniqueness and reliability of thePUFs. Index Terms —Physical unclonable function (PUF), MolecularPUF, Molecular MUX PUF, Molecular multiplexers, dual-railencoding, biomolecular security.
I. I
NTRODUCTION
Physical unclonable functions (PUFs) have attracted in-creased attention in both academia and industry for their uniqueunclonability—these are easy to be created but difficult tobe reproduced due to their intrinsic manufacturing processvariations [1–5]. These variations are sufficient for each PUFto generate unique signatures. For this reason, PUFs are widelyused as hardware security primitives [6].There are a variety of PUFs [7], which can be coarselyclassified based on whether the randomness is introduced ex-trinsically or intrinsically. Extrinsic-based PUFs include opticalPUF and coating PUF, while delay PUF, radio frequency PUF[8] and magnetic PUF are typical intrinsic-based PUFs. Amongvarious PUF implementations, multiplexer-based PUF (MUX-based PUF), a type of delay PUFs, is of great interest as it canbe easily constructed and analyzed.Molecular computing involves computations via molecules,such as DNA, rather than silicon substrate for electrical com-puters. As a programming language for molecular computing[9], chemical reaction networks (CRNs) specify a set ofchemical species and a set of specific chemical reactions.The deduced system behavior is obtained by the CRN modelvia ordinary differential equations (ODEs) [10]. Such a CRNmodel can be applied to real molecular system analysis [11],and also can simplify the process of designing engineeredsystems [12, 13]. Guaranteed by [11], the designed formalCRN can be mapped to real DNA strand displacement reac-tions using unimolecular and bimolecular reactions. Similar toelectronics, approaches to digital computing using molecularlogic have been well established; examples include computingusing combinational logic [5, 14], sequential logic [15, 16] andclock signal [17].For anti-counterfeiting, interesting works for PUF imple-mentation through chemical methods have been presented in[6]. For example, the molecular tags, encrypted by nucleic acid’s sequence, color and length, weaken the unclonability.Molecular PUFs can be used in numerous bio-security ap-plications for authentication. One application might be au-thentication of oligonucleotide Arrays [18]. It is well knownthat a counterfeiter can easily decipher the array sequences.DNA barcoding has been used for preventing seafood fraudby species substitution [19]. With decreasing cost of DNA, theDNA PUFs may find applications in authentication of manychemical and biological substances where using electronicPUFs would be prohibitively expensive.Although various types of PUFs have been presented inthe literature, no molecular PUF has been presented so far.This paper introduces an approach to implement an N -stagemolecular MUX-based PUF where the intrinsic randomness isderived from the change in the rate constant of the molecularreaction. All molecules in the MUX PUF are representedby dual-rail encoding [20]. The paper introduces a proof ofconcept by simulations assuming the rate constant to vary ina Gaussian manner. This proof of concept describes feasibilityof molecular PUF as a molecular security primitive. However,a practical demonstration requires experiment in a test tube. Tobe more specific, the rate constant of each single MUX followsthe Gaussian distribution, i.e., rate ∼ N ( µ, σ ) , which isassumed to be N (16 , throughout the entire paper. Usingother rate constants is likely to lead to different conclusions.After constructing each single MUX with CRN using the one-to-one mapping method proposed in [13], the entire PUF issynthesized by cascading all the multiplexers according to thedesigned PUF configuration. Four MUX PUFs, , , , -stage molecular PUFs, are simulated using molecular reactions,and their uniqueness and reliability properties are investigated.A single PUF should be able to generate different output re-sponses R under different challenges, C , while different PUFsshould generate different R for the same C . In addition, twoPUF metrics, Reliability and
Uniqueness , are calculated.For a synthesized molecular PUF to be feasible, the designmust satisfy the fact that the minimum inter-chip variation islarger than the maximum intra-chip variation. The impact of thenumber of MUX stages on the performance of the molecularPUF is also investigated. Therefore, this paper extends thePUF concept towards molecular computing with the CRNmodel, whose physical implementation could be DNA stranddisplacement reactions.The remainder of this paper is organized as follows. SectionII briefly introduces the preliminaries. Two important metrics,
Reliability and
Uniqueness , are also introduced in thissection. Starting with a single MUX synthesis, Section IIIillustrates how to implement an N -stage molecular PUF byintroducing randomness in the rate constant of each singleMUX CRN. Totally four cases are studied; these include , , and -stage PUFs. Both reliability and uniquenessmetrics, are also calculated in this section. The complexity issummarized in Section IV. Finally, Section V concludes theentire paper. a r X i v : . [ c s . ET ] M a y I. P
RELIMINARIES
A. Chemical Reaction Networks (CRNs)
Formal CRNs can be realized by DNA strand displacementreactions as long as the number of reactants per equation is nomore than two [11]. This paper, however, presents molecularPUFs; these have not been mapped to DNA. The chemicalkinetics of the CRNs can be obtained by numerical simulationof ODEs based on mass-action law.
B. Dual-Rail Representation
Dual-rail representation refers to two species, e.g., X and X , are employed to represent a single bit X [20]. If X is ,then X = 1 and X = 0 . If X is , then X = 0 and X = 1 . All molecules used in this paper are encoded usingdual-rail representation. C. MUX-based PUFs: Uniqueness and Reliability
This paper studies the molecular MUX-based PUFs de-scribed by a linear delay model. The reliability and uniquenessproperties of these PUFs are addressed. The Structure of PUF : Fig. 1 shows the basic con-figuration of an N -stage MUX-based PUF. Given the N -bitchallenge, then the clock signal races through both top pathand bottom path of the cascaded N -stage multiplexers. Thearbiter produces the output response R according to whetherthe clock reaches the top or bottom input of the arbiter first. Ifthe top input is activated first, then the output response R = 1 ,otherwise R = 0 . Challenge Arbiter10 011001 0110 Output0 1 0
Top InputBottom Input
Fig. 1. Configuration of an N -stage MUX-based PUF. The Additive Linear Delay Model : For each singleMUX, the time delay can be defined as an independent identi-cally distributed (i.i.d.) random variable D i , which follows theGaussian distribution N ( µ, σ ) , where µ is the mean and σ isthe standard deviation. At the i th stage, the delay difference ∆ i between top and bottom paths can be described by (1a).The delay difference r N after the last stage is modeled by (1b),where C (cid:48) i = ⊕ Nj = i +1 C j and C (cid:48) N = 0 . The output response R iscomputed by (1c). Interested readers can refer to [4] for moredetails of such a linear delay model. ∆ i = D ti − D bi ∼ N (0 , σ ) , (1a) r N = N (cid:88) i =1 ( − C (cid:48) i ∆ i , (1b) R = sign ( r N ) = (cid:40) , r N ≥ , , r N < . (1c)One thing should be emphasized is that a single N -bitchallenge can produce a -bit response R . Thus, L challengeswith N bits are required to generate a signature of length L .Note that all the multiplexers are designed identically. Themain reason for the clock signal not arriving at the arbitersimultaneously is manufacturing process variations. Althoughall the multiplexers are designed to be identical, their rate con-stants affect the propagation delay. This means in practice we can never produce the same PUF. Therefore, even for the samechallenge, different PUFs with the identical configuration willgenerate different output responses, also referred as signatures.Moreover, a PUF may generate different responses to the samechallenge under various environmental and noise conditions. Intra-Chip and Inter-Chip Variations : In general, envi-ronmental and noise conditions affect the intra-chip variationand the manufacturing process variations lead to inter-chipvariation. a) Intra-chip variation:
This variation refers to variabilityfor a single PUF. This intra-chip variation means that, for acertain PUF, if the challenge bits are fixed, the output response R can vary under different environmental conditions. b) Inter-chip variation: This corresponds to the variationfrom chip to chip, i.e., PUF to PUF. This inter-chip variationimplies that the signatures of different PUFs will be differentfor identical challenge. Reliability and Uniqueness : In this paper, two PUFmetrics, namely reliability and uniqueness, are studied toquantify the PUF performance. a) Reliability:
This metric measures the reliability ofa single PUF when generating response bits under differentenvironmental conditions [21]. The reliability metric is com-puted by (2a), where P intra reflects the intra-chip variationfor the entire L -bit response. P intra is computed by (2b) asthe average Hamming distance (HD) between L -bit responsesgenerated by the same PUF under m different environmentalconditions. The closer the value of Reliability to , the greaterreliability for a PUF. Reliability = 1 − P intra , P intra ∈ [0 , , (2a) P intra = E [ HD intra ]= E (cid:34) m m (cid:88) i =2 HD ( R , R i ) L × (cid:35) . (2b) b) Uniqueness: Computed by (3a), this metric quantifiesthe ability of a PUF to be uniquely distinguished from agroup of PUFs with the same configuration [21], where P inter reflects the inter-chip variation. Assume given K PUF in-stances, P inter is computed using (3b) as the average Hammingdistance of all (cid:0) K (cid:1) possible pairs comparing combinations of L -bit responses. The better Uniqueness , the value is closerto . P inter = 50% represents the best uniqueness for a PUF. Uniqueness = 1 − | P inter − | , P inter ∈ [0 , , (3a) P inter = E [ HD inter ]= E (cid:34) K − K K − (cid:88) i =1 K (cid:88) j = i +1 HD ( R i , R j ) L × (cid:35) . (3b)This paper only covers the reliability and uniqueness metricsfor MUX-based PUF. Other types of PUF configuration likefeed-forward MUX-based PUFs [4] and other metrics likerandomness, unpredictability, and security are also of interest,but these are not investigated in this paper.III. M OLECULAR
MUX-
BASED
PUF
WITH D UAL -R AIL R EPRESENTATION
This section first presents the CRN implementation fora 2-to-1 multiplexer, the unit module for a PUF. Then themolecular PUF synthesis is illustrated in detail. For variousPUF instances, both reliability and uniqueness are calculatedto analyze the molecular PUF performance.2 . CRN for a 2-to-1 MUX
The logic function of a 2-to-1 multiplexer is expressed by Z = A · ¯ S + B · S , where A and B are the two inputs, S represents the select signal and Z represents the output, where ¯ S means the NOT operation of S . The corresponding TruthTable is shown in Fig. 2. AB ZS S S A B Z
A B Z
Fig. 2. The truth table for a 2-to-1 multiplexer.
A 2-to-1 MUX can be synthesized using dual-rail encod-ing with one-to-one [13] mapping from the Truth Table tomolecular reactions. Each molecular MUX is described by reactions listed in (4), where rate is a positive constant.Note chemical species R , R , R and R are intermediatevariables. These are generated to make sure the number ofreactants is no more than two. While bistable reactions [13, 14]have been used in digital logic, we argue that bistable reactionsshould not be used for molecular PUFs. This is because thebistable reactions prevent racing of the input signal as theoutput of every stage is first guaranteed to be stable beforepropagating to next stage in these reactions. Before eachauthentication, the output Z is set to logic , i.e., the initialconcentration for Z is set as nM and for Z is set as nM . A + B −−− (cid:42)(cid:41) −−− R ,A + B −−− (cid:42)(cid:41) −−− R , A + B −−− (cid:42)(cid:41) −−− R ,A + B −−− (cid:42)(cid:41) −−− R , (4a) S + R rate −→ S + A + B + Z (cid:48) ,S + R rate −→ S + A + B + Z (cid:48) ,S + R rate −→ S + A + B + Z (cid:48) ,S + R rate −→ S + A + B + Z (cid:48) ,S + R rate −→ S + A + B + Z (cid:48) ,S + R rate −→ S + A + B + Z (cid:48) ,S + R rate −→ S + A + B + Z (cid:48) ,S + R rate −→ S + A + B + Z (cid:48) , (4b) (cid:40) Z (cid:48) + Z rate −→ Z ,Z (cid:48) + Z rate −→ Z , (cid:40) Z (cid:48) + Z rate −→ Z ,Z (cid:48) + Z rate −→ Z , (4c) B. CRN for a PUF
Based on the configuration shown in Fig. 1, the molecularPUF is synthesized via constructing its all unit modules—multiplexers with the formal CRN as described by (4). Man-ufacturing process variation is introduced by changing rateconstants for each MUX in the PUF. The rate constants for thetwo MUXes in each stage and for different stages are different.This means a slight variation in rate constant is assumed tobe the source of randomness for each stage. Similar to thedelay difference D i , the rate constant of each MUX, rate ,follows a Gaussian distribution N ( µ, σ ) . Specifically, all therate constants in this paper are randomly sampled from the probability density function N (16 , . The response R dependson the top and bottom outputs of the last MUX stage. Thuswe only consider the N th stage outputs of the top and bottompaths, Z t ,N and Z b ,N . If the input signal propagates to Z t ,N first, then R is ; otherwise it is . Since we adopt dual-rail representation, two molecules are employed to representa single bit. Considering the logic value is either or , thenthe corresponding initial species concentration is ideally or nM . The clock signal shown in Fig. 1 reaches the top andbottom inputs of the arbiter but with a delay difference.In this paper, a molecular arbiter has not been implementedas part of the molecular PUF. Instead, the response is computedas R = Z t ,N − Z b ,N in software.
1) Intra-Chip Variation of a PUF under Different Environ-mental Conditions:
Fig. 3 shows four cases where a single PUFis activated by the given challenge under two different envi-ronmental conditions, which produces two different responses.The simulation results show our synthesized molecular PUFhas the ability to produce different responses under variousconditions even when driven by the same challenge.
Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (a) 8-stage PUF (Cond. 1) (R=1). Concentration ( nM ) R [ t ] t: Time ( hrs ) (b) Enlarged version of Cond. 1(R=1). Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (c) 8-stage PUF (Cond. 2) (R=0). Concentration ( nM ) R [ t ] - - - - - t: Time ( hrs ) (d) Enlarged version of Cond. 2(R=0). Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (e) 16-stage PUF (Cond. 1) (R=1). Concentration ( nM ) R [ t ] t: Time ( hrs ) (f) Enlarged version of Cond. 1(R=1). Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (g) 16-stage PUF (Cond. 2) (R=0). Concentration ( nM ) R [ t ] - - - - - - t: Time ( hrs ) (h) Enlarged version of Cond. 2(R=0).Fig. 3. Simulations for a single , , and -stage PUF drivenby the same challenge under two different environmental conditionsdenoted as Cond. 1 and Cond. 2.
2) Inter-Chip Variation of Different PUFs Driven by theSame Challenge:
Fig. 4 shows four cases where two differentPUFs of the same stage produce different responses, positiveand negative, when driven by the same challenge under thesame environmental condition. The rate constant schemes forall PUFs follow the Gaussian distribution N (16 , . Due to the3 ABLE IR
ATE C ONSTANTS FOR T OP AND B OTTOM
MUX
ES FOR T WO TAGE
PUF
S THAT G ENERATE D IFFERENT R ESPONSES ∗ Specific Rate Constants for 8-stage PUF 1 to Generate the Positive ResponseStage 1 2 3 4 5 6 7 8Top
Bottom
Specific Rate Constants for 8-stage PUF 2 to Generate the Negative ResponseStage 1 2 3 4 5 6 7 8Top
Bottom ∗ The -bit challenge=[11101010]. Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (c) 32-stage PUF (Cond.1) (R=1). Concentration ( nM ) R [ t ] t: Time ( hrs ) (d) Enlarged version of Cond. 1(R=1). Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (e) 32-stage PUF (Cond.2) (R=0). Concentration ( nM ) R [ t ] - - - - t: Time ( hrs ) (f) Enlarged version of Cond. 2(R=0). Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (g) 64-stage PUF (Cond.1) (R=1). Concentration ( nM ) R [ t ] t: Time ( hrs ) (h) Enlarged version of Cond. 1(R=1). Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (i) 64-stage PUF (Cond. 2) (R=0). Concentration ( nM ) R [ t ] - - - - - - - t: Time ( hrs ) (j) Enlarged version of Cond. 2(R=0).Fig. 3. Simulations for a single PUF (Continued). page limit, rate constants for only two cases are listed in TableI for 8-stage PUFs and Table II for 16-stage PUFs, respectively.Notice that, as the number of stages increases, the arrivaltime at the N th-stage of both top and bottom inputs is pro-longed. Fig. 4 shows that different PUFs can produce differentresponses when driven by the same challenge. C. PUF Performance Metrics
The results in Section III-B indicate that slight variationsin rate constant of each MUX indeed provide sufficient ran-domness for the PUF. Thus for different PUFs, the responsescan be varied even under the same challenge bits. This subsec-tion calculates the aforementioned two metrics to analyze thesynthesized PUFs’ performance.
Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (a) 8-stage PUF 1. Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (b) 8-stage PUF 2. Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (c) 16-stage PUF 1. Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (d) 16-stage PUF 2. Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (e) 32-stage PUF 1. Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (f) 32-stage PUF 2. Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (g) 64-stage PUF 1. Concentration ( nM ) Z t [ t ] Z b [ t ] R [ t ] t: Time ( hrs ) (h) 64-stage PUF 2.Fig. 4. Simulation results for -stage, -stage, -stage and -stagePUFs activated by the same challenge. Each challenge is applied totwo different PUFs.
1) Reliability:
This metric is measured for a single N -stagePUF under m environmental conditions with N -bit challengesto generate the response R with the length of L bits. Thenoise caused by environment follows the Gaussian distribution N (0 , σ s ) , where the standard deviation σ s = 0 . in the wholepaper. To be more specific, we use m = 200 environmentalconditions to obtain the response R with the bit length L =200 . Thus, the number of possible piecewise comparison for(2) is m − − .
2) Uniqueness:
This metric is evaluated by K different N -stage PUFs where the rate constant is sampled from a Gaussiandistribution N (16 , . Specifically, we use K = 200 differentPUFs to generate the response R with the bit length L = 200 .Therefore, for (3), we have totally (cid:0) K (cid:1) = (cid:0) (cid:1) = 19900 ABLE IIR
ATE C ONSTANTS FOR T OP AND B OTTOM
MUX
ES FOR T WO TAGE
PUF
S THAT G ENERATE D IFFERENT R ESPONSES ∗ Specific Rate Constants for 16-stage PUF 1 to Generate the Positive ResponseStage 1 2 3 4 5 6 7 8Top
Bottom
Stage 9 10 11 12 13 14 15 16Top
Bottom
Specific Rate Constants for 16-stage PUF 2 to Generate the Negative ResponseStage 1 2 3 4 5 6 7 8Top
Bottom
Stage 9 10 11 12 13 14 15 16Top
Bottom ∗ The -bit challenge=[0100000000110001].TABLE IIII NTRA -C HIP AND I NTER -C HIP V ARIATION R ESULTS FOR
PUF
S OF
8, 16, 32
AND
64 S
TAGES
Case 8-stage PUFs 16-stage PUFs 32-stage PUFs 64-stage PUFsTypes Max Min Mean Max Min Mean Max Min Mean Max Min MeanIntra-chip
Inter-chip
Reliability
Uniqueness -stage, -stage, -stage and -stage PUFs. piecewise comparisons.Here we study four types of PUFs with the number ofstages , , and . The corresponding intra-chip andinter-chip variations are shown in Table III. Based on (2) and(3), the calculated two PUF performance metrics, reliabilityand uniqueness, are also listed in this table. Fig. 5 showsthe Gaussian fit curve of inter-chip variation distribution fordifferent stages.Prior to analysis, it should be emphasized that the mini-mum inter-chip variation should be in practice larger than themaximum intra-chip variation [22]. Due to this reason, the -stage PUF in Table III is not feasible as a PUF. Thus, weonly analyze the remaining three cases, , and -stagePUFs. Based on Fig. 5, the shape of Gaussian fit curve getsnarrower and the minimum inter-chip variation increases withthe number of stages. This indicates that the number of stageshas an obvious impact on inter-chip variation.
Comparingthe three PUFs, as the number of stages increases, the intra-chip variation also increases, but is still about , while theinter-chip variation is closer to . With respect to thereliability metric, -stage PUFs show the highest reliability,while -stage PUFs have the least reliability. With respectto the uniqueness metric, -stage PUFs achieve the highestuniqueness. From these cases, we note that larger thenumber of stages, the more likely the synthesized molecularPUF will satisfy that the maximum intra-chip variation is less than the minimum inter-chip variation.According to the derived results for silicon MUX-basedPUFs in [4], we know that with increase in the number ofstages, N , P intra increases, and the reliability decreases. P inter increases and the uniqueness increases. The results fromthe molecular PUFs are consistent with these observations.IV. C OMPLEXITY
In a general case, to synthesize an N -stage molecularPUF, as expressed in (5), a total of N chemical reactionsare required. Each MUX requires reactions and the PUFcontains N multiplexers. reactions = 2 N ×
16 = 32 N (5)V. C ONCLUSION
This paper has investigated the feasibility of a molecularMUX PUF using molecular reactions. The randomness inthe rate constant is an assumption that has been used inthis paper; its validity in an experimental setup remains tobe demonstrated. Although the paper demonstrates feasibilityof a molecular PUF, several limitations exist in the currentimplementation. For the molecular PUF to be complete, amolecular arbiter should be used to compute the response.One possible realization is the use of the D latch presented in[14]. The need to initialize the top and bottom outputs of eachMUX of each stage after each authentication is also a limitation5f the proposed PUF. Molecular PUFs that do not sufferfrom this limitation should be investigated. The rate constantfor MUX stages is sampled from the Gaussian distribution N (16 , in this paper. The performance of a molecular PUFis dependent on the variance of the rate constant. In practicethe variability of the rate constant is dependent on the kineticsof strand displacement that can be modulated by parameterssuch as toeholds [23]. Successful experimental demonstrationof a molecular PUF is a topic of future research. Similar toelectronic PUFs [24], effects of aging, changes in heat, lightand other environmental noise on molecular PUF need to beunderstood.While the feasibility of a molecular MUX PUF is of interest,demonstrating feasibility of DNA PUFs is of greater interest.Thus, molecular reactions need to be mapped to DNA. Similarto molecular PUFs, if the number of stages is small, the DNAPUF may not be useful as a PUF structure. The minimumnumber of stages needed for a DNA PUF to be feasible needsto be investigated.In electronic PUFs, path delay differences for each stage ofthe MUX PUF can be learned by a software model such as anANN [25, 26]. Demonstrating the ability to learn the modelof a molecular PUF by software models from experimentaldata needs to be investigated. In recent work, secure yetreliable electronic MUX PUFs have been demonstrated usinghomogeneous XOR PUFs [27] and heterogeneous XOR PUFs[28]. Investigating security and reliability of homogeneousand heterogeneous molecular XOR PUFs is a topic of futureresearch. A CKNOWLEDGMENT
The authors thank Xingyi Liu for numerous valuable discus-sions. L. Ge has been supported by the Chinese ScholarshipCouncil (CSC). R
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