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Memristor and its Use in Physically Unclonable Functions Circuits
Ludmil Petkov
Computer Science Department, Oxford Brookes University, Oxford, United Kingdom
E-mail: pl205306@gmail.com
Received xxxxxx
Accepted for publication xxxxxx
Published xxxxxx
Abstract
This research investigates the possibility memristors to be used in the design of physical unclonable functions
(PUFs) for cyber-security applications. LTSpice simulation software is used for better understanding of the
processes and the results of the secure circuits.
Keywords: PUF, CRP, LTSpice, TEAM
information stream of bits (set of outputs or
responses)(Joshi, Mohanty, & Kougianos, 2017). PUF
are clearly object-related. This distinguishes them from
biometric factors like fingerprints which relate to
human beings. Nevertheless, some aspects are similar.
The unavoidable variability in the physical world is
taken advantage by the PUF for example different
variations during process fabrication. It is impossible to
make two identical integrated circuits (IC). Exactly these
variations are useful for cyber-security applications
where the size of the device is important. The physical
world variability guarantees the unclonability. A valid
challenge response pairs will not be produced. From
Figure 1-1 It is evident that the challenge question is
presented to the hardware circuit that generates the
response, opposed to the traditional key storing in a
non-volatile memory.
1. Physical Unclonable Functions
The basic concept of Physical Unclonable Function
(PUF) needs to be clearly defined and clarified for this
research. It appeared not an easy task as the variety of
circuits considered PUF is increasing. Some
constructions are not recognised as PUFs because they
were created before the name was created or their
inventors were not aware that they can have
cryptographic applications. It is attempted to class the
constructions according to similarity and then the
observed properties are reaffirmed. An analogy can be
used and resemblance of the PUF can be found to
fingerprint. It is inherent, expressing individualism and
it is unclonable. Another human cannot be created with
the same fingerprint, even the twins have different
fingerprints. The main property of the PUF is that it
cannot be reproduced. In general the natural properties
of hardware devices (set of inputs or challenges) are
mapped by a random function (PUF) to an exclusive
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© xxxx Publishing Ltd
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Ludmil Petkov et al
transistor. The transistor is not a fundamental circuit
element. It is possible to write a Thevenin’s equivalent
circuit for the transistor which has a source of energy
and resistance in series connected to a load. In the
1960s at Berkeley University professor Leon Chua was
the first person to put nonlinear circuit theory on a firm
mathematical foundation. He is compared with his
contribution to circuit theory to Albert Einstein’s theory
of relativity. Professor Chua has a significant
accomplishment as he has a mathematical system to
prove whether a circuit is going to work or not. While
working on his nonlinear circuit theory he noticed that
there was something missing from the circuit
equations. The four fundamental issues are the voltage,
the charge, the current and the flux as shown on fig. 21. There are 6 equations of relating these elements but
only three fundamental passive devices. He postulated
based on his observations that there should be a fourth
circuit element. It can be expressed as: M = dφ/dq.
Figure 1-1 Traditional non-volatile memory key
storage versus the concept of PUF (Joshi et al., 2017).
Reverse engineering and cyber-attacks can be
prevented efficiently by using PUFs. Some other
applications include providing secure keys, random
number generation, key management and generation,
hw/sw binding, secure/trusted boot, trusty anchoring
and root of trust.
PUFs can be used as watchdogs for components with
embedded chip.
Figure 1-2 Smart card application. The card body is
monitored by the chip. Courtesy of NXP.
During the manufacture of the chip, the PUF’s
environment critical parameters are measured and
stored within the chip as exclusive parameter. The
environment is authenticated by calling the stored
metric. An attempt to reverse engineer the chip will
change the environment and will fail the fingerprint
check of the PUF(“Semiconductor Engineering .:.
Physically Unclonable Functions - Semiconductor
Engineering,” 2019).
Figure 2-1 Showing the four fundamental circuit
elements with two-terminals (Kavehei et al., 2010).
2. Memristor
The equations that describe this element are:
The memristor is a device with different applications. It
is the fourth fundamental passive circuit element.
Generally, only three passive circuit elements were
known before: the resistor, the capacitor and the
inductor. All the electronics built was based on only
these 3 devices and the most recent invention the
𝑣𝑣 = 𝑅𝑅(𝑤𝑤, 𝑖𝑖)𝑖𝑖
2
(2-1)
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Ludmil Petkov et al
𝑑𝑑𝑑𝑑
(2-2)
= 𝑓𝑓(𝑤𝑤, 𝑖𝑖)
𝑑𝑑𝑑𝑑
Where, f and R are the function of time, v and i are the
voltage and current, w is the state variable of the
device.
It was named a memristor, because essentially it is a
resistor with memory. The circuit element diagram that
he invented looks like a changed a little bit resistor fig.22.
Figure 2-2 The symbol of memristor.
Figure 2-3 I-V characteristics of memristor “pinched
hysteresis loop”.
The memristor exists in order to relate the flux in a
circuit to the charge. Professor Chua published a paper
“Memristor-the missing circuit element” (Chua, 1971)
where he mathematically proves that such an element
exists and makes a complete set of the known circuit
elements. It was difficult to the other scientists to figure
out what physics could give rise to a coupling between
flux and charge in a circuit. Sung-Mo (Steve) Kang a
graduate student of professor Chua have realised that
an important issue of the memristor is that it is a passive
device with state (Chua & Kang, 1976). There is a
memory within the memristor which essentially is
represented by one or more state variables for the
device. In fact, the memristor is not characterised by
only one equation. Actually, to be completely
characterised two equations are required. One
equation essentially states that there is quasi-static
relationship between voltage and current but the
resistance in that equation depends upon the state of
the device in any given point of time. Then the other
equation is a time dependant equation which shows
how the state variable evolves in time. The two
equation are needed in order to put a model for the
memristor into SPICE simulation to study it. These two
equations don’t say anything about flux. Therefore, if
there is a device that has a state issue that evolves in
time it can be a memristor. It was predicted according
to the two equations that the memristor has an I, V
curve that looks like a bowtie. It was called by Leon
Chua a pinched hysteresis loop. In his paper he stated
that if you see a pinched hysteresis loop I, V curve
probably this is a memristor.
The memristor is a fundamental element because it is
impossible a circuit to be designed with the other three
elements, that will have the same I, V characteristic. It
was proven mathematically by professor Leon Chua.
Even before Chua there were publications showing
pinched hysteresis loops, but the scientists did not
know what they were. They were memristors but they
couldn’t characterise them. They were dynamical
devices that could be only characterised by two
equations. The physical realisation took some time after
the theoretical definition was announced.
Hewlett-Packard research lab reported that has
managed to create a simple device with two platinum
electrodes on either side and a thin film of TiO2 in the
middle (Strukov, Snider, Stewart, & Williams, 2008). The
TiO2 is a semiconductor that is auto doped. If a little bit
of oxygen is taken out from the TiO2 , this essentially is
n-type doping of the material as shown in fig.2-4. The
oxygen vacancies that are left inside the material are
positively charged. Those vacancies are very slightly
mobile. If a large electric field is applied on these
vacancies, they can be pushed toward the left. So, this
reduces the TiO2 film, that lowers the resistance.
When the oxygen vacancies are pulled the other way,
the resistance will increase. In principle this is a device
that can be switched by voltages in a way that gives a
pinched hysteresis loop.
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Ludmil Petkov et al
Figure 2-4 Shows the Hewlett Packard’s ion model of
memristor, where w is the doped region’s length and
D is the length of the device’s channel (Strukov et al.,
2008).
Figure 2-7 Shows the equivalent circuit diagram with
two resistances Ron and Roff (Amdapurkar & V, 2016).
Equations 2-1 and 2-2 can be written as:
v(t) = (𝑅𝑅𝑜𝑜𝑜𝑜
𝑤𝑤(𝑡𝑡)
𝑤𝑤(𝑡𝑡)
+ 𝑅𝑅𝑜𝑜𝑜𝑜𝑜𝑜 −
))𝑖𝑖(𝑡𝑡)
𝐷𝐷
𝐷𝐷
(2-3)
𝑑𝑑𝑑𝑑
𝑅𝑅𝑜𝑜𝑜𝑜
= 𝜇𝜇𝑣𝑣
𝑖𝑖(𝑡𝑡)
𝐷𝐷
𝑑𝑑𝑑𝑑
(2-4)
𝑅𝑅𝑜𝑜𝑜𝑜
𝑞𝑞(𝑡𝑡)
𝐷𝐷
(2-5)
A linear ion drift model is used. It is assumed that the
ions’ average mobility is the same equal to 𝜇𝜇𝑣𝑣 . The
voltage/current relation is defined. The voltage v(t) is
dependent on the current and resistance. The variable
w(t) can take values between zero and D.
Figure 2-5 Idealized representation of the switching
behaviour in the TiO2 layer.
After integrating 2-4 the state variable is:
𝑤𝑤(𝑡𝑡) = 𝜇𝜇𝑣𝑣
Figure 2-6 Crossbar array with TiO2 switch in the
crossing point.
This formula can be used to get the device’s steering
property. A linear dependency is shown of w(t) from
q(t). This results in simple linear drift of dopants model
of first order (Batas & Fiedler, 2011).
Equivalent circuit diagram can be created with two
variable resistors in series with each other. One
represents the TiO2 when is stoichiometric (undoped)
and the other one when it has oxygen vacancies
(doped) in it. The idea is that a dividing line can be
pushed or pulled of the end state between the oxygen
vacancies and the stoichiometric part so that the
resistance of the total device can be varied fig.2-7.
One equation can be written that shows that the
voltage is a function of the current and resistance which
depends on the state variable. The state variable is the
position of the dividing line between the doped and
undoped material of the system. The other equation is
the dynamical state equation of the device. It shows
that the velocity of the dividing line (drift) is equal to the
electric field multiplied by the current through the
device. An equation can be derived for memristance. In
this simple model of linear conduction, the drift is
inversely proportional to the width of the device. It
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Ludmil Petkov et al
Response is the output of the XOR which is dependent
on the Challenge and the random output of the
memory cell. The chance of guessing the result is 50%,
when logic “1” is the output of the memory cell (Rose,
McDonald, Yan, Wysocki, et al., 2013).
shows that the memristance is more important at the
nano scale than the micron scale and at the millimetre
scale is completely unobservable. This explains that the
hysteresis is typical for nano circuits and this
is memristance. Very high fields cause atoms to
move. Memristance is the coupled motion of atoms
and electrons in a device.
3.Memeristor
Architecture
Based
PUF
Hardware Security is an important area of research. It
helps to resolve issues related to side channel attacks,
counterfeiting and piracy. Due to the scaling of the
technology to a nanometre regime the memristor
became a feasible choice to protect integrated circuits.
Memristors are interesting components to be used in
the design of PUFs because they are compatible to the
CMOS manufacturing standards and their sensitivity to
process variations can be controlled. The variation of
the memristor’s thickness D creates different write
times.
Figure 3-2 A one bit PUF based on filament growth
(Rose, McDonald, Yan, Wysocki, et al., 2013).
From fig.3-2 can be noticed that the Challenge is
externally supplied and through it one of the outputs of
the memristors can be selected. The output or the
Response bit will be the hardware explicit part of the
security key. The memristive device keys and values will
not be able to be read because of the requirement of a
forming step without alerting the mechanism for
tamper detection.
There is probability an attacker in theory to influence
the unpredictability of the suggested circuits. For
example, if a voltage potential is applied to a specific
device the time for writing can be successfully reduced.
The protections against such an attack are the required
formation step and multiple reset conditions return the
memristor to a normal state and attempts to overhaul
the memristor are annulled.
(Gao, Ranasinghe, Al-Sarawi, Kavehei, & Abbott, 2015)
are investigating highly secure physical unclonable
functions (PUFs). They are using a nanocrossbar
exploiting the existing high density of information. The
novel design benefits from reliability, uniqueness, great
number of challenge-response pairs (CRPs) and further
desirable characteristics typical for strong PUFs. Further
differently to the existing PUFs it can be reconfigured
(rPUF).
Figure 3-1 Variations in the memristor’s write times
are used in a 1-bit PUF (Rose, McDonald, Yan, Wysocki,
& Xu, 2013).
Fig.3-1 shows implemented a single bit memristive
memory-based PUF cell. Writing or reading of the
� /W). For the
circuit is determined by 2 control signals (𝑹𝑹
polarity of the writing (𝑵𝑵𝑵𝑵𝑵𝑵) is used. To work as a PUF
the circuit first carries out RESET of the memristor
� /W=1 and
𝑹𝑹
NEG=1 for a long enough period to
guarantee that the memristor is in the high resistance
state (HRS). After that NEXT is pulsed out for the write
� /W=1).
nominal time which is 𝒕𝒕𝒘𝒘𝒘𝒘,𝒎𝒎𝒎𝒎𝒎𝒎 (NEG=0 and 𝑹𝑹
Following the SET operation, the memristor can be read
� /W=0.
at the output by applying 𝑹𝑹
The input of an XOR function is used for the Challenge
and the other input is the output of the memristor. The
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Ludmil Petkov et al
reprogramming to revive CRPs of mrSPUF to convert it
into a new PUF instance. In distinction to the
programming control circuit, left analogue
multiplexers block and the top decoder block,
counters and CM-ROs enable the stimulation by a
challenge and the extraction of a corresponding
response. A challenge set as a vector of binary values
(bits) provides the address bits for both the analogue
multiplexers block and the decoder block. (d) CM-RO.
Each current mirror serves only an inverter in the RO
structure, where the bias memristor for each current
mirror, Mi, is carefully chosen from the nanocrossbar
array. Though difference in the oscillation frequency
of each RO is insignificantly inclined by the threshold
voltage difference in the CMOS transistor combining
the starved inverter and current mirror structures, the
complete difference in the oscillation frequency is
mainly determined by the differences in the
memresistance of 𝑴𝑴𝒊𝒊 if the supply voltage,𝑽𝑽𝑫𝑫𝑫𝑫 , is kept
constant (Gao et al., 2015).
Figure 3-3 (a) Memristive devices in a nanocrossbar.
(b) Bottom and top electrodes. (c) Principle of
operation of memristive devices. (d) Model of
interconnection of one memristive device , 𝒓𝒓𝒘𝒘 is the
nanowire segment resistance 𝑹𝑹𝒊𝒊𝒊𝒊 is the memristive
device resistance at j-th top and i-th bottom
conductor. (e) Reading scheme. To the designated
word line, a reading voltage is applied and the current
passing through it is sensed to determine the state of
the memristive device. Throughout the reading there
are some not obvious path currents (purple dashed
line) besides the desired read current (green dashed
line) (Gao et al., 2015).
For the mrSPUF two Current Mirrors – Ring Oscillators
(CM-RO) were used. The nanocrossbar complements
them which helps with reducing the area. The reading
is faster, and the reduced output frequency eases the
precision of the counting. Different to the memristorbased PUF in (Koeberl, Kocabaş, & Sadeghi, 2013)
where the goal was the value of the resistance to be
sensed. This determined the binary value (in ROFF or
RON state) of the targeted memristor in nanocrossbar
array. The memristance analogue value was decoded
(here only RON state was used) into a frequency over
CM-RO. The advantages of this design include: 1)
Significantly smaller number of Ros were used and only
i inverter stages to build each RO. 2) Much improved
number of CRPs were created. 3) The Response bit was
generated using different structure. This mitigated
some of the undesirable Response variations triggered
by power supply and temperature fluctuations 4)
Compared to the design in (Koeberl et al., 2013)
complex circuitry wasn’t needed a memory cell to be
readout. A full physical information (memory’s binary
value) of the all nanocrossbar’s array junctions was not
directly unprotected. These observations are shown in
table 3-1.
The memristors are randomly programmed in 𝑅𝑅𝑜𝑜𝑜𝑜 and
𝑅𝑅𝑜𝑜𝑜𝑜𝑜𝑜 state.
Figure 3-4 Memristive device based strong Physical
Unclonable Function (mrSPUF) architecture. (a) Basic
mrSPUF construction. Very memristor is in the ON
state. The shaded programming control circuit
includes the row programming control circuitry in (b)
and column programming control circuitry in (c) which
is working to program memristors in nanocrossbar
arrangement before it performs as a PUF and enables
reconfigurability
of
mrSPUF
by
successive
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Ludmil Petkov et al
Table 3-1Shows how other memristor PUFs
compare with mrSPUF (Gao et al., 2015).
(Koebe
rl et
al.,
2013)
Uniqueness
≈ 50% ≈ 50%
Uniformity
Number
CRPs
(Rose, mrSPUF
McDo Fig.2-12
nald,
Yan, &
Wysoc
ki,
2013)
N/A
of
Use of
Nanocrossbar
Security
Analysis
MxN
≈ 50%
M
50.07%
50.76%
𝑁𝑁 × �𝑀𝑀
� × �𝑀𝑀−𝑖𝑖
�
1
𝑖𝑖
2
Figure 4-2 some traces of analysing the TEAM single
memristor model.


Kvatinsky introduced a simple and general model of a
memristor, which corresponds to the Simmons model
with an acceptable error. This simplified model is based
on several assumptions. Provisions to simplify the
analysis and facilitate the calculation process. He
assumes the absence of changes in the model variable
parameters in a certain threshold and uses the
polynomial dependence of the current of the memristor
and the thickness of the tunnel barrier instead of the
exponential. The dependent of the derivative of the
thickness of the tunnelling barrier on the current is
divided by multiplying two functions, one of which is a
function of the current, and the other depends on the
variable x.




Strong PUF


Reconfigurabl
e PUF
Figure 4-1 The single TEAM model of a memristor
used by the researchers at Oxford Brookes University.




4. Simulation Using LTSpice
Memristors have great potential for use in various areas
of electronics and hardware cryptography. Therefore,
there is a need to create their qualitative models for the
analysis, design and creation of schematics based on
their newly found properties. Except the model,
provided by the HP lab other mathematical models of
memristors exist. A ThrEshold Adaptive Memristor
model (TEAM) was used as it was already chosen by the
University’s research group and they have created a
working model. It is flexible and can be used for any
practical memristive device and also was further
improved using a new window function (Kvatinsky,
Friedman, Kolodny, & Weiser, 2013),(Zayer, 2018).
Figure 4-3 Shows Memristive Memory PUF circuit
drawn on LTSpice. And gates were used instead of the
original design multiplexers as they have similar
function.
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Ludmil Petkov et al
and a new approach for magnetic flux-controlled memristor
modeling. IEEE Transactions on Nanotechnology, 10(2),
250–255. https://doi.org/10.1109/TNANO.2009.2038051
5. Discussion and Conclusion
Chua, L. O. (1971). Memristor—The Missing Circuit Element. IEEE
Transactions on Circuit Theory, 18(5), 507–519.
https://doi.org/10.1109/TCT.1971.1083337
This research showed that the memristors and the
memristive devices are sensible candidates for applied
PUFs. Some circuits are analysed that use completely
different characteristics of the device. One circuit
exploits the set time of the memristor. It is good that
the noise margin is low for an analogue device. A
satisfactory separation is given between larger noise
margins and memristance states. The second PUF
throughout relies on the ability two devices to be
scanned and written to crosswise. The mrSPUF
compared very well with the other designs and had
some unique features. For more thorough
understanding of the theoretical explanations and the
simulations of the small circuits, more real
experimental work is needed. This is the only way the
initial findings to be verified against protypes and
manufactured chips, circuits and modules. Circuit
designers can work closer with physicists, device
engineers and cyber-security specialist for the creations
of working and verified industrial designs. Reliability,
repeatability, cost, ruggedness, device aging are factors
that will need to be considered for the future projects.
The availability of memristors is still inadequate and
the memristor-based circuits for cyber-security remain
mostly speculative.
Chua, L. O., & Kang, S. M. (1976). Memristive Devices and
Systems. Proceedings of the IEEE, 64(2), 209–223.
https://doi.org/10.1109/PROC.1976.10092
Gao, Y., Ranasinghe, D. C., Al-Sarawi, S. F., Kavehei, O., & Abbott,
D. (2015). Memristive crypto primitive for building highly
secure physical unclonable functions. Scientific Reports, 5, 1–
14. https://doi.org/10.1038/srep12785
Joshi, S., Mohanty, S. P., & Kougianos, E. (2017). Everything You
Wanted to Know about PUFs. IEEE Potentials, 36(6), 38–46.
https://doi.org/10.1109/MPOT.2015.2490261
Kavehei, O., Iqbal, A., Kim, Y. S., Eshraghian, K., Al-Sarawi, S. F.,
& Abbott, D. (2010). The fourth element: Characteristics,
modelling and electromagnetic Theory of the memristor.
Proceedings of the Royal Society A: Mathematical, Physical
and Engineering Sciences, 466(2120), 2175–2202.
https://doi.org/10.1098/rspa.2009.0553
Koeberl, P., Kocabaş, Ü., & Sadeghi, A. R. (2013). Memristor PUFs:
A new generation of memory-based physically unclonable
functions. Proceedings -Design, Automation and Test in
Europe, DATE, 428–431.
Kvatinsky, S., Friedman, E. G., Kolodny, A., & Weiser, U. C. (2013).
TEAM: Threshold adaptive memristor model. IEEE
Transactions on Circuits and Systems I: Regular Papers,
60(1), 211–221. https://doi.org/10.1109/TCSI.2012.2215714
Rose, G. S., McDonald, N., Yan, L. K., & Wysocki, B. (2013). A
write-time based memristive PUF for hardware security
applications. IEEE/ACM International Conference on
Computer-Aided Design, Digest of Technical Papers, ICCAD,
830–833. https://doi.org/10.1109/ICCAD.2013.6691209
Rose, G. S., McDonald, N., Yan, L. K., Wysocki, B., & Xu, K.
(2013). Foundations of memristor based PUF architectures.
Proceedings of the 2013 IEEE/ACM International Symposium
on Nanoscale Architectures, NANOARCH 2013, 52–57.
https://doi.org/10.1109/NanoArch.2013.6623044
Semiconductor Engineering .:. Physically Unclonable Functions Semiconductor Engineering. (2019). Retrieved September 26,
2019,
from
file:///G:/MSc
Thesis
Cyber
Security/Semiconductor Engineering - Physically Unclonable
Functions.html
Acknowledgements
I would like to thank Dr Abusaleh Jabir and Mr Saurabh
Khandelwal for their help with the research.
Strukov, D. B., Snider, G. S., Stewart, D. R., & Williams, R. S.
(2008). The missing memristor found. Nature, 453(7191), 80–
83. https://doi.org/10.1038/nature06932
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