# Singularities, mixing and non-Markovianity of Pauli dynamical maps

Shrikant Utagi<EMAIL_ADDRESS>Theoretical Sciences Division,
Poornaprajna Institute of Scientific Research, Bidalur Bengaluru– 562164,
India Graduate Studies, Manipal Academy of Higher Education, Manipal -576104,
India. Vinod N. Rao Theoretical Sciences Division, Poornaprajna Institute of
Scientific Research, Bidalur Bengaluru– 562164, India R. Srikanth
<EMAIL_ADDRESS>Theoretical Sciences Division, Poornaprajna Institute of
Scientific Research, Bidalur Bengaluru– 562164, India Subhashish Banerjee
<EMAIL_ADDRESS>Interdisciplinary Program on Quantum Information and
Computation (IDRP-QIC), Indian Institute of Technology, Jodhpur-342037, India.

###### Abstract

Quantum non-Markovianity of channels can be produced by mixing Markovian
channels, as observed recently by various authors. We consider an analogous
question of whether singularities of the channel can be produced by mixing
non-singular channels, i.e., ones that lack them. Here we answer the question
in the negative in the context of qubit Pauli channels. On the other hand,
mixing channels with a singularity can lead to the elimination of
singularities in the resultant channel. We distinguish between two types of
singular channels, which lead under mixing to broadly quite different
properties of the singularity in the resultant channel. The connection to non-
Markovianity (in the sense of completely positive indivisibility) is pointed
out. These results impose nontrivial restrictions on the experimental
realization of non-invertible quantum channels by a process of channel mixing.

##  I Introduction

Open quantum systems, which are systems in interaction with an ambient
environment [1], experience an evolution with a rich structure showing the
absence or presence of memory effects [2, 3, 4, 5, 6, 7, 8, 9], unital or non-
unital features [10, 11, 12, 13]. Open system effects have profound
ramifications in areas such as those in quantum thermodynamics [14, 15, 16],
quantum cryptography [17, 18], quantum walks [19, 20], quantum correlations
and coherence [21, 22, 23], among others (cf. [24]).

An open quantum system evolution, under quite general conditions, is known to
be described by the general master equation

$\displaystyle\dot{\rho}(t)$
$\displaystyle=-\frac{i}{\hbar}[H_{S}(t),\rho(t)]+\sum_{j}\gamma_{j}(t)\left(L_{j}(t)\rho(t)L^{\dagger}_{j}(t)\right.$
$\displaystyle\left.-\frac{1}{2}\\{L^{\dagger}_{j}(t)L_{j}(t),\rho(t)\\}\right)$
$\displaystyle\equiv\mathcal{L}(t)[\rho(t)],$ (1)

where $\gamma_{j}(t)$ are the time-dependent decay rates, and $\\{L_{j}\\}$
are the set of orthonormal trace-less operators. The time-independent version
of Eq. (1), was studied in the pioneering works [25, 26].

Equivalently, an open quantum system evolution can be described by a channel,
i.e., a completely positive (CP) dynamical map, which is given by operator-sum
(or Kraus) representation:

$\displaystyle\rho(t)=\mathcal{E}(t)[\rho(0)]=\sum_{j}K_{j}(t)\rho
K^{\dagger}_{j}(t),$ (2)

where $K_{j}(t)$ are the Kraus operators. The dynamical map $\mathcal{E}(t)$
itself obeys the master equation [27, 7]
$\mathcal{\dot{E}}(t)=\mathcal{L}(t)[\mathcal{E}(t)],$ so that the time-
dependent map has the solution

$\displaystyle\mathcal{E}(t,t_{i})=\mathcal{T}{\rm
exp}\biggl{\\{}\int_{t_{i}}^{t}\mathcal{L}(s)ds\biggr{\\}},$ (3)

for all $t_{i}\leq s\leq t$, where $\mathcal{T}$ is the time-ordering
operator. And furthermore,

$\mathcal{L}(t)=\mathcal{\dot{E}}(t)\mathcal{E}^{-1}(t),$ (4)

showing that non-invertibility of the map $\mathcal{E}(t)$ corresponds to a
singularity in the generator $\mathcal{L}(t)$.

Complete positivity of $\mathcal{E}(t,t_{i})$ is the requirement that not only
is $\mathcal{E}(t,t_{i})$ positive, but so is any extension
$\mathcal{E}(t,t_{i})\otimes\mathbb{I}_{d}$, where $\mathbb{I}_{d}$ is the
identity operator in the Hilbert space of a $d$-dimensional ancilla. The map
$\mathcal{E}(t,t_{i})$ is CP if and only if the Choi matrix
$\chi=(\mathcal{E}(t,t_{i})\otimes I)[|\psi^{+}\rangle\langle\psi^{+}|]\geq 0$
for all $t\geq t_{i}$, where $\ket{\psi^{+}}=\ket{00}+\ket{11}$ is an
unnormalized maximally entangled state. Consider the two-parameter composition
of a CP map $\mathcal{E}(t_{f},t_{i})$ given by

$\displaystyle\mathcal{E}(t_{f},t_{i})=\mathcal{E}(t_{f},t)\mathcal{E}(t,t_{i}).$
(5)

If for all $t_{f}\geq t\geq t_{i}$, the intermediate map
$\mathcal{E}(t_{f},t)$ is CP, then the map $\mathcal{E}(t_{f},t_{i})$ is
called CP-divisible [5, 28]. Otherwise, it is CP-indivisible. CP-
indivisibility has been proposed as one of the criteria for non-Markovian
evolution, among a plethora of others [29, 6, 30, 31, 32, 8, 33]. In this
work, we consider the concept of non-Markovianity of a channel in the sense of
CP-indivisibility, as defined above.

Important for our purpose, in this work, are the notions of singular points of
the map and of singular channels.

Definition 1. If there is a time $t=t_{\ast}$ such that the composition Eq.
(5) fails, because the map $\mathcal{E}(t_{\ast},t_{i})$ is non-invertible and
thus
$\mathcal{E}(t_{f},t_{\ast})\equiv\mathcal{E}(t_{f},t_{i})\mathcal{E}(t_{\ast},t_{i})^{-1}$
is undefined, then the point $t_{\ast}$ is called the singularity (or,
singular point) of the channel $\mathcal{E}(t_{f},t_{i})$ [34]. Furthermore,
the channel is called “singular”. If no such singular points $t_{\ast}$ exist,
then the channel is said to be non-singular (or, regular).

Note that the singularity of the channel can be accompanied by perfectly
regular dynamics, i.e., the map $\mathcal{E}(t_{\ast},t_{i})$ itself is well-
defined (albeit non-invertible) [27, 35].

Ref. [36] discusses a method for making singularities tractable in the context
of the definition of CP-divisibility of maps [5]. Building thereon, a measure
of singularities of the maps is presented in [34]. An account of handling the
singularities was reviewed in [29, Sec. 4.3]. These measures based on CP-
indivisibility are equivalent to the one based on decay rate [7] up to a
constant factor.

The effect of mixing different quantum evolutions has attracted attention of
late. References. [37, 38] show that a convex combination of semigroup
dynamical maps can lead to a deviation from the semigroup structure. Quite
interestingly, the convex combination
$\mathcal{E}^{\prime}=(1-p)\mathcal{E}_{1}+p\mathcal{E}_{2}$ of two semigroup
(hence, CP-divisible) maps $\mathcal{E}_{1}={\rm exp}\\{t\mathcal{L}_{1}\\}$
and $\mathcal{E}_{2}={\rm exp}(t\mathcal{L}_{2})$ may give rise to CP-
indivisible (even eternally CP-indivisible) evolution [39]. More recently
various authors have shown that it is possible to obtain a CP-indivisible
Pauli channel by mixing CP-divisible Pauli channels [39, 40, 41], implying
that the set of CP-divisible channels is not convex.

For almost all relevant works in the literature, including those cited above
[36, 34, 29, 35, 42], instances of singularity of a channel are always
accompanied by non-Markovianity in the sense of CP-indivisibility (though the
converse is not true). In this light, our above observation concerning the
mixing of Markovian channels prompts the question of whether an analogous
behavior holds with respect to mixing singular channels. This will be
important for understanding the geometry of quantum channels.

In particular, restricting to the context of mixing Pauli channels, we ask
whether singular channels can be produced by mixing non-singular ones, and
answer the question in the negative. This negative result implies that non-
singular channels form a convex set. On the other hand, mixing singular
channels does not necessarily result in a singularity of the resultant
channel, showing that singular channels do not form a convex set. Finally, we
explain why in the context of mixing Pauli channels, singularities of the
channel imply CP-indivisibility, but the converse is not true. This connection
between singularity and CP-indivisibility does not hold in general.

This work is organized as follows. In Sec. II, we show that it is not possible
to produce singularities of the channel by mixing non-singular channels. In
Sec. III and IV, we discuss the results pertaining to mixing singular channels
of two broad types. The interplay of singularities and non-Markovianity is
discussed in Sec. V. In all cases, the results are illustrated with examples.
Finally, we conclude in Sec. VI.

##  II Mixing non-singular Pauli channels

A general Pauli dynamical map is given by

$\displaystyle\mathcal{E}(t)[\rho]=\sum_{i=0}^{3}k_{i}(t)\sigma_{i}\rho\sigma^{\dagger}_{i},$
(6)

where $\sigma_{0}=I$, and $\sigma_{i},\;i\in\\{1,2,3\\}$ are Pauli X, Y, Z
operators respectively, and $\sum_{i=0}^{3}k_{i}(t)=1$. The canonical form of
master equation corresponding to the map (6) has the form

$\displaystyle\dot{\rho}(t)=\mathcal{L}(t)[\rho(t)]=\sum_{j=1}^{3}\gamma_{j}(t)(\sigma_{j}\rho(t)\sigma_{j}^{\dagger}-\rho(t))$
(7)

where $\gamma_{j}(t)$ are the rates.

The decay rates may be readily obtained using Eq. (4) [43]. Noting that

$\displaystyle\mathcal{E}(t)[\sigma_{j}]=\lambda_{j}(t)\sigma_{j}.$ (8)

we have

$\displaystyle\dot{\mathcal{E}}(t)[\sigma_{j}]$
$\displaystyle=\dot{\lambda}_{j}(t)\sigma_{j}\;\;;\;\;\mathcal{E}^{-1}(t)[\sigma_{j}]=\frac{1}{\lambda_{j}}\sigma_{j},$
(9)

showing that the vanishing of a $\lambda_{j}$ at some time $t_{\ast}$
corresponds to non-invertibility of the map at that instant, and thus to a
singularity of the Pauli channel, per the argument following Eq. (4). This is
made more explicit below.

Now, from (9) and (4), we readily obtain the rates in the master equation Eq.
(7):

$\displaystyle\gamma_{1}(t)$
$\displaystyle=\frac{1}{4}\bigg{(}\frac{\dot{\lambda}_{1}(t)}{\lambda_{1}(t)}-\frac{\dot{\lambda}_{2}(t)}{\lambda_{2}(t)}-\frac{\dot{\lambda}_{3}(t)}{\lambda_{3}(t)}\bigg{)},$
$\displaystyle\gamma_{2}(t)$
$\displaystyle=\frac{1}{4}\bigg{(}\frac{\dot{\lambda}_{2}(t)}{\lambda_{2}(t)}-\frac{\dot{\lambda}_{1}(t)}{\lambda_{1}(t)}-\frac{\dot{\lambda}_{3}(t)}{\lambda_{3}(t)}\bigg{)},$
$\displaystyle\gamma_{3}(t)$
$\displaystyle=\frac{1}{4}\bigg{(}\frac{\dot{\lambda}_{3}(t)}{\lambda_{3}(t)}-\frac{\dot{\lambda}_{1}(t)}{\lambda_{1}(t)}-\frac{\dot{\lambda}_{2}(t)}{\lambda_{2}(t)}\bigg{)}.$
(10)

Note in particular that the $\gamma_{j}$’s have a singular point when any of
the $\lambda_{i}$ vanishes. Our first result, below, essentially asserts the
convexity of non-singular Pauli channels.

Lemma 1. It is impossible to produce a singular Pauli channel by mixing only
non-singular Pauli channels.

Proof. Let the Pauli channels that are being mixed be given by

$\displaystyle\mathcal{E}_{1}(\rho)$
$\displaystyle\equiv(1-p(t))\rho(0)+p(t)\sigma_{1}\rho\sigma_{1},$
$\displaystyle\mathcal{E}_{2}(\rho)$
$\displaystyle\equiv(1-q(t))\rho(0)+q(t)\sigma_{2}\rho\sigma_{2},$
$\displaystyle\mathcal{E}_{3}(\rho)$
$\displaystyle\equiv(1-r(t))\rho(0)+r(t)\sigma_{3}\rho\sigma_{3},$ (11)

where the functions $p,q,$ and $r$ quantify the degree of decoherence of the
channels and must satisfy $0\leq p,q,r\leq 1$ to ensure complete positivity of
the maps. The corresponding individual Lindblad rates are

$\gamma_{\eta}=\frac{-\dot{\eta}}{1-2\eta},$ (12)

where $\eta\in\\{p(t),q(t),r(t)\\}$. Let the three channels in Eq. (11) be
mixed with probabilities $a,b$ and $c$, where $0\leq a,b,c\leq 1$ and
$a+b+c=1$. This gives rise to the channel:

$\displaystyle\tilde{\mathcal{E}}(\rho)$
$\displaystyle=a\mathcal{E}_{1}(\rho)+b\mathcal{E}_{2}(\rho)+c\mathcal{E}_{3}(\rho)$
$\displaystyle=(1-ap-bq-
cr)\rho+ap\sigma_{1}\rho\sigma_{1}+bq\sigma_{2}\rho\sigma_{2}$
$\displaystyle+cr\sigma_{3}\rho\sigma_{3}.$ (13)

By assumption, the mixing maps $\mathcal{E}_{1},\mathcal{E}_{2}$ and
$\mathcal{E}_{3}$ are non-singular. In view of Eq. (12), this implies that

$0\leq p(t),q(t),r(t)<\frac{1}{2}$ (14)

for finite time $t$. The time-dependent eigenvalues of the map
$\tilde{\mathcal{E}}$ from Eq. (13) read

$\displaystyle\lambda_{1}(t)$ $\displaystyle=1-2(bq+cr),$ (15a)
$\displaystyle\lambda_{2}(t)$ $\displaystyle=1-2(ap+cr),$ (15b)
$\displaystyle\lambda_{3}(t)$ $\displaystyle=1-2(ap+bq).$ (15c)

The condition for a singularity in the resultant channel is that one or more
of $\lambda_{j}$ in Eq. (15) should vanish at a certain finite time(s)
$t_{s}$. For example, consider $\lambda_{1}$ in Eq. (15a). Given the range
restriction Eq. (14) on the decoherence functions $p(t)$ and $q(t)$, we have

$\lambda_{1}(t)>1-(b+c)\geq 0$ (16)

for finite $t$. Repeating the argument for $\lambda_{2}$ and $\lambda_{3}$, we
conclude that there can be no singularity in the mixed channel. $\blacksquare$

It follows from Eq. (16) and analogous results for $\lambda_{2}$ and
$\lambda_{3}$ that the non-singular mixing channels necessarily have positive
decay rates $\gamma_{j}$, and therefore are CP-divisible. Thus, as a corollary
of Lemma 1, we find that it is impossible to produce a singularity by mixing
CP-divisible Pauli channels.

Lemma 1 does not address the question of whether mixing singular channels
produces a singularity in the resultant channel. To address this question, it
is convenient to distinguish two types of singular channels. It is clear from
Eq. (12) that for a Pauli channel to be singular, the decoherence function
$p(t),q(t)$ or $r(t)$, as the case may be, should attain the value of
$\frac{1}{2}$ at some finite time $t$. Accordingly, the two types of Pauli
singular channels are those where the value $\frac{1}{2}$ is the maximum or is
exceeded. It turns out that they evince quite different behaviors under
mixing.

Definition 2. Channels of Type I: Those in which the maximum value attained by
the decoherence function $p(t),q(t)$ or $r(t)$ in Eq. (11) is $\frac{1}{2}$.

In this case, the occurrence of non-Markovianity (CP-indivisibility) can be
attributed to the non-monotonicity of the decoherence functions $p(t)$ etc.,
leading to recoherence in the negative slope region of the functions. Typical
instances of interest here would be channels for which the decoherence
function is non-monotonic, oscillating between $0$ and $\frac{1}{2}$. In Sec.
III, we shall show that: (a) mixing two such channels with singular points
$t_{\ast}^{p}$ and $t_{\ast}^{q}$ produces a singularity only if their
singularities are simultaneous ($t_{\ast}^{p}=t_{\ast}^{q}$), and moreover the
resultant singularity occurs at the same time; and furthermore, (b) three-way
mixing of such channels can never produce a singularity.

Definition 3. Channels of type II: Those in which the maximum value attained
by the decoherence function $p(t),q(t)$ or $r(t)$ in Eq. (11) can exceed
$\frac{1}{2}$.

Typical instances of interest here would be channels for which the decoherence
function is monotonic, reaching an asymptotic value in the interval
$(\frac{1}{2},1]$. The positive slope region of the decoherence function
$p(t)$ etc., when they exceed half, corresponds to recoherence of the system,
leading to non-Markovianity. Unlike in the case of Type I channels, we will
find, in Sec. IV, that the features (a) and (b) do not hold in this case,
i.e., singularities need not be simultaneous, and the restriction to two-way
mixing is not needed.

We shall find below that the conditions under which mixing of channels leads
to a singularity in the resultant channel depends on the type of the channels
being mixed.

We may note that Type I is a more usual occurrence, and can for example result
when a qubit system and its qubit environment evolve according to a
Hamiltonian given by $\omega(\ket{01}\bra{10}+\ket{10}\bra{01})$ acting on the
initial state $\ket{01}$, where $\omega$ is a real number. The joint system
remains in the subspace spanned by $\\{\ket{01},\ket{10}\\}$, and the reduced
state of the system is $\cos^{2}(\omega t)\ket{0}\bra{0}+\sin^{2}(\omega
t)\ket{1}\bra{1}$.

##  III  Mixing of Type I channels

In Eq. (15), suppose $\lambda_{1}(t)$ vanishes at some finite time $t_{\ast}$.
Observe that this can happen only if $q(t_{\ast})=r(t_{\ast})=\frac{1}{2}$. In
other words, the mixing channels $\mathcal{E}_{2}$ and $\mathcal{E}_{3}$ must
each possess a singularity such that these singularities occur simultaneously
at $t_{\ast}$, which coincides with the singularity in the resultant channel.
Moreover, we require that the mixing parameter $a=0$, meaning that only two of
three channels should be mixed. A similar argument holds for $\lambda_{2}(t)$
and $\lambda_{3}(t)$.

To summarize, in the context of mixing Type I channels to produce a
singularity, precisely two channels should be mixed, and they should be
synchronized in the occurrence of their singularities. If they are not
synchronized, then the singularity will be eliminated in the resultant
channel.

The fact that the mixing of two singular channels can produce a non-singular
channel can be compared to the situation that mixing CP-indivisible channels
can result in a CP-divisible channel - even a semigroup [39, 40]. A
consequence is that, like CP-indivisible channels, singular ones also form a
nonconvex set.

Example 1. Let the mixing channels be $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$,
with $p(t)=\frac{1}{2}[1-\cos^{2}(\mu t)]$ and
$q(t)=\frac{1}{2}[1-\cos^{2}(\nu t)]$, $a,b>0$ and $a+b=1$ in Eq. (13). In
view of Eq. (10), the eigenvalues $\lambda_{i}(t)$ of the resultant channel
read

$\displaystyle\lambda_{1}(t)$ $\displaystyle=1-b\sin^{2}(\nu t),$
$\displaystyle\lambda_{2}(t)$ $\displaystyle=1-a\sin^{2}(\mu t),$
$\displaystyle\lambda_{3}(t)$ $\displaystyle=1-a\sin^{2}(\mu t)-b\sin^{2}(\nu
t),$ (17)

showing that there is a singularity only from the zeros of $\lambda_{3}(t)$,
and furthermore this happens if and only if the two trigonometric terms attain
1 at the same time $t_{\ast}$, which will also be the singular point of the
resultant channel. A simple way to ensure this is by having $\mu=\nu$, in
which case singularities occur in the resultant channel for
$t=n\frac{\pi}{2}$. $\blacklozenge$

It is not hard to show that this behavior, of the singularities of the mixing
channels to be simultaneous at some time $t_{\ast}$, and leading to a
singularity at the same time $t_{\ast}^{R}=t_{\ast}$ in the resultant channel,
is general for Type I channels.

The following example illustrates the idea that the number of mixing channels
should not exceed 2. Otherwise the singularity is eliminated.

Example 2. A depolarizing colored noise is the Random telegraph noise (RTN)
non-Markovian depolarizing channel $\mathcal{E}[\rho]=\sum_{i}A_{i}\rho
A_{i}^{\dagger}$, with the Kraus operators [44]
$A_{i}=\sqrt{P_{i}}\sigma_{i},$ where
$\sigma_{0}=I,\sigma_{x}=\sigma_{1},\sigma_{y}=\sigma_{2},\sigma_{z}=\sigma_{3}$
are Pauli operators. Here,

$\displaystyle P_{0}$
$\displaystyle=\frac{1}{4}[1+\Lambda_{1}+\Lambda_{2}+\Lambda_{3}],$
$\displaystyle P_{1}$
$\displaystyle=\frac{1}{4}[1+\Lambda_{1}-\Lambda_{2}-\Lambda_{3}],$
$\displaystyle P_{2}$
$\displaystyle=\frac{1}{4}[1-\Lambda_{1}+\Lambda_{2}-\Lambda_{3}],$
$\displaystyle P_{3}$
$\displaystyle=\frac{1}{4}[1-\Lambda_{1}-\Lambda_{2}+\Lambda_{3}],$ (18)

where

$\displaystyle\Lambda_{i}=\exp(-wt)\left[\frac{\sin(wt\mu_{i})}{\mu_{i}}+\cos(wt\mu_{i})\right],$
(19)

The quantity $w=\frac{1}{2\tau}$ is the spectral bandwidth while $\tau$ is the
rate of fluctuation of the environment affecting the qubit, and
$\mu_{i}=\sqrt{\left(\frac{2d_{i}}{w}\right)^{2}-1}$, with $d_{i}$
representing the coupling strengths corresponding to the $i$th Pauli channel.
For the present purpose, let all $d_{i}$’s in Eq. (19) be taken to be equal,
given by $d$. This corresponds to equal mixing of the $X,Y,$ and $Z$ Pauli RTN
channels, as a result of which we obtain an isotropic RTN Pauli channel.

Now consider individual RTN Pauli channels of Type I with their respective
decoherence function being

$\displaystyle p(t)=q(t)=r(t)=\frac{1-\Lambda(t)}{2},$ (20)

where $\Lambda(t)$ is given by Eq. (19) with $\mu_{1}=\mu_{2}=\mu_{3}$. We now
consider the question of whether the above RTN non-Markovian depolarizing
channel can be reproduced by mixing the individual RTN Pauli channels of Type
I. For $d\gg w$, the zeros of $\Lambda$ occur periodically, making the channel
possess an infinite number of singularities. For $d<w$, $\Lambda(t)$ attains
zero only at $t=\infty$, making the channel non-singular.

Eq. (15) yields the following eigenvalues of the resultant channel:

$\displaystyle\lambda_{1}(t)$ $\displaystyle=1-(b+c)(1-\Lambda),$
$\displaystyle\lambda_{2}(t)$ $\displaystyle=1-(a+c)(1-\Lambda),$
$\displaystyle\lambda_{3}(t)$ $\displaystyle=1-(a+b)(1-\Lambda).$ (21)

Whilst in general $\Lambda$ takes values in the range $(-1,+1]$, but to
conform to Type I, the parameters $d$ and $w$ must be so chosen that
$\Lambda(t)$ is confined in the range $[0,1]$, with the singularity occurring
when $\Lambda(t)=0$. If $a,b,c>0$, then the sum of any two of them is strictly
less than 1. It follows from Eq. (21) and the Type I restriction (requiring
that $\Lambda$ is bounded below by 0) that each $\lambda_{j}$ in Eq. (21)
never vanishes. $\blacklozenge$

We now consider an analogous result when type II channels are mixed, and show
how it contrasts with the above two examples.

##  IV Mixing of Type II channels

In Equation (15), if we relax the requirement that $p,q$ and $r$ are bounded
by $\frac{1}{2}$, then we obtain Type II channels. In this case, we will find
that neither the synchronization nor restriction of the mixing channels to
two, is required, for producing a singular channel.

Example 3. Consider the same system as in Example 2, but letting $p,q,$ and
$r$ to exceed $\frac{1}{2}$. Accordingly, $\Lambda$ takes values in the range
$(-1,+1]$. For simplicity, let $a=b=c=\frac{1}{3}$, in which case the
eigenvalues become

$\displaystyle\lambda_{1,2,3}=\frac{1}{3}[1+2\Lambda(t)].$ (22)

The singularity of the resulting RTN depolarizing channel occurs when
$\Lambda(t)=-\frac{1}{2}.$ From Eq. (20), we find that this singularity
happens when $p(t)=q(t)=r(t)=\frac{3}{4}$, which is permissible for mixing
channels of type II. $\blacklozenge$

As a final example, consider the three mixing channels to be of type II,
having possibly different functional forms, but all reaching an asymptotic
value greater than (say) $\frac{4}{5}$.

Example 4. At some time $t_{\ast}^{R}$, let $q(t_{\ast}^{R})=\frac{3}{5}$ and
$r(t_{\ast}^{R})=\frac{4}{5}$. The singularities of $\mathcal{E}_{Y}$ and
$\mathcal{E}_{Z}$ occur, respectively, at $t_{\ast}^{2}$ and $t_{\ast}^{3}$,
where $q(t_{\ast}^{2})=r(t_{\ast}^{3})=\frac{1}{2}$, where in general we don’t
require $t^{2}_{\ast}=t^{3}_{\ast}$, i.e., the singularities of the mixing
channels are not necessarily synchronized. Furthermore let the mixing fraction
$b=\frac{1}{6}$ and $c=\frac{1}{2}$, so that $a=1-b-c=\frac{1}{3}$, implying
that there is a finite fraction of the channel $\mathcal{E}_{X}$ in the
mixing. It follows from Eq. (15a) that $\lambda_{1}=0$ at $t_{\ast}^{R}$,
meaning that this is a singularity of the resultant channel. Note that
$t^{R}_{\ast}$ need not coincide with either $t^{j}_{\ast}$ $(j=2,3)$.
$\blacklozenge$

##  V Interplay of singularities and non-Markovianity

It turns out that for the resultant channels considered here, singular
channels are necessarily non-Markovian (in the CP-indivisible sense). In Eq.
(15), consider the point $t_{\ast}$ where the first singularity is
encountered, i.e., one of the $\lambda_{j}(t_{\ast})$ vanish, say
$\lambda_{1}(t_{\ast})=0$. From Eq. (15a), we have
$\dot{\lambda}_{1}(t)\equiv-2(b\dot{q}+c\dot{r})$. In the case of Type I
channels, both $q(t)$ and $r(t)$ reach $\frac{1}{2}$ and fall off at the same
time. Therefore, $\dot{\lambda}_{1}(t)$ is negative just before $t_{\ast}$,
and flips sign to positive just after $t_{\ast}$. On the other hand,
$\lambda_{1}$ remains positive for all time. Thus:

$\displaystyle\lim_{t\rightarrow
t_{\ast}^{\pm}}\frac{\dot{\lambda_{1}}}{\lambda_{1}}=\pm\infty.$ (23)

This implies that $\gamma_{2}$ and $\gamma_{3}$ flip the sign from positive to
negative at the singularity, and $\gamma_{1}$ the other way.

For Type II channels, by virtue of monotonic increase of $q(t)$ and $r(t)$,
$\dot{\lambda}_{1}(t)\equiv-2(b\dot{q}+c\dot{r})<0$ for all time. On the other
hand, $\lambda_{1}$ flips its sign from positive to negative at the
singularity. Thus, Eqs. (23) and the attendant consequences for the decay
rates hold here too. Therefore, curiously, despite the contrasting behavior in
the eigenvalues and the rate of change, yet in both Type I and Type II
channels, singularities signal CP-indivisibility in a similar way.

It may be worth pointing out that singularities do not necessarily imply CP-
indivisibility. For illustration, consider a CP-indivisible dephasing channel
described by $\mathcal{L}[\rho]\equiv\gamma(t)(\sigma_{3}\rho\sigma_{3}-\rho)$
with the decay rate $\gamma(t)=\tan(\omega t)$, where $\omega$ is some real
number. The channel has an infinitely many singularities at
$t_{\ast}=\frac{(2n+1)\pi}{2\omega}$ for integer $n$, which will flip the sign
of the rate, and thus signal CP-indivisibility in a similar manner as
discussed above. By contrast, consider the same channel, but with decay rate
$\gamma(t)=\tan^{2}(\omega t)$. This channel contains singularities at the
same instants as the above channel, but the sign of the rate never flips from
positive to negative at any of these singularities and thus corresponds to a
CP-divisible process.

##  VI Discussions and Conclusions

This work discusses the problem of producing a singular general Pauli
dynamical map by the mixing non-singular (or, regular) Pauli channels. We
point out that it is impossible to do so. Different conditions on the classes
of mixing singular channels are considered in order to guarantee that the
resultant channel is singular. In particular, we show that: (i) for Type I
channels it is possible to produce a singular channel by mixing two singular
Pauli channels, provided the occurrence of their singularities is
synchronized; (ii) mixing three Type I singular channels results in the
elimination of singularities i.e., such a convex combination results in a
regular channel; and (iii) by contrast, in the case of Type II channels, we
have shown that mixing two or three singular channels can result in a singular
channel, and the singularities need not be synchronized in their occurrence.

A further question that may be considered here is the power of mixing weaker
forms of non-Markovianity than CP-indivisibility (cf. Reference [8] and
references therein), in terms of generating singularities and/or stronger
forms of non-Markovianity. An aspect of this for the measure of CP-indivisible
maps produced by mixing a class of Pauli maps was considered in Reference
[41]. A future direction would be to explore the results of this paper from a
geometric point of view [45, 46, 47], in particular to quantify the measure of
non-singular channels produced by mixing singular ones, analogous to results
for CP-divisible channels in References [40, 41]. Reference [48] shows how an
operation of coarse-graining in time while transforming a master equation from
a nonlocal integro-differential form to a time-local one can modify the CP-
indivisibility property of the dynamics [48]. An interesting question here
would be whether this approximation procedure can also modify the
(non-)singular property of the dynamics.

Ref. [49] reports on an experimental implementation of producing non-
Markovianity by two-way mixing of Pauli semigroups in a linear quantum optical
platform, an extension of which to three-way mixing of more general CP-
divisible maps has recently been proposed [41]. This idea may be adapted for
mixing singular channels, which can be produced by suitable bath engineering,
possibly with appropriate modifications to the experimental implementation in
Ref. [50] in the case of a photonic realization, or in Reference [51], in the
case of a semiconductor quantum optical realization.

Our result here, that singularities cannot be produced by mixing non-singular
channels, is shown only for qubit Pauli channels. We expect that this is true
quite generally, and in particular, applicable to non-Pauli maps.

###### Acknowledgements.

SU and VNR acknowledge Admar Mutt Education Foundation for support from a
scholarship. RS acknowledges the support of Department of Science and
Technology (DST), India, under Grant No. MTR/2019/001516. RS and SB
acknowledge, respectively, the support from the Interdisciplinary Cyber
Physical Systems (ICPS) program of the Department of Science and Technology
(DST), India, under Grants No.: DST/ICPS/QuEST/Theme-1/2019/14 and No.
DST/ICPS/QuEST/Theme-1/2019/6.

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L PREPRINT - accepted by 30th IFIP/IEEE International Conference on Very Large
Scale Integration 2022 (VLSI-SoC 2022)3.5cm,1cm1.001.8

# PA-PUF: A Novel Priority Arbiter PUF

Simranjeet Singh1, Srinivasu Bodapati2, Sachin Patkar1, Rainer Leupers4,
Anupam Chattopadhyay3, Farhad Merchant4 {simranjeet<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>{leupers<EMAIL_ADDRESS>aachen.de  1Indian Institute of Technology, Bombay, 2Indian Institute of
Technology, Mandi,
3Nanyang Technological University, Singapore, 4RWTH Aachen University, Germany

###### Abstract

This paper proposes a 3-input arbiter-based novel physically unclonable
function (PUF) design. Firstly, a 3-input priority arbiter is designed using a
simple arbiter, two multiplexers (2:1), and an XOR logic gate. The priority
arbiter has an equal probability of 0’s and 1’s at the output, which results
in excellent uniformity (49.45$\%$) while retrieving the PUF response.
Secondly, a new PUF design based on priority arbiter PUF (PA-PUF) is
presented. The PA-PUF design is evaluated for uniqueness, non-linearity, and
uniformity against the standard tests. The proposed PA-PUF design is
configurable in challenge-response pairs through an arbitrary number of feed-
forward priority arbiters introduced to the design. We demonstrate, through
extensive experiments, reliability of 100$\%$ after performing the error
correction techniques and uniqueness of 49.63$\%$. Finally, the design is
compared with the literature to evaluate its implementation efficiency, where
it is clearly found to be superior compared to the state-of-the-art.

###### Index Terms:

arbiter, priority arbiter, PUF, device authentication, security, LFSR

##  I Introduction

Physical unclonable functions (PUFs) have great prominence in today’s secure
device authentication and secure communication [1]. A PUF takes advantage of
randomness due to manufacturing process variation in integrated circuits and
generates a unique response for every device by tapping into the sources of
entropy such as variations in path delay or timing [1]. An integrated circuit
within the PUF maps the input challenge with a unique response, thus creating
a challenge-response pair (CRP). These CRPs are utilized to design various
security protocols, ranging from device attestation to data encryption. The
concept of a PUF was first reported in [2]. Based on the parameters of how the
response is derived, various kinds of PUFs have been proposed in the
literature. These parameters can be delay, memory, or the difference between
the current in the rails. In the literature, PUF designs are categorized as
delay-based and memory-based PUF - of which the prominent examples are Arbiter
PUF [3], ring oscillator PUF [4], SRAM PUF [5], Butterfly PUF [6], Glitch PUF
[7] and MEmory Cell-based Chip Authentication (MECCA) PUF [8]. These advanced
PUFs are being threatened with various attacks [9, 10]. Therefore, we need to
keep studying new constructions of PUFs that potentially thwart such attacks
with minimalistic overhead.

Arbiter PUF (APUF) is one of the delay-based PUF designs, where an arbiter
decides the response of a PUF between the two data paths comprised of cross-
coupled multiplexers [11]. For this purpose, the arbiter PUF uses the analog
timing difference between the two data paths and decides the output based on
the timing difference between the two lines. Several modifications to the
classical arbiter PUF are presented in the literature. In arbiter PUF, it is
possible to predict the relation between the challenge and response through
software modeling and programs. To prevent the modeling-based attacks, various
modifications were proposed, such as multi-arbiter PUF [12], [13] and double
arbiter PUF [14]. The multi-arbiter PUF design consists of arbiters in each
multiplexer stage and a multiplexer to choose the arbiter response. An
alternative is to take the PUF response as XOR of the arbiter outputs, which
improves the uniqueness, reliability, and robustness of the PUF.

The arbiter is an electronic circuit that can identify a signal’s first
occurrence. A simple D flip-flop can be used as an arbiter where one signal is
connected as the clock and the other as the data signal. Priority arbiter is
typically used in the Network-On-Chip (NoC) [15], in order to determine the
priority of the data request among the several requests [15]. Different kinds
of arbiters are proposed in literature [16], such as daisy chain arbiter,
round-robin arbiter, and dynamic arbiter. Based on the application of the NoC,
the arbiters are designed. The concept of priority arbiter from the NoC
communication can be adopted into the PUF to improve the design efficiency.
The advantage of having a priority arbiter is that the design has more non-
linearity compared to the simple arbiter PUF. In this paper, a new priority
arbiter is proposed, which demonstrates good uniformity. Based on that, a
novel PA-PUF is designed. The major contributions of this work are reported as
follows:

  * •

A new PUF using the priority arbiter called PA-PUF is designed. The PA-PUF
offers a uniqueness of 49.63 $\%$ and uniformity of 49.45$\%$ at the output.
The non-linearity in the output of the PUF is increased with the use of a
priority arbiter.

  * •

We demonstrate the configurability of PA-PUF by varying the number of CRPs.
The number of CRPs can be increased by increasing the length of the data path
by introducing more feed-forward arbiters.

  * •

The performance of the proposed priority arbiter PUF is studied as a function
of the number of feed-forward arbiters in the data path and the length of the
data path. For example, the uniqueness of the PUF can be increased by
increasing the length of the data path. It offers a reliability of 94.5$\%$
for a 128-bit response, which can be increased to 100$\%$ by implementing
Bose-Chaudhuri-Hocquenghem (BCH) error-correcting codes [17].

Figure 1: Classical arbiter PUF design, red lines indicate the data path of a
given challenge.

The rest of the paper is structured as follows. Section II provides the design
insights of the newly proposed PUF. The experimental results are given in
section III. Section IV compares the proposed design with state-of-the-art
designs from literature. Finally, Section V provides conclusions and outlook
of this work.

##  II Proposed Priority Arbiter PUF

Figure 2: (a), (b) possible input wave forms, and (c) D flip-flop.

A classical arbiter PUF with multiplexers and an arbiter is depicted in Fig.
1. The arbiter PUF is a delay-based design and the delay difference is taken
from the cross-coupled multiplexers as shown in the design. The challenge of
the arbiter PUF is given as select signals of the multiplexers in the data
path, a low to high transition given at the input of the multiplexer will be
propagated to the arbiter in the path selected by the challenge of the PUF.
The arbiter is implemented using a D flip-flop, and the same is presented with
the possible operations in Fig. 2, with two signals as Top ($T$) and Center
($C$). When the signal $C$ arrives first, then the output of the arbiter is
led to ‘0’; otherwise, the output is ‘1’. The same is explained in the Fig. 2
(a) $\&$ (b). Fig. 1 presents the symmetric path of the circuit through the
multiplexers. The multiplexers’ select signals (challenge) generate different
delay paths, resulting in a unique response for every combination. The major
disadvantage with the arbiter PUF is that the uniqueness of the PUF is less.
There are several modifications to the classical arbiter PUF by adding a feed-
forward path [18] in the design to introduce the non-linearity in the results.

In arbiter PUF, it is possible to predict the relation between the challenge
and response through software modeling and programs. To prevent this modeling
prediction, various modifications have been proposed in the past. One
alternative of arbiter PUF is multi-arbiter PUF [12], [13] and double arbiter
PUF [14]. All these methods are proposed to improve the uniqueness,
reliability, and robustness of the PUF output. Since the order of non-
linearity in the arbiter is less, so by considering multi-arbiter designs, the
non-linearity can be increased in the design. Next, we will propose a priority
arbiter-based PUF design.

(a)

(b)

Figure 3:  (a) Feed-forward arbiter for the proposed PA-PUF; T, C and B as the
top, center and bottom data lines; $F0,F1$ and $F2$ are the three outputs from
the feed-forward arbiter (b) Three-input priority arbiter; T, C and B as the
top, center and bottom data lines and R as response bit.

A new three-input priority arbiter is proposed to decide the response as per
the priority of the input’s arrival time. The arbiter used in the classical
arbiter is a two-input circuit. Now, an extra input is added to increase the
non-linearity in the output. The proposed three-input priority arbiter diagram
is shown in Fig. 3b, which is designed with three D-flip flops, two 2-to-1
multiplexers, and an XOR logic gate. The three inputs, T, C, and B (top,
center, and bottom) are given to the D flip-flops. Next, the select signal and
the data line of the multiplexer are taken from the outputs of the D flip-
flops. Finally, the output of multiplexers is applied to the input of the XOR
gate to generate a response bit.

Figure 4: Priority arbiter possible output (R) combinations with T, C and B as
the top, center and bottom data lines.

The operation of the proposed priority arbiter is shown in Fig. 4, where the
output is decided based on the priority of the input arrival times. Some of
the possible conditions are considered in the plot. Consider a case where the
arrival times of these signals are as follows, T, C, and B (first case in Fig.
4). Since T comes first in time, the outputs of the bottom and center D flip-
flops (DFF B and DFF C) are ‘0’ and ‘0’, while the top D flip-flop (DFF T)
output is ‘1’. After multiplexer output, the input of the XOR gate will be ‘1’
and ‘0’. This results in the output of the arbiter being ‘1’. The Fig. 4
demonstrates the possible conditions
$\\{T,B,C\\},\\{T,C,B\\},\\{B,T,C\\},\\{B,C,T\\},\\{C,B,T\\}$ and
$\\{C,T,B\\}$ out of these six conditions, the output is ‘0’ in three cases
and ‘1’ in three cases. This further results in achieving good uniformity in
the output of the priority arbiter. The same circuit with different
connections of the inputs will lead to non-uniform outputs (with non-equal
probability of 0’s and 1’s).

Figure 5: Schematic of proposed PA-PUF which includes three parallel
multiplexer lines (T, C, and B) and a priority arbiter at the end to generate
the response bit.

The PA-PUF is presented in Fig. 5, where the modifications to the classical
arbiter PUF have been done by adding a third data path in addition to the two
data paths. Fig. 5 shows the working of the PA-PUF for a given challenge
(marked in red). This shows that just by increasing the hardware by one-third,
the non-linearity in the output can be increased to an extreme. In the case of
arbiter PUF, when the output is ‘1’, then it can be understood that signal $T$
arrives before signal $C$. While in the case of priority arbiter PUF, as
explained in Fig. 4, the output is ‘1’, in three cases. Hence, it is difficult
to find which signal comes first. Moreover, the design is extended by
introducing the feed-forward paths to increase the robustness in the design.
The priority arbiter PUF with the feed-forward path is shown in Fig. 6, while
the feed-forward arbiter for the priority arbiter PUF is shown in Fig. 3b.

Figure 6:  Proposed PA-PUF with feed-forward arbiter.

##  III Experimental results

The proposed PA-PUF is designed using Verilog programming and implemented on
Nexys Video Artix-7 FPGA board. The responses of the PUF are collected using
the universal asynchronous receiver/transmitter (UART) protocol. Next, we will
discuss the proposed PA-PUF (with a feed-forward arbiter) and the implications
of deriving a security key. For a good PUF design, it should satisfy some
important criteria such as intra-chip Hamming distance, inter-chip Hamming
distance. Based on the results of the inter-chip and intra-chip Hamming
distance, we can analyse the other important parameters such as uniformity,
robustness, uniqueness, bit-aliasing, and reliability. Intra Hamming distance
(HD) is the Hamming distance between the responses of the PUF design within
the chip. Ideally, a single bit change in the challenge should result in a 50%
Hamming distance in the responses bits. The intra HD can be calculated by the
formula given in equation 1.

$Intera\ HD=\sum_{i=1}^{k}\frac{HD(R_{i},R_{i+1})}{n}\times 100$ (1)

Here, ‘k’ is the total number of challenges given to the PUF, $R_{i}$ and
$R_{i+1}$ are the responses to the challenges $C_{i}$ and $C_{i+1}$
respectively. The challenges are differed by one-bit change, and Fig. 7 shows
the Hamming distance in responses of the 128-bit PA-PUF. Fig. 7 also shows the
HD has a maximum at the half of the responses, and the plot follows the
Gaussian distribution.

Figure 7: Intra-chip Hamming distance plot of 128-bit PA-PUF.

Inter HD is the Hamming distance of the responses between two chips of the
same family. We have used three FPGAs of the same family/configuration to
produce the results on inter HD. Fig. 8 shows the inter HD between responses
of two FPGA boards, which is calculated by the formula given in equation 2.

$Inter\
HD=\frac{2}{k(k-1)}\sum_{i=1}^{k-1}\sum_{j=i+1}^{k}\frac{HD(R_{i},R_{j})}{n}$
(2)

Where, $i$ and $j$ are two different FPGAs. $R_{i}$ and $R_{j}$ are the
responses from $FPGA_{i}$ and $FPGA_{j}$ for the challenge C respectively. $k$
is number of PUF designs (3 in our case).

Figure 8: Inter-chip Hamming distance plot of 128-bit PA-PUF. Figure 9:
Uniformity of 128-bit PA-PUF along with 50% probability line.

###  III-A Robustness

Any response bit of the PUF should not be stuck at logic ‘0’ or ‘1’. The
stable ‘0’, stable ‘1’ and unstable bits are calculated for various sizes of
the PA-PUF design. The results are given in Table I and the results reveal
that the design has good robustness.

TABLE I: Robustness results of stable ‘0’ , ‘1’ and unstable bits. Board | Number of | Stable ‘0’ | Stable ‘1’ | Unstable
---|---|---|---|---
Artix-7 | bits | % | % | %
8-bit proposed PA-PUF
1 |  0.5 x $10^{6}$ | 25.93 | 24.64 | 49.43
2 |  0.5 x $10^{6}$ | 27.65 | 21.71 | 50.64
16-bit proposed PA-PUF
1 |  1 x $10^{6}$ | 26.65 | 24.24 | 49.11
2 |  1 x $10^{6}$ | 25.07 | 25.00 | 49.93
32-bit proposed PA-PUF
1 |  2 x $10^{6}$ | 26.24 | 24.92 | 48.84
2 |  2 x $10^{6}$ | 27.66 | 23.19 | 49.15
64-bit proposed PA-PUF
1 |  4 x $10^{6}$ | 30.52 | 21.32 | 48.16
2 |  4 x $10^{6}$ | 28.03 | 23.01 | 48.96
128-bit proposed PA-PUF
1 |  8 x $10^{6}$ | 31.80 | 19.79 | 48.41
2 |  8 x $10^{6}$ | 30.49 | 20.58 | 48.93
128-bit Ring oscillator PUF
1 |  8 $\times 10^{6}$ | 28.1 | 23.3 | 48.6
2 |  8 $\times 10^{6}$ | 27.6 | 23.6 | 48.8
Mean |  | 27.85 | 23.45 | 48.7
Mean [19] |  | 29.51 | 30.25 | 40.22

###  III-B Uniformity and Bit-aliasing

The response should have an equal number of 0‘s and 1‘s. This can be
calculated using the ‘uniformity’ of the PUF. The ideal value of uniformity is
50$\%$, which means the PUF response is uniformly distributed. Fig. 9 shows
the distribution of 0‘s and 1‘s in response bits. Table II gives the values
for various sizes of the PUF response. The other parameter of interest is the
‘bit-aliasing’, which is calculated over different chips/devices. The
experiment is conducted on three Nexys Video Artix-7 FPGA boards of the same
family. The PUF design is placed in the exact physical locations across
different boards to find whether any particular bit position is permanently
connected to the logic ‘0’ or ‘1’ irrespective of the challenge or a board.
The ideal value of the bit aliasing is 50$\%$, which is near to the values
recorded in Table II.

TABLE II: Uniformity and bit-aliasing for the proposed PUF. Parameter | Uniformity | Bit-alising
---|---|---
Ideal value |  50 % | 50 %
8-bit proposed PA-PUF
Maximum |  100$\%$ |  52.41 $\%$
Minimum |  0 $\%$ |  42.74 $\%$
Average |  49.3 $\%$ |  49.17 $\%$
16-bit proposed PA-PUF
Maximum |  100 $\%$ |  64.37 $\%$
Minimum |  0 $\%$ |  39.35 $\%$
Average |  49.25 $\%$ |  48.77 $\%$
32-bit proposed PA-PUF
Maximum |  84.37 $\%$ |  60.79 $\%$
Minimum |  9 $\%$ |  34.87 $\%$
Average |  47.8 $\%$ |  47.76 $\%$
64-bit proposed PA-PUF
Maximum |  78.75 $\%$ |  54.82 $\%$
Minimum |  21.87 $\%$ |  32.39 $\%$
Average |  47.47 $\%$ |  47.41 $\%$
128-bit proposed PA-PUF
Maximum |  67.9 $\%$ |  51.69 $\%$
Minimum |  25 $\%$ |  33.03 $\%$
Average |  49.45 $\%$ |  49.66 $\%$
128-bit Ring oscillator PUF
Maximum |  67.2 $\%$ |  52.82 $\%$
Minimum |  0 $\%$ |  0 $\%$
Average |  48.3 $\%$ |  47.8 $\%$

###  III-C Reliability and Uniqueness

Figure 10: Reliability plot of 128-bit priority arbiter PUF.

Reliability and uniqueness are the most important metrics in deciding the
design of the PUF. The uniqueness is defined as how uniquely the design can
identify from one chip to the other chip of the same family. The ideal value
of the uniqueness is 50$\%$; hence the response of the PUF from one chip to
the other is differed by half of its length. Uniqueness is calculated using
the ‘inter Hamming distance’ of the response, and the calculated values are
tabulated in Table III for various sizes of the proposed PUF. Further, the
reliability is also given in Table III, which is calculated using the ‘intra-
Hamming distance’ of the response by giving the same challenge over million
times, and Fig. 10 depicts the Hamming distance plot to calculate the
reliability. The error that occurred in the response of the PUF overages can
be corrected using the error correction mechanisms [20] and [21]. It has been
shown in Fig. 10 that after BCH error correction codes, reliability can
increase from 94.5% to 100% in PA-PUF.

TABLE III: Uniqueness and Reliability for various sizes of the proposed PA-PUF. PUF size | Uniqueness | Reliability
---|---|---
8-bit |  50 $\%$ |  99.45 $\%$
16-bit |  50 $\%$ |  99.05 $\%$
32-bit |  50 $\%$ |  98.84 $\%$
64-bit |  48.4 $\%$ |  98.15 $\%$
128-bit |  49.63 $\%$ |  95.37$\%$
Figure 11: Hamming distance of the proposed PA-PUF with and without feed-
forward arbiters and the classical arbiter PUF.

###  III-D Machine Learning-based Modelling Attacks

The key purpose of PUFs can be defeated by modeling the PUF structure and by
being able to predict its output. In a series of works [22, 9, 10], it has
been shown that nearly all variants of delay-based PUFs can be efficiently
modeled using machine learning. To further boost security, XOR-based arbiter
PUFs are introduced. It was also shown in a recent study [23] that learning
XOR-based arbiter PUFs is possible up to a limit on the number of parallel
arbiter chains. Complementing this research direction, alternative PUF design
frameworks [24] or restricted visibility of the PUF outputs [25] are proposed.
Since our proposed design applies to the general studies on arbiter-based
PUFs, it will come under the same purview of attacks and resilience presented
earlier. Hence, we focus on the lightweight PUF design itself and reserve the
study of detailed analysis of modeling attacks for the future. In that
context, it will be interesting to juxtapose the priority arbiter design
against the XOR-based arbiter chain merger.

##  IV Comparisons

Based on the performance metrics calculation, the comparisons of the proposed
PUF with the existing designs is summarized in Table IV. It is evident from
the results that the proposed priority arbiter PUF has good results compared
to the existing designs.

Figure 12: Comparison of uniqueness and reliability with respect to data path’s length (16-bit challenge + feed-forward arbiters). TABLE IV: Comparison of the performance metrics of the ring oscillator PUF; Ideal value of reliability is 100$\%$ while the remaining parameters have an ideal value of 50$\%$. Parameter | Proposed | RO | Original | CHAR | CHAR& MAJ
---|---|---|---|---|---
% | PA-PUF | PUF |  Design [19]
Uniqueness | 49.63 | 49.22 | 48.52 | 45.60 | 45.60
Uniformity | 49.45 | 48.5 | 51.06 | 50.60 | 50.54
Reliability | 100 | 99.99 | 92.00 | 98.87 | 99.58
Bit-aliasing | 49.66 | 47.89 | 43.52 | 43.52 | 43.52

The length of the data path plays a prominent role in deriving the response.
In addition to the length of the multiplexer-data path, the number of feed-
forward arbiters also plays a key role. Fig. 11 shows the comparison of the
classical arbiter PUF, proposed priority arbiter PUF, and the priority arbiter
PUF with the feed-forward arbiter. The plot reveals that the classical arbiter
PUF is not able to produce a various number of responses while the proposed
designs have more number of CRPs. Since the intra-Hamming distance plot has a
maximum at half of its response length only for the priority arbiter with the
feed-forward arbiter. The number of CRPs in a PA-PUF can be increased by using
the feed-forward arbiters.

Further, the number of feed-forward arbiters used as a select signal to the
multiplexer in the data path also influences the response. The plot shown in
Fig. 12 reveals that as the number of feed-forward arbiters increases in the
design, the reliability of the PUF reduces and the uniqueness of the PUF
improves. Since the number of CRPs can be increased by increasing the data
path length and by introducing more feed-forward arbiters, the PUF is able to
generate more variations in the output. Hence, the reliability is going to
reduce. So, one can decide the number of feed-forward arbiters as per their
requirements of uniqueness and reliability.

TABLE V: Comparisons of different PUF designs; ‘U’ is the uniqueness and ‘R’ is the reliability. PUF design ${}^{(\textbf{response~{}size})}$ |  U (%) |  R (%) | Platform | Area
---|---|---|---|---
Proposed PA-PUF ${}^{\textbf{(128)}}$ |  $>$ 49.63 | 100 | Artix-7 |  47 slices per PUF
SRAM PUF [5] (128) | 49.97 |  $>$ 88 | FPGA |  4800 SRAM memory bits
Latch PUF [26] (128) | 46 |  $>$ 87 | Spartan-3 |  2$\times$128 slices
Flip flop PUF [27] (4096) |  $\approx$50 |  $>$ 95 | Virtex-2 |  4096 flip-flops
Butterfly PUF [6] (64) |  $\approx$50 | 94 | Virtex-5 | 130 slices
RO PUF [4] (128) | 46.15 | 99.52 | Virtex-4 | 1024 RO’s
CRO PUF [28] (127) | 43.50 | 96 | Spartan-3 | 64 slices
PUF ID [29] (128) | 49.90 | 93.93 | Artix-7 | 128 slices
Ultra-compact PUF ID [30] (128) | 49.93 | 93.96 | Spartan-6 | 40 slices
Improved PUF ID [19] (128) | 45.60 | 99.42 | Spartan-6 | 128 slices

##  V Conclusions

This paper proposed a new priority arbiter with a uniform output with an equal
probability of 0’s and 1’s. We also proposed a new priority arbiter-based PUF
(PA-PUF) with three data path lines and a priority arbiter. Further, the feed-
forward arbiters are introduced to get new select signals to the multiplexers
in the data path to improve the number of challenge-response pairs. Finally,
we demonstrated the results of the proposed PA-PUF on Nexys Video Artix-7 FPGA
board. The experimental results show that the proposed PA-PUF has a
reliability of 94.5$\%$, which can be improved to 100% by implementing error
correction techniques. Moreover, the PA-PUF improves the PUF uniformity to
49.45% and uniqueness to 49.63$\%$. We plan to study the attacks on PA-PUF in
more detail, practically and theoretically, in the future.

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