# New approach to $\lambda$-Stirling numbers

Dae San Kim  Department of Mathematics, Sogang University, Seoul 121-742,
Republic of Korea<EMAIL_ADDRESS>,  Hye Kyung Kim2,∗ Department of
Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of
Korea<EMAIL_ADDRESS>and  Taekyun Kim 3,∗ Department of Mathematics,
Kwangwoon University, Seoul 139-701, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to study the $\lambda$-Stirling numbers of both kinds
which are $\lambda$-analogues of Stirling numbers of both kinds. Those numbers
have nice combinatorial interpretations when $\lambda$ are positive integers.
If $\lambda=1$, then the $\lambda$-Stirling numbers of both kinds reduce to
the Stirling numbers of both kinds. We derive new types of generating
functions of the $\lambda$-Stirling numbers of both kinds which are related to
the reciprocals of the generalized rising factorials. Furthermore, some
related identities are also derived from those generating functions. In
addition, all the corresponding results to the $\lambda$-Stirling numbers of
both kinds are obtained also for the $\lambda$-analogues of $r$-Stirling
numbers of both kinds which are generalizations of those numbers.

###### Key words and phrases:

$\lambda$-Stirling numbers of the first kind; $\lambda$-Stirling numbers of
the second kind; $\lambda$-analogues of $r$-Stirling numbers of the first
kind; $\lambda$-analogues of $r$-Stirling numbers of the second kind;
$\lambda$-analogues of binomial coefficients

###### MSC2020 Mathematics Subject Classification:

11B73; 05A18

* is corresponding author

##  1\. Introduction

It is remarkable that explorations for degenerate versions of quite a few
special numbers and polynomials, which began with the study of degenerate
Bernoulli and degenerate Euler polynomials by Carlitz, have led to the
discoveries of many interesting results (see [10,12-14,15] and references
therein). Especially, the degenerate Stirling numbers of both kinds are
degenerate versions of the usual Stirling numbers of both kinds and occur very
frequently when we study degenerate versions of many special numbers and
polynomials. In addition, $\lambda$-analogues, which are different from
degenerate versions, of some special numbers and polynomials have also been
studied (see [5, 6, 9, 11]). Here we consider the $\lambda$-analogues of
Stirling numbers of both kinds.

The Stirling number of the second kind ${{n}\brace{k}}$ enumerates the number
of partitions of the set $[n]=\left\\{1,2,\dots,n\right\\}$ into $k$ nonempty
disjoint sets, while the unsigned Stirling number of the first kind
${{n}\brack{k}}=(-1)^{n-k}S_{1}(n,k)$ counts the number of permutations of $n$
elements with $k$ disjoint cycles. Here we consider the $\lambda$-analogues of
the Stirling numbers of both kinds. The $\lambda$-analogues of
${{n}\brace{k}}$ are called the $\lambda$-Stirling numbers of the second kind
and denoted by ${{n}\brace{k}}_{\lambda}$, while those of ${{n}\brack{k}}$ are
called the $\lambda$-Stirling numbers of the first kind and denoted by
${{n}\brack{k}}_{\lambda}$. Here $\lambda$ is any real number and we note that
${{n}\brack{k}}_{1}={{n}\brack{k}},\,\,\textrm{and}\,\,{{n}\brace{k}}_{1}={{n}\brace{k}}$.
Now, let $\lambda$ be any positive integer. Then it is possible to give
combinatorial interpretations of ${{n}\brace{k}}_{\lambda}$ and
${{n}\brack{k}}_{\lambda}$. Indeed, ${{n}\brace{k}}_{\lambda}$, called the
translated Whitney numbers of the second kind in [2, 16], enumerates the
number of partitions of the set $[n]$ into $k$ subsets such that each element
of each subset can mutate in $\lambda$ ways, except the dominant one. While
${{n}\brack{k}}_{\lambda}$, called the translated Whitney numbers of the first
kind in [2, 16], counts the number of permutations of $n$ elements with $k$
cycles such that the element of each cycle can mutate in $\lambda$ ways,
except the dominant one. We also consider generalizations of the
$\lambda$-Stirling numbers of both kinds, namely the $\lambda$-analogues of
$r$-Stirling numbers of the second kind ${{n+r}\brace{k+r}}_{r,\lambda}$ and
those of (unsigned) $r$-Stirling numbers of the first kind
${{n+r}\brack{k+r}}_{r,\lambda}$. Here we note that
${{n}\brace{k}}_{0,\lambda}={{n}\brace{k}}_{\lambda}$ and
${{n}\brack{k}}_{0,\lambda}={{n}\brack{k}}_{\lambda}$.

The aim of this paper is to derive new types of generating functions of the
$\lambda$-Stirling numbers of both kinds (see Theorem 3, Corollary 4, Remark
5). We note that the new type of generating function of the $\lambda$-Stirling
numbers of the second kind is equivalent the ordinary generating function of
those numbers (see Theorem 3, Remark 5). The generating function of the
$\lambda$-Stirling numbers of the first kind follows from that of the second
kind by inversion (see Corollary 4). In Theorem 6, we find expressions of the
integral over $(a,\infty)$ of the reciprocals of the generalized rising
factorials as finite sums by using an identity in (18) and the fact that the
limit as $b$ tends to $\infty$ of a certain finite sum is zero (see (26)).
Also, we derive an expression of $\frac{1}{ka^{k}}$ as an infinite sum by
integrating the generating function of the $\lambda$-Stirling numbers of the
second kind in Corollary 4. In Section 3, we derive the corresponding results
to those ones in Section 2 for the $\lambda$-analogues of $r$-Stirling numbers
of both kinds by using similar methods. We are indebted to [3] for many ideas
of this paper.

The outline of this paper is as follows. In Section 1, we recall the
generalized falling factorial sequence, the Stirling numbers of the first
kind, the Stirling numbers of the second kind and the unsigned Stirling
numbers of the first kind. We remind the reader of the $\lambda$-Stirling
numbers of the first kind, the $\lambda$-Stirling numbers of the second kind
and the unsigned $\lambda$-Stirling numbers of the first kind. We recall the
$\lambda$-analogues of (unsigned) $r$-Stirling numbers of the first kind and
the $\lambda$-analogues of $r$-Stirling numbers of the second kind. We recall
an explicit expression of the $\lambda$-analogues of $r$-Stirling numbers of
the second kind and the $\lambda$-analogues of binomial coefficients. Then we
remind the reader of orthogonality and inverse relations for the
$\lambda$-analogues of $r$-Stirling numbers. Section2 is the main result of
this paper. In Theorem 3, we obtain an expression of the reciprocal of the
generalized rising factorial as an infinite series involving the
$\lambda$-Stirling numbers of the second kind, which is equivalent to the
ordinary generating function of those numbers. It is noted in Corollary 4 that
we get, by inversion, a generating function of the unsigned $\lambda$-Stirling
numbers of the first kind in terms of the reciprocals of the generalized
rising factorials. By applying a Frullani’s integral in (23), we show that the
limit as $b$ tends to $\infty$ of a certain finite sum is equal to zero (see
(26)). By using this fact and an expression of the reciprocals of the
generalized rising factorials in (18), we get explicit expressions as finite
sums of the integral of those reciprocals over $(a,\infty)$ in Theorem 6 and
those as infinite sums of $\frac{1}{ka^{k}}$ in Corollary 7. In Section 3, all
the results in Section 2 are extended to the $\lambda$-analogues of
$r$-Stirling numbers and similar results to those ones in Section 2 are
obtained. In the rest of this section, we recall the necessary facts that will
be used throughout this paper.

For any $\lambda\in\mathbb{R}$, the generalized falling factorial sequence is
defined by

(1) $\begin{split}(x)_{0,\lambda}=1,\
(x)_{n,\lambda}=x(x-\lambda)(x-2\lambda)\ \cdots\ (x-(n-1)\lambda),\quad(n\geq
1).\end{split}$

For $n\geq 0$, it is known that the Stirling numbers of the first kind are
defined by

(2) $\begin{split}(x)_{n}=\sum_{k=0}^{n}S_{1}(n,l)x^{l},\quad(\rm{see}\ \
[2-16,18]).\end{split}$

As the inversion formula of (2), the Stirling numbers of the second kind are
defined by

(3) $\begin{split}x^{n}=\sum_{k=0}^{n}{n\brace k}(x)_{k},\quad(n\geq
0),\quad(\rm{see}\ \ [2-16,18,19]).\end{split}$

The unsigned Stirling numbers of the first kind are defined by ${n\brack
k}=(-1)^{n-k}S_{1}(n,k),\quad(n,k\geq 0)$. In [9, 11], the $\lambda$-Stirling
numbers of the first kind are given by

(4)
$\begin{split}(x)_{n,\lambda}=\sum_{k=0}^{n}S_{1,\lambda}(n,k)x^{k},\quad(n\geq
0).\end{split}$

The unsigned $\lambda$-Stirling numbers of the first kind are defined by

(5)
$\begin{split}<x>_{n,\lambda}=(-1)^{n}(-x)_{n,\lambda}=\sum_{k=0}^{n}{n\brack
k}_{\lambda}x^{k},\end{split}$

where the generalized rising factorial sequence is given by

(6) $\begin{split}<x>_{0,\lambda}=1,\ <x>_{n,\lambda}=x(x+\lambda)\ \cdots\
(x+(n-1)\lambda),\quad(n\geq 1).\end{split}$

From (4) and (5), we note that ${n\brack
k}_{\lambda}=(-1)^{n-k}S_{1,\lambda}(n,k)$.

The $\lambda$-Stirling numbers of the second kind are defined by

(7) $\begin{split}x^{n}=\sum_{k=0}^{n}{n\brace
k}_{\lambda}(x)_{k,\lambda},\quad(n\geq 0),\quad(\text{see
\cite[cite]{[\@@bibref{}{9, 11}{}{}]}}).\end{split}$

For $r\in\mathbb{N}\cup\\{0\\}$, the $\lambda$-analogues of (unsigned)
$r$-Stirling numbers of the first kind are defined by

(8)
$\begin{split}<x+r>_{n,\lambda}=\sum_{k=0}^{n}{{n+r}\brack{k+r}}_{r,\lambda}x^{k},(\text{see
\cite[cite]{[\@@bibref{}{9, 11}{}{}]}}),\end{split}$

where we note that ${{n}\brack{k}}_{0,\lambda}={{n}\brack{k}}_{\lambda}$, for
$r=0$, (see (5), (8)).

Thus, by (8), we get

(9)
$\begin{split}(x)_{n,\lambda}=\sum_{k=0}^{n}(-1)^{n-k}{{n+r}\brack{k+r}}_{r,\lambda}(x+r)^{k},\quad(n\geq
0).\end{split}$

The $\lambda$-analogues of $r$-Stirling numbers of the second kind are defined
by

(10)
$\begin{split}(x+r)^{n}=\sum_{k=0}^{n}{{n+r}\brace{k+r}}_{r,\lambda}(x)_{k,\lambda},\quad(n\geq
0),\quad(\text{see \cite[cite]{[\@@bibref{}{9, 11}{}{}]}}).\end{split}$

Note that

(11)
$\begin{split}{{n+r}\brace{k+r}}_{r,\lambda}=\frac{1}{\lambda^{k}}\frac{1}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}(l\lambda+r)^{n}.\end{split}$

This follows easily from Theorem 1 of [9]. As a special case of (11), we
observe that

(12)
${{n}\brace{k}}_{\lambda}={{n}\brace{k}}_{0,\lambda}=\frac{1}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}\lambda^{n-k}l^{n},\quad(\text{see}\,\,\eqref{eq06},\eqref{eq09}).$

For $n\geq 0$, the $\lambda$-analogues of binomial coefficients are defined by

(13)
$\begin{split}{\binom{x}{n}}_{\lambda}=\frac{(x)_{n,\lambda}}{n!}=\frac{x(x-\lambda)\
\cdots\ (x-(n-1)\lambda))}{n!},\quad(\text{see
\cite[cite]{[\@@bibref{}{15}{}{}]}}).\end{split}$

We are going to use the next theorem and its corollary several times in the
sequel. Both (a) and (b) follow from (9) and (10), while (c), (d) and (e) can
be derived from (a) and (b).

######  Theorem 1.

The $\lambda$-analogues of $r$-Stirling numbers enjoy the following
orthogonality and inverse relations:

$\displaystyle(a)\,\,\sum_{k=l}^{n}(-1)^{n-k}{{n+r}\brack{k+r}}_{r,\lambda}{{k+r}\brace{l+r}}_{r,\lambda}=\delta_{n,l},$
$\displaystyle(b)\,\,\sum_{k=l}^{n}(-1)^{k-l}{{k+r}\brack{l+r}}_{r,\lambda}{{n+r}\brace{k+r}}_{r,\lambda}=\delta_{n,l},$
$\displaystyle(c)\,\,a_{n}=\sum_{k=0}^{n}{{n+r}\brace{k+r}}_{r,\lambda}c_{k}\Longleftrightarrow
c_{n}=\sum_{k=0}^{n}(-1)^{n-k}{{n+r}\brack{k+r}}_{r,\lambda}a_{k},$
$\displaystyle(d)\,\,a_{n}=\sum_{k=n}^{m}{{k+r}\brace{n+r}}_{r,\lambda}c_{k}\Longleftrightarrow
c_{n}=\sum_{k=n}^{m}(-1)^{k-n}{{k+r}\brack{n+r}}_{r,\lambda}a_{k},$
$\displaystyle(e)\,\,a_{n}=\sum_{k=n}^{\infty}{{k+r}\brace{n+r}}_{r,\lambda}c_{k}\Longleftrightarrow
c_{n}=\sum_{k=n}^{\infty}(-1)^{k-n}{{k+r}\brack{n+r}}_{r,\lambda}a_{k}.$

In the special case of $r=0$, we obtain the next orthogonality and inverse
relations for the $\lambda$-Stirling numbers.

######  Corollary 2.

The $\lambda$-Stirling numbers enjoy the following orthogonality and inverse
relations:

$\displaystyle(a)\,\,\sum_{k=l}^{n}(-1)^{n-k}{{n}\brack{k}}_{\lambda}{{k}\brace{l}}_{\lambda}=\delta_{n,l},$
$\displaystyle(b)\,\,\sum_{k=l}^{n}(-1)^{k-l}{{k}\brack{l}}_{\lambda}{{n}\brace{k}}_{\lambda}=\delta_{n,l},$
$\displaystyle(c)\,\,a_{n}=\sum_{k=0}^{n}{{n}\brace{k}}_{\lambda}c_{k}\Longleftrightarrow
c_{n}=\sum_{k=0}^{n}(-1)^{n-k}{{n}\brack{k}}_{\lambda}a_{k},$
$\displaystyle(d)\,\,a_{n}=\sum_{k=n}^{m}{{k}\brace{n}}_{\lambda}c_{k}\Longleftrightarrow
c_{n}=\sum_{k=n}^{m}(-1)^{k-n}{{k}\brack{n}}_{\lambda}a_{k},$
$\displaystyle(e)\,\,a_{n}=\sum_{k=n}^{\infty}{{k}\brace{n}}_{\lambda}c_{k}\Longleftrightarrow
c_{n}=\sum_{k=n}^{\infty}(-1)^{k-n}{{k}\brack{n}}_{\lambda}a_{k}.$

##  2\. New approach to $\lambda$-Stirling numbers

Let

(14) $\begin{split}\frac{1}{x(x+\lambda)(x+2\lambda)\ \cdots\
(x+k\lambda)}=\sum_{l=0}^{k}\frac{A_{l,\lambda}}{x+l\lambda}.\end{split}$

Then, for $0\leq m\leq k$, we have

(15)
$\begin{split}\lim_{x\rightarrow-m\lambda}(x+m\lambda)\frac{1}{x(x+\lambda)(x+2\lambda)\
\cdots\
(x+k\lambda)}=\lim_{x\rightarrow-m\lambda}(x+m\lambda)\sum_{l=0}^{k}\frac{A_{l,\lambda}}{x+l\lambda}=A_{m,\lambda}.\end{split}$

On the other hand, by (15), we get

(16)
$\begin{split}\lim_{x\rightarrow-m\lambda}&\frac{x+m\lambda}{x(x+\lambda)\
\cdots\ (x+(m-1)\lambda)(x+m\lambda)(x+(m+1)\lambda)\ \cdots\ (x+k\lambda)}\\\
&\hskip 56.9055pt=\frac{1}{(-m\lambda)\cdot(-m+1)\lambda\ \cdots\ (-\lambda)\
\cdot\ \lambda\ \cdot\ 2\lambda\ \cdots\ (k-m)\lambda}\\\ &\hskip
56.9055pt=\frac{(-1)^{m}}{{\lambda}^{m}m!\lambda^{k-m}(k-m)!}=\frac{(-1)^{m}}{\lambda^{k}}\frac{1}{k!}\binom{k}{m}.\end{split}$

From (15) and (16), we note that

(17)
$\begin{split}A_{m,\lambda}=\frac{1}{\lambda^{k}}\frac{(-1)^{m}}{k!}\binom{k}{m}.\end{split}$

By (14) and (17), we get

(18) $\begin{split}\frac{1}{x(x+\lambda)\ \cdots\
(x+k\lambda)}&=\sum_{l=0}^{k}\frac{1}{\lambda^{k}}\frac{(-1)^{l}}{k!}\binom{k}{l}\frac{1}{x+l\lambda}\\\
&=\frac{\lambda^{-k}}{k!}\sum_{l=0}^{k}(-1)^{l}\binom{k}{l}\sum_{n=0}^{\infty}(-1)^{n}l^{n}\lambda^{n}\bigg{(}\frac{1}{x}\bigg{)}^{n+1}\\\
&=\sum_{n=0}^{\infty}\bigg{(}\frac{(-1)^{n}}{k!}\sum_{l=0}^{k}(-1)^{l}\binom{k}{l}\lambda^{n-k}l^{n}\bigg{)}\bigg{(}\frac{1}{x}\bigg{)}^{n+1}\\\
&=\sum_{n=0}^{\infty}\frac{(-1)^{n-k}}{k!}\bigg{(}\sum_{l=0}^{k}(-1)^{k-l}\binom{k}{l}\lambda^{n-k}l^{n}\bigg{)}\bigg{(}\frac{1}{x}\bigg{)}^{n+1}\\\
&=\sum_{n=k}^{\infty}(-1)^{n-k}{{n}\brace{k}}_{\lambda}\bigg{(}\frac{1}{x}\bigg{)}^{n+1},\end{split}$

where we used (12).

Therefore, by (18), we obtain the following theorem.

######  Theorem 3.

For $k\geq 0$, we have

(19) $\begin{split}\frac{1}{x(x+\lambda)\ \cdots\
(x+k\lambda)}=\sum_{n=k}^{\infty}(-1)^{n-k}{{n}\brace{k}}_{\lambda}\bigg{(}\frac{1}{x}\bigg{)}^{n+1}.\end{split}$

Then, from Corollary 2 (e), we obtain the following result, which is the
generating function of the unsigned $\lambda$-Stirling numbers of the first
kind in terms of the inverses of the generalized rising factorials.

######  Corollary 4.

For $k\geq 0$, we have

(20)
$\begin{split}\bigg{(}\frac{1}{x}\bigg{)}^{k+1}&=\sum_{n=k}^{\infty}(-1)^{n-k}S_{1,\lambda}(n,k)\frac{1}{x(x+\lambda)\
\cdots\ (x+n\lambda)}\\\ &=\sum_{n=k}^{\infty}{n\brack
k}_{\lambda}\frac{1}{x(x+\lambda)\ \cdots\ (x+n\lambda)}.\end{split}$

######  Remark 5.

It is immediate to see that (19) is equivalent to the ordinary generating
function of the $\lambda$-Stirling numbers of the second kind given by

$\frac{x^{k}}{(1-\lambda x)(1-2\lambda x)\cdots(1-k\lambda
x)}=\sum_{n=k}^{\infty}{{n}\brace{k}}_{\lambda}x^{n}.$

On the other hand, the exponential generating function of the
$\lambda$-Stirling numbers of the second kind is shown in Equation (11) of [9]
to be equal to

$\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}=\sum_{n=k}^{\infty}{n\brace k}_{\lambda}\frac{t^{n}}{n!}.$

Let $f$ be a nonnegative real valued function that has value at $\infty$,
which is denoted by $f(\infty)$.
Then, for $a,b>0$, we have

(21)
$\begin{split}\int_{0}^{\infty}\frac{f(ax)-f(bx)}{x}dx&=\int_{0}^{\infty}\int_{b}^{a}\frac{\partial}{\partial
t}f(xt)dtdx\\\ &=\int_{b}^{a}\int_{0}^{\infty}\frac{\partial}{\partial
t}f(xt)dxdt=\int_{b}^{a}\frac{f(\infty)-f(0)}{t}dt\\\
&=(f(\infty)-f(0))\log\frac{a}{b}.\end{split}$

The integral in (21) is the Frullani’s integral and its evaluation in (22) was
first published by Cauchy in 1823. The equation (21) is valid for ‘nice’
functions $f$ (see [1, 4, 17]).

Thus, by (21), we get

(22)
$\begin{split}\int_{0}^{\infty}\frac{f(ax)-f(bx)}{x}dx=(f(\infty)-f(0))\log\frac{a}{b},\quad(\text{see
\cite[cite]{[\@@bibref{}{4}{}{}]}}).\end{split}$

Let us take $f(x)=e^{-x}$ and $a=1$. Then we have

(23)
$\begin{split}\int_{0}^{\infty}\frac{e^{-x}-e^{-bx}}{x}dx=-\log\frac{1}{b}=\log
b,\quad(b>0).\end{split}$

Let $k$ be any positive integer, and let $b,\lambda>0$. Then, from (23), we
note that

(24) $\begin{split}\sum_{l=0}^{k}&\binom{k}{l}(-1)^{l}\log(b+l\lambda)\\\
&=\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}\int_{0}^{\infty}\frac{e^{-x}-e^{-(b+l\lambda)x}}{x}dx\\\
&=\int_{0}^{\infty}\left\\{\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}\frac{e^{-x}}{x}-\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}\frac{e^{-(b+l\lambda)x}}{x}\right\\}dx\\\
&=\int_{0}^{\infty}\left\\{(1-1)^{k}\frac{e^{-x}}{x}-\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}\frac{e^{-(b+l\lambda)x}}{x}\right\\}dx\\\
&=-\int_{0}^{\infty}\frac{e^{-bx}}{x}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}e^{-l\lambda
x}dx=-\int_{0}^{\infty}\frac{e^{-bx}}{x}(1-e^{-\lambda x})^{k}dx.\end{split}$

As $\frac{(1-e^{-\lambda x})^{k}}{x}$ is bounded on $[0,\infty)$, say by $M$,
we have

(25) $\left|\int_{0}^{\infty}\frac{e^{-bx}}{x}(1-e^{-\lambda
x})^{k}dx\right|\leq M\int_{0}^{\infty}e^{-bx}dx=\frac{M}{b},$

and hence the integral $\int_{0}^{\infty}\frac{e^{-bx}}{x}(1-e^{-\lambda
x})^{k}dx$ is convergent for any $b,\lambda>0$ and any positive integer $k$.

Moreover, for any positive integer $k$ and any $\lambda>0$, by (25) we get

(26)
$\begin{split}\lim_{b\rightarrow\infty}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}\log(b+l\lambda)=-\lim_{b\rightarrow\infty}\int_{0}^{\infty}\frac{e^{-bx}}{x}(1-e^{-\lambda
x})^{k}dx=0.\end{split}$

From (26) and (18), for any $\lambda,a>0$ and any positive integer $k$, we
note that

(27) $\begin{split}\int_{a}^{\infty}\frac{1}{x(x+\lambda)\ \cdots\
(x+k\lambda)}dx&=\lim_{b\rightarrow\infty}\int_{a}^{b}\frac{1}{x(x+\lambda)\
\cdots\ (x+k\lambda)}dx\\\
&=\frac{\lambda^{-k}}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}\lim_{b\rightarrow\infty}\int_{a}^{b}\frac{1}{x+l\lambda}dx\\\
&=\frac{\lambda^{-k}}{k!}\lim_{b\rightarrow\infty}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}(\log(b+l\lambda)-\log(a+l\lambda))\\\
&=\frac{\lambda^{-k}}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l+1}\log(a+l\lambda).\end{split}$

Thus, by (26), we get the following theorem.

######  Theorem 6.

For any $\lambda,a>0$ and any positive integer $k$, the following identity
holds.

(28) $\begin{split}\int_{a}^{\infty}\frac{1}{x(x+\lambda)\ \cdots\
(x+k\lambda)}dx=\frac{\lambda^{-k}}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l+1}\log(a+l\lambda).\end{split}$

From Corollary 4 and (28), for any positive integer $k$ and any $\lambda,a>0$,
we have

(29)
$\begin{split}\frac{1}{ka^{k}}=\int_{a}^{\infty}\frac{1}{x^{k+1}}dx&=\sum_{n=k}^{\infty}{n\brack
k}_{\lambda}\int_{a}^{\infty}\frac{1}{x(x+\lambda)\ \cdots\ (x+n\lambda)}dx\\\
&=\sum_{n=k}^{\infty}{n\brack
k}_{\lambda}\frac{1}{n!\lambda^{n}}\sum_{l=0}^{n}\binom{n}{l}(-1)^{l+1}\log(a+l\lambda).\end{split}$

We state this as a corollary.

######  Corollary 7.

For any positive integer $k$ and any $\lambda,a>0$, the following holds true.

$\frac{1}{ka^{k}}=\sum_{n=k}^{\infty}{n\brack
k}_{\lambda}\frac{1}{n!\lambda^{n}}\sum_{l=0}^{n}\binom{n}{l}(-1)^{l+1}\log(a+l\lambda).$

By Theorem 3, we get

(30) $\begin{split}\bigg{(}\frac{1}{x}\bigg{)}^{k}=\sum_{n=k}^{\infty}{n\brack
k}_{\lambda}\frac{1}{(x+\lambda)(x+2\lambda)\ \cdots\
(x+n\lambda)}.\end{split}$

Replacing $x$ by $x-\lambda$ and $n$ by $n+1$, we get

(31)
$\begin{split}\bigg{(}\frac{1}{x-\lambda}\bigg{)}^{k}=\sum_{n=k-1}^{\infty}{{n+1}\brack
k}_{\lambda}\frac{1}{x(x+\lambda)\ \cdots\ (x+n\lambda)}.\end{split}$

In particular, for $k=1$, we have

(32)
$\begin{split}\frac{1}{x-\lambda}&=\sum_{n=0}^{\infty}\frac{\lambda^{n}n!}{x(x+\lambda)\
\cdots\
(x+n\lambda)}=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{\binom{x+(n-1)\lambda}{n}_{\lambda}(x+n\lambda)}.\end{split}$

##  3\. $\lambda$-analogues of $r$-Stirling numbers

Let

(33) $\begin{split}\frac{1}{(x+r)(x+r+\lambda)\ \cdots\
(x+r+k\lambda)}=\sum_{l=0}^{k}\frac{K_{l,\lambda}}{x+r+l\lambda}.\end{split}$

For $0\leq m\leq k$, we have

(34)
$\begin{split}K_{m,\lambda}&=\lim_{x\rightarrow{-r-m\lambda}}\sum_{l=0}^{k}\frac{K_{l,\lambda}}{x+r+l\lambda}(x+r+m\lambda)\\\
&=\lim_{x\rightarrow{-r-m\lambda}}\frac{x+r+m\lambda}{(x+r)(x+r+\lambda)\
\cdots\ (x+r+k\lambda)}\\\
&=\frac{(-1)^{m}}{\lambda^{k}}\frac{k!}{m!(k-m)!}\frac{1}{k!}=\frac{(-1)^{m}}{\lambda^{k}}\frac{1}{k!}\binom{k}{m}.\end{split}$

By (33) and (34), we get

(35) $\begin{split}\frac{1}{(x+r)(x+r+\lambda)\ \cdots\
(x+r+k\lambda)}&=\sum_{l=0}^{k}\frac{(-1)^{l}}{\lambda^{k}k!}\binom{k}{l}\frac{1}{x+r+l\lambda}\\\
&=\frac{1}{\lambda^{k}k!}\sum_{l=0}^{k}(-1)^{l}\binom{k}{l}\frac{1}{x}\frac{1}{1+\frac{r+l\lambda}{x}}\\\
&=\sum_{n=0}^{\infty}(-1)^{n}\frac{1}{\lambda^{k}k!}\sum_{l=0}^{k}(-1)^{l}\binom{k}{l}(r+l\lambda)^{n}\bigg{(}\frac{1}{x}\bigg{)}^{n+1}\\\
&=\sum_{n=0}^{\infty}(-1)^{n-k}\bigg{(}\frac{1}{\lambda^{k}}\frac{1}{k!}\sum_{l=0}^{k}(-1)^{k-l}\binom{k}{l}(r+l\lambda)^{n}\bigg{)}\bigg{(}\frac{1}{x}\bigg{)}^{n+1}\\\
&=\sum_{n=k}^{\infty}(-1)^{n-k}{{n+r}\brace{k+r}}_{r,\lambda}\bigg{(}\frac{1}{x}\bigg{)}^{n+1},\end{split}$

where we used (11).

Therefore, by (35), we obtain the following theorem.

######  Theorem 8.

For $k\geq 0$, we have

(36) $\begin{split}\frac{1}{(x+r)(x+r+\lambda)\ \cdots\
(x+r+k\lambda)}=\sum_{n=k}^{\infty}(-1)^{n-k}{{n+r}\brace{k+r}}_{r,\lambda}\bigg{(}\frac{1}{x}\bigg{)}^{n+1}.\end{split}$

Now, from Corollary 2 (e) and using the notation in (13), we obtain the
following result..

######  Corollary 9.

For $k\geq 0$, we have

$\begin{split}\bigg{(}\frac{1}{x}\bigg{)}^{k+1}&=\sum_{n=k}^{\infty}{{n+r}\brack{k+r}}_{r,\lambda}\frac{1}{(x+r)(x+r+\lambda)\
\cdots\ (x+r+n\lambda)}\\\
&=\sum_{n=k}^{\infty}{{n+r}\brack{k+r}}_{r,\lambda}\frac{1}{n!{\binom{x+r+(n-1)\lambda}{n}}_{\lambda}(x+r+n\lambda)}.\end{split}$

######  Remark 10.

The equation (36) is equivalent to the ordinary generating function of the
$\lambda$-analogues of $r$-Stirling numbers of the second kind given by

$\frac{x^{k}}{(1-rx)(1-(r+\lambda)x)\cdots(1-(r+k\lambda)x)}=\sum_{n=k}^{\infty}{{n+r}\brace{k+r}}_{r,\lambda}x^{n}.$

On the other hand, it is shown in Theorem 1 of [9] that the exponential
generating function of the $\lambda$-analogues of $r$-Stirling numbers of the
second kind is given by

$\frac{1}{\lambda^{k}}\frac{1}{k!}\big{(}e^{\lambda
t}-1\big{)}^{k}e^{rt}=\sum_{n=k}^{\infty}{n+r\brace
k+r}_{r,\lambda}\frac{t^{n}}{n!},\quad(k\geq 0).$

Let $k$ be any positive integer and let $\lambda,a>0$. By using (26) and (35),
we obtain

(37) $\begin{split}\int_{a}^{\infty}\frac{1}{(x+r)(x+r+\lambda)\ \cdots\
(x+r+k\lambda)}dx=\frac{1}{k!}\frac{1}{\lambda^{k}}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l+1}\log(a+r+l\lambda).\end{split}$

From (37) and Corollary 6, we have

(38) $\begin{split}\frac{1}{ka^{k}}&=\int_{a}^{\infty}\frac{1}{x^{k+1}}dx\\\
&=\sum_{n=k}^{\infty}{{n+r}\brack{k+r}}_{r,\lambda}\frac{1}{n!}\int_{a}^{\infty}\frac{1}{n!{\binom{x+r+(n-1)\lambda}{n}}_{\lambda}(x+r+n\lambda)}dx\\\
&=\sum_{n=k}^{\infty}{{n+r}\brack{k+r}}_{r,\lambda}\frac{1}{n!\lambda^{n}}\sum_{l=0}^{n}\binom{n}{l}(-1)^{l+1}\log(a+r+l\lambda).\end{split}$

Thus we have shown the following theorem.

######  Theorem 11.

For any positive integer $k$ and any $\lambda,a>0$, the following hold true.

$\displaystyle\int_{a}^{\infty}\frac{1}{(x+r)(x+r+\lambda)\ \cdots\
(x+r+k\lambda)}dx=\frac{1}{k!}\frac{1}{\lambda^{k}}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l+1}\log(a+r+l\lambda),$
$\displaystyle\frac{1}{ka^{k}}=\sum_{n=k}^{\infty}{{n+r}\brack{k+r}}_{r,\lambda}\frac{1}{n!\lambda^{n}}\sum_{l=0}^{n}\binom{n}{l}(-1)^{l+1}\log(a+r+l\lambda).$

##  4\. Conclusion

We have witnessed that studying both degenerate versions and
$\lambda$-analogues of some special numbers and polynomials yielded many
fascinating results. Here we note that these two are different in nature.
Indeed, the degenerate versions tend to the non-degenerate ones as $\lambda$
tends to 0, while the $\lambda$-analogues as $\lambda$ tends to 1. In this
paper, we studied the $\lambda$-Stirling numbers of both kinds which are
$\lambda$-analogues of the usual Stirling numbers. We derived new types of
generating functions for those numbers and also for the $\lambda$-analogues of
$r$-Stirling numbers of both kinds. All of these generating functions are
related to reciprocals of the generalized rising factorials. Furthermore, from
those new types of generating functions we obtained expressions of the
integral over $(a,\infty)$ of the reciprocals of the generalized rising
factorials as finite sums and an expression of $\frac{1}{ka^{k}}$ as an
infinite sum involving the $\lambda$-Stirling numbers of the first kind or the
$\lambda$-analogues of $r$-Stirling numbers of the first kind.

It is one of our future projects to continue to study various
$\lambda$-analogues of many special numbers and polynomials and to find their
applications to physics, science and engineering as well as to mathematics.

## References

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# Accelerating Tactile Internet with QUIC: An Exploration of its Security and
Privacy Attacks

Jayasree Sengupta1, Debasmita Dey2, Simone Ferlin3, Nirnay Ghosh2, and Vaibhav
Bajpai4
1CISPA Helmholtz Center for Information Security, Germany
2Indian Institute of Engineering Science and Technology, Shibpur, India;
1RedHat, Sweden
4Hasso Plattner Institute, Germany  Jayasree Sengupta, Debasmita Dey, Simone
Ferlin, Nirnay Ghosh and Vaibhav Bajpai J. Sengupta is with CISPA Helmholtz
Center for Information Security, Germany; D. Dey and N. Ghosh are with Indian
Institute of Engineering Science and Technology, Shibpur, India; S. Ferlin is
with RedHat and Karlstad University, Sweden; V.Bajpai is with Hasso Plattner
Institute, Germany.

# Accelerating Tactile Internet with QUIC: An Exploration of the Security $\&$
Privacy Attack Surface

Jayasree Sengupta1, Debasmita Dey2, Simone Ferlin3, Nirnay Ghosh2, and Vaibhav
Bajpai4
1CISPA Helmholtz Center for Information Security, Germany
2Indian Institute of Engineering Science and Technology, Shibpur, India;
1RedHat, Sweden
4Hasso Plattner Institute, Germany  Jayasree Sengupta, Debasmita Dey, Simone
Ferlin, Nirnay Ghosh and Vaibhav Bajpai J. Sengupta is with CISPA Helmholtz
Center for Information Security, Germany; D. Dey and N. Ghosh are with Indian
Institute of Engineering Science and Technology, Shibpur, India; S. Ferlin is
with RedHat and Karlstad University, Sweden; V.Bajpai is with Hasso Plattner
Institute, Germany.

# Accelerating Tactile Internet with QUIC: An Exploration of the Security,
Privacy Attack Surface

Jayasree Sengupta1, Debasmita Dey2, Simone Ferlin3, Nirnay Ghosh2, and Vaibhav
Bajpai4
1CISPA Helmholtz Center for Information Security, Germany
2Indian Institute of Engineering Science and Technology, Shibpur, India;
1RedHat, Sweden
4Hasso Plattner Institute, Germany  Jayasree Sengupta, Debasmita Dey, Simone
Ferlin, Nirnay Ghosh and Vaibhav Bajpai J. Sengupta is with CISPA Helmholtz
Center for Information Security, Germany; D. Dey and N. Ghosh are with Indian
Institute of Engineering Science and Technology, Shibpur, India; S. Ferlin is
with RedHat and Karlstad University, Sweden; V.Bajpai is with Hasso Plattner
Institute, Germany.

# Accelerating Tactile Internet with QUIC: An Exploration of the Security
Attack Surface

Jayasree Sengupta1, Debasmita Dey2, Simone Ferlin3, Nirnay Ghosh2, and Vaibhav
Bajpai4
1CISPA Helmholtz Center for Information Security, Germany
2Indian Institute of Engineering Science and Technology, Shibpur, India;
1RedHat, Sweden
4Hasso Plattner Institute, Germany  Jayasree Sengupta, Debasmita Dey, Simone
Ferlin, Nirnay Ghosh and Vaibhav Bajpai J. Sengupta is with CISPA Helmholtz
Center for Information Security, Germany; D. Dey and N. Ghosh are with Indian
Institute of Engineering Science and Technology, Shibpur, India; S. Ferlin is
with RedHat and Karlstad University, Sweden; V.Bajpai is with Hasso Plattner
Institute, Germany.

# Accelerating Tactile Internet with QUIC:
A Cybersecurity Perspective

Jayasree Sengupta1, Debasmita Dey2, Simone Ferlin3, Nirnay Ghosh2, and Vaibhav
Bajpai4
1CISPA Helmholtz Center for Information Security, Germany
2Indian Institute of Engineering Science and Technology, Shibpur, India;
1RedHat, Sweden
4Hasso Plattner Institute, Germany  Jayasree Sengupta, Debasmita Dey, Simone
Ferlin, Nirnay Ghosh and Vaibhav Bajpai J. Sengupta is with CISPA Helmholtz
Center for Information Security, Germany; D. Dey and N. Ghosh are with Indian
Institute of Engineering Science and Technology, Shibpur, India; S. Ferlin is
with RedHat and Karlstad University, Sweden; V.Bajpai is with Hasso Plattner
Institute, Germany.

# Accelerating Tactile Internet with QUIC:
A Security and Privacy Perspective

Jayasree Sengupta1, Debasmita Dey2, Simone Ferlin3, Nirnay Ghosh2, and Vaibhav
Bajpai4
1CISPA Helmholtz Center for Information Security, Germany
2Indian Institute of Engineering Science and Technology, Shibpur, India;
1RedHat, Sweden
4Hasso Plattner Institute, Germany  Jayasree Sengupta, Debasmita Dey, Simone
Ferlin, Nirnay Ghosh and Vaibhav Bajpai J. Sengupta is with CISPA Helmholtz
Center for Information Security, Germany; D. Dey and N. Ghosh are with Indian
Institute of Engineering Science and Technology, Shibpur, India; S. Ferlin is
with RedHat and Karlstad University, Sweden; V.Bajpai is with Hasso Plattner
Institute, Germany.

###### Abstract

The Tactile Internet paradigm is set to revolutionize human society by
enabling skill-set delivery and haptic communication over ultra-reliable, low-
latency networks. The emerging sixth-generation (6G) mobile communication
systems are envisioned to underpin this Tactile Internet ecosystem at the
network edge by providing ubiquitous global connectivity. However, apart from
a multitude of opportunities of the Tactile Internet, security and privacy
challenges emerge at the forefront. We believe that the recently standardized
QUIC protocol, characterized by end-to-end encryption and reduced round-trip
delay would serve as the backbone of Tactile Internet. In this article, we
envision a futuristic scenario where a QUIC-enabled network uses the
underlying 6G communication infrastructure to achieve the requirements for
Tactile Internet. Interestingly this requires a deeper investigation of a wide
range of security and privacy challenges in QUIC, that need to be mitigated
for its adoption in Tactile Internet. Henceforth, this article reviews the
existing security and privacy attacks in QUIC and their implication on users.
Followed by that, we discuss state-of-the-art attack mitigation strategies and
investigate some of their drawbacks with possible directions for future work.

##  I Introduction

The Tactile Internet [1], driven by recent technological advancements, focuses
on transmitting touch in real-time via suitable haptic equipment and robotics
over a robust communication network. Tactile Internet promises to deliver
real-time control through ultra-responsive and ultra-reliable network
connectivity over a distance to create a more sustainable ecosystem. This is
expected to revolutionize the entire human society by shifting the focus from
mere content delivery to a new system of skill-set delivery over the Internet.
However, its implementation at the infrastructure level demands ensuring low
latency with highly reliable communication and privacy coupled with a very
high level of security. This is because Tactile Internet is designed to serve
highly critical applications where human senses (touch, video, and audio) and
muscular movements directly interact with machines, i.e., robots. Hence,
cybersecurity and quick reaction time ($<1$ ms) are two primary requirements
of the Tactile Internet [1].

Any significant delay can induce cyber-sickness in users due to unsynchronized
senses, occurring as a result of a time lag ($>1$ ms) between visual and
tactile movements. Achieving global connectivity with a high data rate and
minimizing jitter are additional challenges. Given such non-elastic
requirements, the emerging sixth-generation (6G) communication system,
offering ubiquitous connectivity, is poised to serve as the Tactile Internet’s
network backbone. While existing 5G systems meet the throughput requirements,
the integration of terahertz (THz) communication and millimeter-wave (mmWave)
in 6G ensures higher data rates and ultra-low-latency (0.1 ms delay), making
it ideal for the Tactile Internet [2].

Figure 1: A high-level architectural view of a QUIC-enabled futuristic Tactile
Internet infrastructure using 6G as the underlying mobile communication
network.

It is evident from the above discussion that security/privacy breaches and
packet losses in the Tactile Internet environment can lead to detrimental
effects, particularly if it is used in mission-critical applications. In this
regard, the traditional transport layer protocols such as TCP and UDP are
inadequate as they are not designed to ensure security. Moreover, the “TCP
head-of-line blocking” problem [3] poses a significant bottleneck for Tactile
Internet applications, as packet loss from one source leads to buffering of
successive packets from all sources (e.g., different sensors). Given the high
redundancy in sensory data, receiving every generated packet is not mandatory.
Hence, the Internet Engineering Task Force’s (IETF) recently standardized
transport protocol, QUIC (see: §II), is a suitable alternative.

QUIC is a connection-oriented, end-to-end encrypted protocol based on UDP and
built on top of TLS 1.3, thus supporting $0-RTT$. QUIC allows sending parallel
and independent data streams which are logically separate from one another.
This ensures fast and reliable in-order delivery of independent data streams,
thereby avoiding the head-of-line blocking problem faced by TCP [3]. Though
QUIC was primarily designed to handle Web traffic, its unique features (low-
latency delivery, security, $0-RTT$ performance benefits) are suitable for
interactive Tactile Internet applications that need to transmit kinesthetic
and tactile data collected from various sensors with potential low delay and
high security. Moreover, QUIC is already under the deployment stage across
multiple application scenarios such as DNS, VPNs and beyond other than just
the Web [3]. Inspired by these facts, we envision QUIC as a potential
candidate for 6G-based Tactile Internet applications as depicted in Fig. 1.
Consequently, a deeper investigation of the various security (see: §II) and
privacy (see: §III) challenges in QUIC is needed.

This article provides a comprehensive overview of the security and privacy
issues in QUIC. We believe this will help practitioners and system designers
take appropriate measures before deploying QUIC in a Tactile Internet
environment. We also discuss a few mitigation strategies and highlight some of
their drawbacks (see: §IV), providing guidance for the research community to
delve deeper and propose alternative or improved solutions to the challenges
outlined in this article.

Figure 2: QUIC connection establishment with attack vectors. The packets or
tuples under attack are represented in blue, whereas the attacks are
highlighted in red.

##  II Security Attacks on QUIC

We begin this section by providing a deep dive into QUIC handshakes, as
several security attacks exploit the packets exchanged during QUIC connection
establishment. We then discuss the relevant security attacks on QUIC (see:
Table I) applicable to a Tactile Internet environment.

QUIC employs end-to-end encryption, necessitating agreement on multiple
parameters between endpoints (client and server). This agreement occurs
through a TLS 1.3 handshake, as depicted in Fig. 2. A typical QUIC segment’s
header fields are also illustrated. When a client lacks prior knowledge of a
server, it initiates the process by sending an Initial Client Hello (CHLO)
message to the server. Upon receiving CHLO, the server responds with a reject
reject (REJ) packet to the client, containing tuples: (i) server config (scfg)
with Diffie-Hellman public value, (ii) source-address token (stk) comprising
timestamp from the server and client’s IP address, (iii) server authentication
certificate chain, and (iv) server config signature. After receiving the
server config, the client authenticates it using the certificate chain and the
server signature. The client then creates a Complete CHLO, including its
ephemeral Diffie-Hellman key, and sends it back to the server. With the
completion of the handshake, the client obtains the initial keys for
communication, enabling it to send an encrypted connection request to the
server. The server responds with a server hello (SHLO) packet, encrypted with
initial keys and the server’s ephemeral keys. Both client and server then
calculate their forward secure keys to be used for subsequent communication
rounds. The client can now engage in a $0$-RTT handshake, bypassing the
initial handshake mechanism, by sending a Complete CHLO.

###  II-A 0-RTT Attack

During the initiation of a connection setup between the client and server, a
$0$-RTT attack can occur when the attacker intercepts the unencrypted initial
CHLO packet and spoofs it (Fig. 2). This opens the door to two potential
attacks [4]:
QUIC RST Attack: Attacker sends a public reset packet to the client, deceiving
it into thinking that the server has rejected the connection. This leads the
client to proactively abandon the connection.
Version Forgery Attack: The attacker poses as a server and transmits a version
negotiation packet to the client, containing a version unsupported by the
client. This prompts the client to either downgrade or abandon the connection.

In this regard, Cao _et al._ [4] have detailed the core concepts and
implementation of the $0$-RTT attack, emphasizing its potential to result in
denial-of-service. They have introduced a mathematical attack description
model based on finite state machines to demonstrate the QUIC protocol’s
vulnerability, showcasing the attack process and confirming QUIC’s
susceptibility to the described attacks.

###  II-B Replay Attack

A replay attack occurs when an intruder intercepts a legitimate network
transmission and later resends it to deceive the system into treating the re-
transmitted data as authentic. Lychev _et al._ [5] conducted two types of
replay attacks (Fig. 2) on the QUIC implementation in Chromium:
Server Config Replay Attack: This occurs when an adversary replays the
server’s public value (scfg) to clients who have sent an initial connection
request to the server, without being detected by the server. Consequently,
these clients generate the initial key and send it to the server, which on
being unable to verify their identity, rejects the packets. While the secrecy
of the entities involved remains largely unaffected, this attack results in
unnecessary consumption of computational resources.
Source Address Token Replay Attack: In this attack, the intruder replays the
source-address token (stk) of a client to the server that issued it, allowing
the creation of multiple additional connections. This will make the server
establish both initial keys and final forward-secure keys. Although subsequent
handshake steps would fail, the attacker could potentially launch a denial-of-
service (DoS) attack on the server by generating numerous connections on
behalf of various clients, leading to a depletion of computational and memory
resources.

Fischlin _et al._ [6] addressed replay attacks in the $0$-RTT QUIC handshake.
While $0$-RTT uses registers for storing nonces to prevent repetition, in
QUIC, once the server rejects the $0$-RTT message, the attacker resends the
rejected message using a second key to ensure proper delivery. This results in
duplicate processing of the same data by the server.

TABLE I: Overview of various Security and Privacy Attacks and their Effects on
QUIC

Category | Attack Surface | Attack Name | Effects | Reference
---|---|---|---|---
Security Attack | Initial CHLO message | 0-RTT Attack |  | Downgrade; Client-side
---
Connection Rejection
[4]
REJ message | Replay Attack |  | Connection Failure;
---
Server DoS
[5, 6]
| Source-Address Token (stk)
---
Manipulation Attack
Connection Failure; Server Load | [5]
Packet Header |  | Connection ID (cid)
---
Manipulation Attack
[5]
Request Forgery Attack | Traffic Amplification | [7]
| Handshake and
---
Packet Header
| State-Overflow and
---
Resource Exhaustion Attacks
| Fake connections;
---
Resource Wastage
[8]
QUIC Server | QUIC Flood DDoS Attack |  | Resource Exhaustion;
---
System breakdown
[8, 9]
Privacy Attack |  Traffic traces and Communication Channel | User Tracking via QUIC |  | User Profiling;
---
Targetted Advertising
[10]
Website Fingerprinting | User Privacy breach | [11, 12]

###  II-C Manipulation Attack

A manipulation attack aims to compromise the key agreement process, leading
the client and server to agree on distinct keys. It is achieved by altering
unprotected packet fields, such as connection id (cid) or the source-address
token (stk) (Fig. 2) that are utilized as input to the key derivation process.
Lychev _et al._ [5] describe the following two types of manipulation attacks
in QUIC:

Connection ID Manipulation Attack: Occurs when an attacker generates a fresh
cid, causing the server and client to perceive different cid values. Though
the handshake begins normally, the server and the client have distinct
encryption keys due to different cid values. Consequently, decryption fails,
and failed packets are buffered until the handshake is completed. In due time,
this situation leads to the disconnection of the link. However, the error
message gets encoded using the original encryption key, thereby preventing its
decryption and preserving the state until it times out.
Source-Address Token Manipulation Attack: In this attack, the attacker
generates separate stk values for both the client and server, resembling a CID
manipulation attack. The handshake typically begins with the transmission of
CHLO messages. However, since encryption keys rely on stk as an input, the
client and server generate distinct encryption keys, leading to decryption
failure. As a result, the client buffers failed CHLO messages and retransmits
them for 10 seconds. After a timeout, the client sends an encrypted error
message to the server. Yet, as the server possesses initial encryption keys,
the connection remains intact, and this state may persist for 10 minutes.

###  II-D Request Forgery Attack

This attack involves instigating a client to send deceptive requests to
another client. In [7], Gbur _et al._ assume a scenario where the attacker has
complete control over packets delivered to the victim, enabling three types of
forgery attacks:
Server Initial Request Forgery (SIRF): The attacker initiates a QUIC handshake
with the server, where the packet’s source IP address and port fields are
spoofed. This tricks the victim to assume that the connection is from a
genuine server.
Version Negotiation Request Forgery (VNRF): In this attack, the attacker sends
an unknown version in the client’s packet, prompting the QUIC server to
respond with a version negotiation packet. This action triggers the version
negotiation functionality.
Connection Migration Request Forgery (CMRF): While QUIC’s connection migration
feature offers advantages, a drawback exists: the server is unable to
differentiate between a genuinely migrated client address and a spoofed one.
In this scenario, an attacker conducts a legitimate handshake with the server
to establish a new connection, then spoofs an arbitrary packet to obtain the
source address and sends it to the server. Consequently, the server, upon
detecting a new source address, unwittingly establishes the connection and
transmits UDP packets to the manipulated address.

###  II-E State-Overflow and Resource Exhaustion Attacks

In these attacks, adversaries masquerade as legitimate clients and initiate
full handshakes with QUIC servers. The server responds with a unique Source
Connection ID (SCID) and the corresponding TLS certificate, thus allocating
resources to maintain connection states. To overwhelm the server, the
adversarial client randomly spoofs source ports and IP addresses, flooding the
server with multiple handshake requests. This results in the creation of
numerous unique SCIDs and concurrent connection states, leading to server
overload, state overflow, and resource exhaustion — similar to TCP SYN floods,
where legitimate client requests may also be rejected. The QUIC design’s
primary vulnerability lies in the lack of client verification during the
initial round-trip of the QUIC full handshake.

A study [8] evaluates these attacks on real-world QUIC traffic based on active
measurements using the UCSD network telescope. Results show that, similar to
TCP SYN floods, the QUIC handshake is susceptible to resource exhaustion
attacks. Notably, 98% of these attacks target QUIC servers of well-known
companies, with Google being the primary target (58%) and Facebook accounting
for 25% of the attacks. While QUIC originally supports RETRY messages to
mitigate resource exhaustion attacks, their implementation adds an extra
round-trip time (RTT), conflicting with QUIC’s claimed performance gains. This
happens as a RETRY message typically precedes a QUIC handshake and forces the
client to prove its authenticity by responding with a unique token. The study
[8] emphasizes that during experimentation, no RETRY messages were observed,
indicating a lack of practical defense mechanisms in QUIC deployments.

###  II-F QUIC Flood DDoS Attack

As previously explained, a QUIC flood DDoS attack occurs when an attacker
overwhelms a QUIC server with a large volume of data, causing the victim
server to slow down significantly or crash. Defending against DDoS attacks in
QUIC is challenging because QUIC is based on UDP, which provides minimal or no
information for blocking illegitimate traffic. Additionally, as QUIC packets
are encrypted, the victim server cannot easily verify the legitimacy of the
data source. While some Content Distribution Networks (CDNs) like Cloudflare
have been successful in mitigating QUIC floods [9], a study in [8] indicates
that these attacks persist in the real world. In [8], the authors reported
that the Internet is vulnerable to four QUIC flood attacks every hour. Among
these, 51% of the attacks occur concurrently with TCP/ICMP floods, while
another 40% target the victim sequentially. However, QUIC floods have a
shorter duration, lasting only $255\ seconds$, compared to their TCP/ICMP
counterparts. However, such shorter duration may lead to faster resource
exhaustion or system breakdown, presenting an aspect that is yet to be
explored.

##  III Privacy Attacks on QUIC

This section reviews two specific privacy threats that are posed to QUIC-based
user applications.

###  III-A User Tracking via QUIC

Online tracking threatens user privacy by exploiting users’ browsing habits,
which may reveal sensitive information. It can then be used for profiling, web
analytics, and targeted ads. To counter this, browsers need robust privacy
protection. While QUIC, being encrypted is widely adopted across Web browsers
as it aims to ensure user privacy, it remains susceptible to tracking through
the following two mechanisms:
Linking several website visits by the same user: The source-address-token is a
unique data block included in the reject (REJ) message sent by the server to a
client during initial connection setup. The client caches it and presents to
the server for every new $0$-RTT connection setup with the same server. This
enables the server to identify subsequent connection requests from the same
user which enables linking different website visits to the same usernames.
User tracking across multiple sessions: QUIC clients cache server-config (part
of REJ message), containing a unique 16-byte server config identifier (SCID)
assigned by the server to each user. This SCID allows Web servers and
potential attackers to link the initial connection request to subsequent
requests using the same SCID within CHLO messages.

The study [10] reveals QUIC’s privacy violation, enabling trackers to map
users based on tokens provided during a $0$-RTT connection setup attempt.
Analysis of popular browsers (Chrome) revealed insufficient protective
measures against Web user tracking. Such monitoring was particularly effective
in resource-constrained scenarios due to lower bandwidth requirements and
delays compared to HTTP cookies or traditional browser fingerprinting.

In a use-case scenario (Fig. 3), despite using encrypted QUIC connection, an
attacker observing traffic traces during a real-time haptic signal transfer
session (such as remote music training), can still gather information about
students and the teacher. This information could be strategically exploited
for user profiling and collecting sensitive details like gender and location.

Figure 3: A use-case scenario of Online Education where a network attacker can
eavesdrop on the QUIC connection to launch privacy attacks.

###  III-B Website Fingerprinting

An encrypted transmission protocol is susceptible to website fingerprinting
(WFP) attacks if adversaries can deduce a user’s visited websites by
monitoring the transmission channel. As QUIC encrypts data, adversaries target
unencrypted handshake packets. Further, as QUIC is based on a request-response
model, it allows attackers to distinguish traffic from different Web resources
due to the relatively fixed browser rendering sequence [11]. This weakens
privacy benefits achieved through encryption under WFP attacks.

Zhan _et al._ [11] explored WFP attack vulnerabilities in both QUIC variants:
(i) GQUIC (Google’s QUIC) and (ii) IQUIC (IETF’s QUIC). GQUIC is more
vulnerable in restricted traffic scenarios, while both protocols exhibit
similar vulnerabilities under typical traffic. Multiple WFP attacks with only
40 packets achieved 95.4% and 95.5% accuracy for GQUIC and IQUIC,
respectively.

To counter WFP attacks, IQUIC introduces a PADDING frame, but studies suggest
network-level padding inefficiency against adversaries capable of observing
traffic traces. Current research [12] focuses on integrating defenses directly
into client applications (e.g., browsers) by effectively using the PADDING
frame to thwart adversary’s attempt of traffic analysis in Virtual Private
Networks (VPNs).

A summary of various security and privacy attacks on QUIC is presented in
Table I.

##  IV Potential Mitigation Strategies

In QUIC, connection establishment between the endpoints is achieved through a
TLS 1.3 handshake where the client sends an initial ClientHello to the server
over a UDP socket. However, this introduces several security and privacy
challenges as sensitive parameters are negotiated between the endpoints during
such TLS connections. Though TLS 1.3 encrypts most of the information
exchanged, an on-path attacker can still learn certain sensitive private
information such as Server Name Identification (SNI) extension (see: § II)
available in the ClientHello message. It is also evident from § II that most
of the attacks occur due to the unencrypted packet exchange during connection
establishment as eavesdroppers get access to a plethora of metadata during
this phase.

This motivated the IETF to explore and encrypt the handshake messages apart
from just the application data. In a work-in-progress
document111https://datatracker.ietf.org/doc/draft-ietf-tls-esni/16/, IETF
proposes the inclusion of the Encrypted Client Hello (ECH) extension into the
TLS to mitigate the security risks and improve the privacy of the Internet as
a whole. The goal of ECH is to protect all sensitive handshake parameters such
as preshared key, server name, etc. by encrypting the entire ClientHello,
thereby closing the gap left by TLS 1.3 and Encrypted Server Name
Identification (ESNI), the predecessor of ECH.

An extensive study was performed by Bhargavan _et al._ [13] to prove that TLS
1.3 with ECH indeed ensures basic security goals (like confidentiality,
authenticity etc.), handshake privacy, and downgrade resistance. They provide
a formal analysis of the privacy properties of TLS 1.3 in the presence of ECH
using a symbolic protocol analyzer, named “ProVerif” to justify the security
of ECH. However, the authors didn’t study privacy-preservation of ECH in the
presence of traffic analysis. Therefore, any attack that relies on observing
traffic patterns including the length of application messages and their
timings would be something interesting to study in the future.

In another recent study [14], Trevisan _et al._ explore whether an attacker
can recover the domain names of websites visited by a user in the presence of
ECH by passively observing the traffic traces. The study reveals that simple
machine learning models are capable of recovering a user’s visited websites
with high accuracy. Their experiments reveal that ECH is only able to protect
$\approx 7.5\%$ flows. In the end, they also evaluate the efficacy of standard
padding-based techniques to thwart attacks on ECH. However, it is observed
that the attackers are still able to gain a sufficiently large amount of
information with just a minimal decrease in the effectiveness of the attacks
launched. Hence, as a future work, it is important to improve ECH to increase
its resistance to traffic analysis. Designing more robust techniques is
necessary for offering better protection to end-users in terms of privacy.

While ECH can be a potential mitigation strategy against some of the security
attacks (discussed in see: § II), Oblivious
HTTP222https://datatracker.ietf.org/doc/draft-ietf-ohai-ohttp/ and MASQUE
(Multiplexed Application Substrate over QUIC
Encryption)333https://datatracker.ietf.org/doc/draft-ietf-masque-connect-ip/
are other potential work in progress technologies for the privacy attacks
highlighted in § III. Private relay [15] offers privacy-preserving
communication by establishing multiple proxy servers between the client and
the server. This ensures that no attacker has visibility over both the user’s
IP address and their access history. In this way, it separates the client’s
IPs from servers and uses multi-hop MASQUE for proxying IP packets. On the
contrary, Oblivious HTTP offers to send encrypted HTTP messages from a client
to a gateway via a trusted relay service. This ensures that nobody sees the
client’s IP address as well as the plaintext HTTP message remains hidden from
the relay resources.

Overall, as a potential future work, researchers should consider designing
privacy-preserving extensions or modifications to the QUIC protocol in order
to mitigate potential privacy risks, such as by employing encryption for
sensitive metadata and exploring differential privacy techniques. It is also
crucial to integrate multi-factor authentication to fortify the
trustworthiness of QUIC connections.

##  V Conclusion

The Tactile Internet is expected to bring a paradigm shift in all spheres of
the connected society. However, this is only possible with tight integration
of the network infrastructure, transport, and application (Tactile Internet)
layers. Moreover, addressing the security and privacy challenges is crucial
for the success of the Tactile Internet. In this article, we envisioned a
QUIC-accelerated Tactile Internet that leverages 6G as the underlying network
infrastructure for achieving low-latency, resilient global connectivity. We
explored this ecosystem from a security and privacy perspective and presented
several attacks that can make Tactile Internet vulnerable by targeting QUIC.
To this end, we presented a few potential mitigation strategies that can be
the future directions to plug the present security and privacy gap in this
ecosystem, to help the Tactile Internet transition the connected society into
a more sustainable future.

## References

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Jayasree Sengupta is a Postdoctoral Researcher at CISPA, Germany. Her research
interests include network security and data privacy.
---
Debasmita Dey is pursuing Ph.D. at IIEST Shibpur. She works on security
aspects of IoT systems.
---
Simone Ferlina is a Performance Engineer at RedHat and an Adjunct Senior
Lecturer at Karlstad University, Sweden. She works in areas of 5G, network and
system management.
---
Nirnay Ghosh is an Assistant Professor at IIEST, Shibpur. He works in areas of
IoT security, blockchains and its applications.
---
Vaibhav Bajpai is a Professor at HPI, Germany. He is interested in performance
and management of next-generation networked systems.
---


# Quantum Neural Networks: Concepts, Applications, and Challenges

∘Yunseok Kwak, ∘Won Joon Yun, ∘Soyi Jung, and ∘Joongheon Kim
∘Department of Electrical and Computer Engineering, Korea University, Seoul
02841, Republic of Korea
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Quantum deep learning is a research field for the use of quantum computing
techniques for training deep neural networks. The research topics and
directions of deep learning and quantum computing have been separated for long
time, however by discovering that quantum circuits can act like artificial
neural networks, quantum deep learning research is widely adopted. This paper
explains the backgrounds and basic principles of quantum deep learning and
also introduces major achievements. After that, this paper discusses the
challenges of quantum deep learning research in multiple perspectives. Lastly,
this paper presents various future research directions and application fields
of quantum deep learning.

##  I Introduction

As quantum computing and deep learning have recently begun to draw attentions,
notable research achievements have been pouring over past decades. In the
field of deep learning, the problems which were considered as their inherent
limitations like gradient vanishing, local minimum, learning inefficiencies in
large-scale parameter training are gradually being conquered [1]. On the one
hand, innovative new deep learning algorithms such as quantum neural network
(QNN), convolutional neural network (CNN), and recurrent neural network (RNN)
are completely changing the way various kinds of data are processed.
Meanwhile, the field of quantum computing has also undergone rapid
developments in recent years. Quantum computing, which has been recognized
only for its potential for a long time, has opened up a new era of enormous
potentials with the recent advances of variational quantum circuits (VQC). The
surprising potentials of the variational quantum algorithms were made clear by
solving various combinatorial optimization problems and the intrinsic energy
problems of molecules, which were difficult to solve using conventional
methods, and further extensions are considered to design machine learning
algorithms using quantum computing. Among them, quantum deep learning fields
are growing rapidly, inheriting the achievements of existing deep learning
research. Accordingly, numerous notable achievements related to quantum deep
learning have been published, and active follow-up studies are being conducted
at this time. In this paper, we first briefly introduce the background
knowledge, basic principles of quantum deep learning, and look at the current
research directions. We then discuss the various directions and challenges of
future research in quantum deep learning.

###  I-A Quantum Computing

Quantum computers use qubits as the basic units of computation, which
represent a superposition state between $|0\rangle$ and $|1\rangle$ [2, 3, 4,
5]. A single qubit state can be represented as a normalized two-dimensional
complex vector, i.e.,

$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle,\|\alpha\|^{2}+\|\beta\|^{2}=1$
(1)

and $\|\alpha\|^{2}$ and $\|\beta\|^{2}$ are the probabilities of observing
$|0\rangle$ and $|1\rangle$ from the qubit, respectively. This can be also
geometrically represented using polar coordinates $\theta$ and $\phi$,

$|\psi\rangle=\cos(\theta/2)|0\rangle+e^{i\phi}\sin(\theta/2)|1\rangle,$ (2)

where $0\leq\theta\leq\pi$ and $0\leq\phi\leq\pi$. This representation maps a
single qubit state into the surface of 3-dimensional unit sphere, which is
called Bloch sphere. A multi qubit system can be represented as the tensor
product of $n$ single qubits, which exists as a superposition of $2^{n}$ basis
states from $|00...00\rangle$ to $|11...11\rangle$. Quantum entanglement
appears as a correlation between different qubits in this system. For example,
in a 2-qubit system
$\frac{1}{\sqrt{2}}|00\rangle+\frac{1}{\sqrt{2}}|11\rangle$, the observation
of the first qubit directly determines that of the second qubit. Those systems
are controlled by quantum gates in a quantum circuit to perform a quantum
computation on its purpose [6, 7].

Quantum gates are unitary operators mapping a qubit system into another one,
and as classical computing, it is known that every quantum gate can be
factorized into the combination of several basic operators like rotation
operator gates and CX gate [8]. Rotation operator gates
$R_{x}(\theta),R_{y}(\theta),R_{z}(\theta)$ rotates a qubit state in Bloch
sphere around corresponding axis by $\theta$ and CX gate entangles two qubits
by flipping a qubit state if the other is $|1\rangle$. Those quantum gates
utilizes quantum superposition and entanglement to take an advantage over
classical computing, and it is well known that quantum algorithms can obtain
an exponential computational gain over existing algorithms in certain tasks
such as prime factorization [9].

##  II Quantum Deep Learning

###  II-A Variational Quantum Circuits (VQC)

A variational quantum circuit (VQC) is a quantum circuit using rotation
operator gates with free parameters to perform various numerical tasks, such
as approximation, optimization, classification. An algorithm using a
variational quantum circuit is called variational quantum algorithm (VQA),
which is a classical-quantum hybrid algorithm because its parameter
optimization is often performed by a classical computer. Since its universal
function approximating property [10], many algorithms using VQC [11] are
designed to solve various numerical problems [12, 13, 2, 8, 14]. This flow led
to many applications of VQA in machine learning and is also for replacing the
artificial neural network of the existing model with VQC [15, 16, 17, 18]. VQC
is similar to artificial neural networks in that it approximates functions
through parameter learning, but has differences due to the several
characteristics of quantum computing. Since all quantum gate operations are
reversible linear operations, quantum circuits use entanglement layers instead
of activation functions to have multilayer structures. These VQCs are called
quantum neural networks, and this paper will look at them through
classification according to their structure and characteristics.

###  II-B Quantum Neural Networks

Figure 1: Illustration of QNN with the input $|\psi\rangle$, the parameter
$\theta$ and linear entanglement structure.

In this section, we try to demonstrate how a basic quantum neural network(QNN)
works with a simple example described in the Fig. 1. The way a QNN processes
data is as follows. First, the input data is encoded into the corresponding
qubit state of an appropriate number of qubits. Then, the qubit state is
transformed through the parameterized rotation gates and entangling gates for
a given number of layers. The transformed qubit state is then measured by
obtaining expected value of a hamiltonian operator, such as Pauli gates. These
measurements are decoded back into the form of appropriate output data. The
parameters are then updated by an optimizer like Adam optimizer. A neural
network constructed in the form of VQC can perform various roles in various
forms, which will be explored as quantum neural networks.

####  II-B1 Quantum Convolutional Neural Networks

Figure 2: Illustration of QCNN with the input $|\psi\rangle$, the parameter
$\theta$ with single convolution and pooling layer.

Quantum convolutional neural network (QCNN) was proposed in [16], implementing
the convolution layer and pooling layer on the quantum circuits. According to
the previous research results in [5, 19], the QCNN circuit computation
proceeds as follows. The first step is same as any other QNN models, encoding
input data into a qubit state with rotation operator gates. Then the
convolution layer with quasi-local unitary gates filters the input data into a
feature map. The pooling layer with controlled rotation operators then
downsizes the feature map. By repeating this process sufficiently, the fully
connected layer acts on the qubit state as classical CNN models. Finally, the
measurement of the qubit state is decoded into an output data with desired
sizes. The circuit parameters are updated with gradient descent based
optimizer after each measurements.

Unfortunaltely, in the current quantum computing environment [20], QCNN is
difficult to perform better than the existing classical CNN. However, it is
expected that the QCNN will be able to obtain sufficient computational gains
over the classical ones in future quantum computing environment where larger-
size quantum calculations are possible [16, 5].

##  III Future Work Directions and Challenges

###  III-A Applications of Quantum Deep Learning to Reinforcement Learning

There are many research results applying deep learning to reinforcement
learning to derive optimal actions from a complex state space [21, 22, 23,
24]. However, reinforcement learning research using quantum deep learning [18,
25, 26] is still in its infancy. The current approach step is to replace the
policy training network with a quantum neural network from the existing deep
neural network, but there remains the possibility of many algorithms applying
various ideas of classical deep reinforcement learning researches. In
particular, if it is proved that quantum computational gains can be obtained
through QNN in a situation of high computational complexity due to the complex
Markov decision process environment, quantum reinforcement learning will open
a new horizon for reinforcement learning research.

###  III-B Applications of Quantum Deep Learning to Communication Networks

The QNN and quantum reinforcement learning algorithms can be used in various
research fields, and this paper considers the applications in terms of
communications and networks. In terms of the acceleration of computation in
fully distributed platforms, e.g., blockchain [27, 28], QNN can be used.

In addition, various advanced communication technologies such as Internet of
Things (IoT) [29, 30], millimeter-wave networks [31, 32], caching networks
[33, 34, 35, 36], and video streaming/scheduling [37, 38, 39, 40] are good
applications of QNN and quantum reinforcement learning algorithms.

###  III-C Challenges

####  III-C1 Gradient Vanishing

Vanishing gradient is a crucial problem in quantum deep learning as of
classical deep learning. The problem of gradient disappearance while
backpropagating many hidden layers has been considered a chronic problem in
deep neural network computation. Since quantum neural networks also use
gradient descent method training their parameters as classical ones, they have
to solve the same problem. Classical deep learning models solve this problem
by utilizing an appropriate activation function, but quantum deep learning
does not use an activation function, thus eventually, a different solution is
needed. A former research [41] called this quantum gradient vanishing
pheonomena as barren plateaus, while proving that when the number of qubits
increases, the probability of occurring barren plateaus increases
exponentially. This can be avoided by setting good initial parameters in
small-scale QNN, but it is unavoidable to deal with this problem when
designing large-scale QNN. This is an open problem for which a solution is not
yet clear.

####  III-C2 Near-Term Device Compatibility

Noisy intermediate scale quantum (NISQ) [20], which means fewer qubits and a
lot of computational error of near-term quantum devices, has already become a
familiar term to quantum researchers. Many algorithms designed to implement
quantum computational gains do not work at all in this NISQ environment, and
are expected to be implemented at least several decades later. For example, a
practical implementation of the Shor’s algorithm requires at least thousand of
qubits even without an error correction processes, current quantum devices
have only a few tens of qubits with non-negligible computational error rate of
several percent. However, due to the relatively small circuit depth and qubit
requirements, VQA and QNN based on them are tolerant to these environmental
constraints. Nevertheless, in order to increase the data processing capability
of quantum neural network, it is necessary to consider near-term device
compatibility. For example, using many multi-qubit controlling gates for
quantum entanglement is theoretically thought to increase the performance of
QNN, but it entails a large error rate and a complicated error correction
process. Therefore, it is essential to design an algorithm regarding these
tradeoffs in quantum deep learning research.

####  III-C3 The Quantum Advantage

The term quantum supremacy may lead to the illusion that quantum algorithms
are always better than classical algorithms performing the same function.
However, given the inherent limitations of quantum computing, quantum
computing benefits can only be realized through well-thought-out algorithms
under certain circumstances. In fact, especially among variational quantum-
based algorithms, only a few of them have proven their quantum advantage in a
limited situation.

Due to the universal approximation property of QNN, it is known that quantum
deep learning can perform most of the computations performed in classical deep
learning [10]. Nevertheless, if one approaches simply based on this fact
without the consideration of quantum gain, the result may be much inefficient
compared to the existing classical algorithm. Therefore, in designing a new
QNN-based deep learning algorithm, it is necessary to justify it by
articulating its advantages over the corresponding classical models.

##  IV Conclusion

This paper introduces the basic concepts of quantum neural networks and their
applications scenarios in various fields. Furthermore, this paper presents
research challenges and potential solutions of quantum neural network
computation.

## Acknowledgment

This work was supported by the National Research Foundation of Korea
(2019M3E4A1080391). Joongheon Kim is a corresponding author of this paper.

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