Measurement device–independent quantum secure direct communication with user authentication

Abstract

Quantum secure direct communication (QSDC) and deterministic secure quantum communication (DSQC) are two important branches of quantum cryptography, where one can transmit a secret message securely without encrypting it by a prior key. In the practical scenario, an adversary can apply detector-side-channel attacks to get some non-negligible amount of information about the secret message. Measurement device–independent (MDI) quantum protocols can remove this kind of detector-side-channel attacks, by introducing an untrusted third party, who performs all the measurements during the protocol with imperfect measurement devices. In this paper, we put forward the first MDI-QSDC protocol with user identity authentication, where both the sender and the receiver first check the authenticity of the other party and then exchange the secret message. Then, we extend this to an MDI quantum dialogue protocol, where both the parties can send their respective secret messages after verifying the identity of the other party. Along with this, we also report the first MDI-DSQC protocol with user identity authentication. Theoretical analyses prove the security of our proposed protocols against common attacks.

Appendix: Proof of Lemma 1

Lemma 1

For a probability distribution $\{\delta _i, 1\le i \le 4\}$, $-\sum _{i=1}^4 \delta _i log \delta _i \le h(\delta _2+ \delta _4)+ h(\delta _3+ \delta _4)$, where $h(\cdot )$ represents the binary entropy function.

Proof

Let X be a random variable such that

$$\begin{aligned} {X} = {\left\{ \begin{array}{ll} 00 &{}\text {with probability}\, {\delta _1},\\ 01 &{}\text {with probability}\, {\delta _2},\\ 10 &{}\text {with probability}\,{\delta _3},\\ 11 &{}\text {with probability }\,{\delta _4}. \end{array}\right. } \end{aligned}$$

Let Y and Z be the following events,

$$\begin{aligned} {Y}= & {} {\left\{ \begin{array}{ll} 1, &{}\text {if the least significant bit of}\,{X=1 },\\ 0, &{}\text {otherwise}. \end{array}\right. }\\ {Z}= & {} {\left\{ \begin{array}{ll} 1, &{}\text {if the most significant bit of}\,{X=1 },\\ 0, &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

In other words,

$$\begin{aligned} {Y} = {\left\{ \begin{array}{ll} 1 &{}\text {with probability}\,{\delta _2+ \delta _4 },\\ 0 &{}\text {with probability}\,{\delta _1+ \delta _3}. \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {Z} = {\left\{ \begin{array}{ll} 1 &{}\text {with probability}\,{\delta _3 + \delta _4 },\\ 0 &{}\text {with probability}\,{\delta _1+ \delta _2}. \end{array}\right. } \end{aligned}$$

(10)

Then, the entropy of the events Y and Z is as follows

$$\begin{aligned} H(Y)= & {} -\sum _{y \in \{0,1\}} \Pr (Y=y)log[\Pr (Y=y)]=h(\delta _2+ \delta _4).\\ H(Z)= & {} -\sum _{z \in \{0,1\}} \Pr (Z=z)log[\Pr (Z=z)]=h(\delta _3+ \delta _4). \end{aligned}$$

The joint entropy H(Y, Z) of the events Y and Z is

$$\begin{aligned} \begin{aligned} H(Y,Z)&= -\sum _{y \in \{0,1\}} \sum _{z \in \{0,1\}} \Pr (Y=y,Z=z)log[\Pr (Y=y,Z=z)] \\&=-\sum _{x \in \{00,01,10,11\}} \Pr (X=x)log[\Pr (X=x)]\\&=-\sum _{i=1}^4 \delta _i log \delta _i. \end{aligned} \end{aligned}$$

Now, using sub-additivity property of entropy, i.e., the fact that the joint entropy of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set. Therefore,

$$\begin{aligned} \begin{aligned}&H(Y,Z) \le H(Y)+H(Z) \\&\text {or, } -\sum _{i=1}^4 \delta _i log \delta _i \le h(\delta _2+ \delta _4)+ h(\delta _3+ \delta _4). \end{aligned} \end{aligned}$$