# A Resource-Friendly Certificateless Proxy Signcryption Scheme for Drones in Networks beyond 5G

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## Abstract

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## 1. Introduction

- We propose a resource-friendly certificateless proxy signcryption scheme for drones in B5G networks. The proposed scheme is based on the elliptic curve cryptography (ECC) algorithm and enjoys some of its favorable features, such as no key escrow and no secure channel.
- The proposed scheme has a clear distribution of roles: the control center acts as the original signer, the network provider serves as the key generation center (KGC), the ground control station acts as a proxy, and the drones perform the task of un-signcryption.
- The proposed protocol guarantees anonymity for both senders and receivers by employing a mechanism wherein participants (${PP}_{i}$), where $i=(CC,GCS,drone)$, send their identities in an encrypted form while requesting a partial private key.
- The proposed scheme is capable of withstanding a wide variety of commonly known attacks under ROM. Additionally, it was found that this scheme is efficient in terms of both computation and communication costs when compared to other existing schemes.

## 2. Literature Review

## 3. Preliminaries

#### 3.1. Random Oracle Model

#### 3.2. Adversarial or Threat Model

#### 3.3. Syntax of Certificateless Proxy Signcryption

#### 3.4. Network Model

## 4. Construction of the Proposed Scheme

**Setup:**Here, the network provider (${NP}_{KC}$) assumes the role of KGC; when it receives the security parameter ${k}_{KC}$, ${NP}_{KC}$ executes the steps outlined below.

- Selects the group ${G}_{KC}$ of order ${q}_{KC}$ and ${\gamma}_{KC}$, which will be the generator of ${G}_{KC}$;
- Selects four hash functions ${H}_{KC1}:{\left\{\mathrm{0,1}\right\}}^{*}\times {G}_{KC}\to {{Z}_{{q}_{KC}}}^{*},{H}_{KC2}:{\left\{\mathrm{0,1}\right\}}^{*}\times {G}_{KC}\to {{Z}_{{q}_{KC}}}^{*},{H}_{KC3}:{\left\{\mathrm{0,1}\right\}}^{*}\times {G}_{KC}\to {{Z}_{{q}_{KC}}}^{*},{H}_{KC4}:{\left\{\mathrm{0,1}\right\}}^{*}\times {G}_{KC}\to {{Z}_{{q}_{KC}}}^{*}$;
- Sets ${{\rm Y}}_{m}$ is the plaintext length and $\left|{{Z}_{{q}_{KC}}}^{*}\right|$ will be the length of selected parameter;
- Selects the system private key as ${\Phi}_{KC}\in {{Z}_{{q}_{KC}}}^{*}$ and computes the public key ${\delta}_{KC}={\Phi}_{KC}.{\gamma}_{KC}$;
- ${NP}_{KC}$ can made ${PAR}_{KC}=\{{H}_{KC1},{H}_{KC2},{H}_{KC3},{H}_{KC4},{\delta}_{KC},{\gamma}_{KC},{G}_{KC},{{Z}_{{q}_{KC}}}^{*}\}$ as the public parameter and distributes it throughout a network.

**Partial Key Generation (PCGU):**If a participant (${PP}_{i}$), where $i=(CC,GCS,drone)$, desires a partial private key (${P}_{i}$) from ${NP}_{KC}$, it first selects ${\beta}_{{PP}_{i}}\in {{Z}_{{q}_{KC}}}^{*}$, computes ${V}_{{PP}_{i}}={\beta}_{{PP}_{i}}.{\delta}_{KC}$, computes ${U}_{{PP}_{i}}={\beta}_{{PP}_{i}}.{\gamma}_{KC},$ calculates ${EID}_{{PP}_{i}}={ID}_{{PP}_{i}}\oplus {V}_{{PP}_{i}}$, and then, sends $({EID}_{{PP}_{i}},{U}_{{PP}_{i}})$ through insecure network to ${NP}_{KC}$. Alternatively, when ${NP}_{KC}$ receives $({EID}_{{PP}_{i}},{U}_{{PP}_{i}})$, it executes the following calculations: Computes ${V}_{{PP}_{i}}={\Phi}_{KC}.{U}_{{PP}_{i}}$, recovers ${PP}_{i}$ identity as ${ID}_{{PP}_{i}}={EID}_{{PP}_{i}}\oplus {V}_{{PP}_{i}}$, and then, ${NP}_{KC}$ selects ${\alpha}_{KC}\in {{Z}_{{q}_{KC}}}^{*}$ and computes ${X}_{KC}={\alpha}_{KC}.{\gamma}_{KC}$. In addition, ${NP}_{KC}$ computes ${P}_{i}={\alpha}_{KC}+{\Phi}_{KC}{H}_{KC1}({X}_{KC},{ID}_{{PP}_{i}})$, calculates ${EP}_{i}={(P}_{i},{X}_{KC})\oplus {V}_{{PP}_{i}}$, and sends ${EP}_{i}$ as an encrypted partial private and public key to ${PP}_{i}$ over an insecure network.

**Public and Private Key Generation (PBCGU):**When ${PP}_{i}$ receives ${EP}_{i}$, it computes ${(P}_{i},{X}_{KC})={EP}_{i}\oplus {V}_{{PP}_{i}}$, sets (${U}_{{PP}_{i}},{X}_{KC}$) to his public key and sets (${P}_{i},{\beta}_{{PP}_{i}})$ to his private key.

**Delegation Generation (DG):**This phase is run by the CC and when it receives ${PAR}_{KC}=\left\{{H}_{KC1},{H}_{KC2},{H}_{KC3},{H}_{KC4},{\delta}_{KC},{\gamma}_{KC},{G}_{KC},{{Z}_{{q}_{KC}}}^{*}\right\},$ (${ID}_{\mathrm{C}\mathrm{C}},{ID}_{\mathrm{C}\mathrm{G}},{P}_{CC},{\beta}_{\mathrm{C}\mathrm{C}},{U}_{\mathrm{C}\mathrm{G}},{X}_{CG}$), where ${ID}_{\mathrm{C}\mathrm{C}}$ and ${ID}_{\mathrm{G}\mathrm{C}\mathrm{C}}$ are the identities of CC and GCS, respectively, $({P}_{CC},{\beta}_{\mathrm{C}\mathrm{C}})$ is CC’s private key pair, and (${U}_{\mathrm{C}\mathrm{G}},{X}_{CG}$) is GCS’s public key pair.

- It selects ${A}_{\mathrm{C}\mathrm{C}}\in {{Z}_{{q}_{KC}}}^{\mathrm{*}}$, computes ${O}_{\mathrm{C}\mathrm{C}}={A}_{\mathrm{C}\mathrm{C}}.{\gamma}_{KC}$, and ${H}_{2}={H}_{KC2}({U}_{\mathrm{G}\mathrm{C}\mathrm{S}},{U}_{\mathrm{C}\mathrm{C}},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}},{ID}_{\mathrm{C}\mathrm{C}},{O}_{\mathrm{C}\mathrm{C}},{m}_{\mathrm{w}})$;
- Computes ${S}_{{m}_{\mathrm{w}}}=\frac{{\beta}_{\mathrm{C}\mathrm{C}}+{A}_{\mathrm{C}\mathrm{C}}}{{H}_{2}+{\beta}_{\mathrm{C}\mathrm{C}}+{P}_{CC}}modq$ and sends the triple ${(m}_{\mathrm{w}},{S}_{{m}_{\mathrm{w}}},{O}_{\mathrm{C}\mathrm{C}})$ as a delegation to the GCS through an open network.

**Delegation Verification (DV):**When ${(m}_{\mathrm{w}},{S}_{{m}_{\mathrm{w}}},{O}_{\mathrm{C}\mathrm{C}})$ is received by the GCS, the following verification procedures are carried out.

- Computes ${H}_{2}={H}_{KC2}\left({U}_{\mathrm{G}\mathrm{C}\mathrm{S}},{U}_{\mathrm{C}\mathrm{C}},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}},{ID}_{\mathrm{C}\mathrm{C}},{O}_{\mathrm{C}\mathrm{C}},{m}_{\mathrm{w}}\right)$ and ${H}_{1}={H}_{KC1}({ID}_{\mathrm{C}\mathrm{C}},{X}_{CC})$;
- If ${S}_{{m}_{\mathrm{w}}}\left({X}_{CC}+{U}_{\mathrm{C}\mathrm{C}}+{H}_{1}.{\delta}_{KC}+{H}_{2}.{\gamma}_{KC}\right)={U}_{\mathrm{C}\mathrm{C}}+{O}_{\mathrm{C}\mathrm{C}}$, then accept ${(m}_{\mathrm{w}},{S}_{{m}_{\mathrm{w}}},{O}_{\mathrm{C}\mathrm{C}});$ otherwise, an error message is returned.

**CL-Proxy Signcryption Generation (CL-PSG):**This phase is executed by the GCS, which generates a certificateless proxy signcryption using the procedures below.

- It selects ${F}_{GCS}\in {{Z}_{{q}_{KC}}}^{*}$, computes ${Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}={F}_{\mathrm{G}\mathrm{C}\mathrm{S}}.{\gamma}_{KC}$;
- Computes ${H}_{3}={H}_{KC3}({U}_{GCS},{U}_{CC},{ID}_{GCS},{ID}_{CC},{Q}_{GCS},m)$;
- Computes ${H}_{1}={H}_{KC1}({ID}_{drone},{X}_{drone})$ and $K={F}_{GCS}({U}_{drone}+{X}_{drone}+{\delta}_{KC}.{H}_{1})$;
- Computes ${C}_{GCS}={H}_{KC4}\left(K\right)\oplus m$ and ${S}_{GCS}=\frac{{\beta}_{GCS}+{F}_{GCS}}{{H}_{3}+{\beta}_{GCS}+{P}_{GCS}}modq$;
- Finally, it sends the triple (${C}_{GCS},{S}_{GCS},{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}})$ as a proxy signcryption to drone via an open network.

**CL-Proxy Un-Signcryption (CL-PU-S):**When $({C}_{GCS},{S}_{GCS},{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}})$ is received by the drone, it performs the following verifications steps.

- Computes $K={Q}_{GCS}\left({\beta}_{drone}+{P}_{drone}\right)$ and $m={H}_{KC4}\left(K\right)\oplus {C}_{GCS}$ ;
- Computes ${H}_{3}={H}_{KC3}({U}_{GCS},{U}_{CC},{ID}_{GCS},{ID}_{CC},{Q}_{GCS},m)$ and ${{H}_{1}}^{\$}={H}_{KC1}({ID}_{GCS},{X}_{GCS})$ ;
- If ${S}_{GCS}\left({X}_{GCS}+{U}_{GCS}+{{H}_{1}}^{\$}.{\delta}_{KC}+{H}_{3}.{\gamma}_{KC}\right)={U}_{GCS}+{Q}_{GCS}$, then accept $({C}_{GCS},{S}_{GCS},{Q}_{GCS}),$ otherwise, an error message is returned.

**Correctness**

## 5. Security Analysis

**Elliptic Curve Diffie–Hellman Problem (ECDHP):**Given (${\gamma}_{KC},a.{\gamma}_{KC},b.{\gamma}_{KC}$), finding the values of $a,b$ from $a.{\gamma}_{KC},b.{\gamma}_{KC}$ is hard and is reported to ECDHP.

**Elliptic Curve Discrete Logarithm Problem (ECDLP):**Given (${\gamma}_{KC},a.{\gamma}_{KC}$), finding the value of $a$ from $a.{\gamma}_{KC}$ is hard and is reported to ECDLP.

**Theorem**

**1.**

**Proof.**

**,**the task of ${NP}_{{A}_{1}}$ is to extract the value $a,b$ from $a.{\gamma}_{KC},b.{\gamma}_{KC}$ with the help of ${NP}_{CR}$. The following is the process in which ${NP}_{{A}_{1}}$ with the help of ${NP}_{CR}$ could solve the above problem.

**Setup:**Here, ${NP}_{CR}$ selects ${{\Phi}_{KC}}^{*}\in {{Z}_{{q}_{KC}}}^{*},$ computes ${\delta}_{KC}$, makes a param ${PAR}_{KC}$, and sends ${PAR}_{KC}$ to ${NP}_{{A}_{1}}.$ Then, ${NP}_{{A}_{1}}$ can ask for the following queries.

**Find Stage:**Here, in this section, ${NP}_{{A}_{1}}$ can ask for the following polynomial bounded queries.

**Query:**If ${NP}_{CR}$ receives $\left({X}_{j},{ID}_{\mathrm{j}}\right)$ as a query from ${NP}_{{A}_{1}}$, ${NP}_{CR}$ checks for $\left({X}_{j},{ID}_{\mathrm{j}},{H}_{1},l\right)$ in the list ${L}_{{H}_{KC1}}$, if it is available, it sends ${H}_{1}$ to ${NP}_{{A}_{1}};$ otherwise, ${NP}_{CR}$ choose $l\in \left\{\mathrm{0,1}\right\}$, here, its probability as $\mathrm{Pr}\left(l=1\right)=\frac{1}{{q}_{PS}+1}$. Then, it checks, if $\left(l=0\right)$, and then chooses ${H}_{1}\in {{Z}_{{q}_{KC}}}^{*},$ sends ${H}_{1}$ to ${NP}_{{A}_{1}}$ and adds $\left({X}_{j},{ID}_{\mathrm{j}},{H}_{1},l\right)$ into ${L}_{{H}_{KC1}}$. If $\left(l=1\right)$, ${NP}_{CR}$ sets ${k}_{KC}={H}_{1}$, and returns ${k}_{KC}$ to ${NP}_{{A}_{1}}$.

**Query:**If ${NP}_{CR}$ receives $\left({U}_{j},{ID}_{j},{O}_{j},{m}_{\mathrm{w}}\right)$ as a query from ${NP}_{{A}_{1}}$, checks for $\left({U}_{j},{ID}_{j},{O}_{j},{m}_{\mathrm{w}},{H}_{2}\right)$ in the list ${L}_{{H}_{KC2}}$, if it is available, it sends ${H}_{2}$ to ${NP}_{{A}_{1}};$ otherwise, ${NP}_{CR}$ chooses ${H}_{2}\in {{Z}_{{q}_{KC}}}^{*},$ sends ${H}_{2}$ to ${NP}_{{A}_{1}}$ and adds $\left({U}_{j},{ID}_{j},{O}_{j},{m}_{\mathrm{w}},{H}_{2}\right)$ into ${L}_{{H}_{KC1}}$.

**Query:**If ${NP}_{CR}$ receives $\left({U}_{j},{ID}_{j},{O}_{j},m\right)$ as a query from ${NP}_{{A}_{1}}$, ${NP}_{CR}$ checks for $\left({U}_{j},{ID}_{j},{O}_{j},m,{H}_{3}\right)$ in the list ${L}_{{H}_{KC3}}$, if it is available, it sends ${H}_{3}$ to ${NP}_{{A}_{1}};$ otherwise, ${NP}_{CR}$ chooses ${H}_{3}\in {{Z}_{{q}_{KC}}}^{*},$ sends ${H}_{3}$ to ${NP}_{{A}_{1}}$ and adds $\left({U}_{j},{ID}_{j},{O}_{j},m,{H}_{3}\right)$ into ${L}_{{H}_{KC3}}$.

**Query:**If ${NP}_{CR}$ receives $\left({ID}_{j},K\right)$ as a query from ${NP}_{{A}_{1}}$, ${NP}_{CR}$ checks for $\left({ID}_{j},K,{H}_{4}\right)$ in the list ${L}_{{H}_{KC4}}$, if it is available, it sends ${H}_{4}$ to ${NP}_{{A}_{1}};$ otherwise, ${NP}_{CR}$ chooses ${H}_{4}\in {{Z}_{{q}_{KC}}}^{*},$ sends ${H}_{4}$ to ${NP}_{{A}_{1}}$ and adds $({ID}_{j},K,{H}_{4})$ into ${L}_{{H}_{KC4}}$.

**PCGU Query**$:$ If ${NP}_{CR}$ receives $({X}_{j},{ID}_{j},{P}_{j})$ as a query from ${NP}_{{A}_{1}},$ ${NP}_{CR}$ checks for $({X}_{j},{ID}_{j},{P}_{j})$ in the list ${L}_{PCGU}$, if it is available, it sends $({X}_{j},{P}_{j})$ to ${NP}_{{A}_{1}};$ otherwise, ${NP}_{CR}$ chooses ${\alpha}_{j},{\Phi}_{j}\in {{Z}_{{q}_{KC}}}^{*},$ computes ${P}_{j}={\alpha}_{j}+{\Phi}_{j}{H}_{KC1}({X}_{j},{ID}_{\mathrm{j}})$, sends $({X}_{j},{P}_{j})$ to ${NP}_{{A}_{1}}$, and adds $({X}_{j},{ID}_{j},{P}_{j})$ into ${L}_{PCGU}$.

**Private Key Query**$:$ If ${NP}_{CR}$ receives $({\beta}_{j},{ID}_{j},{P}_{j})$ as a query from ${NP}_{{A}_{1}},$ ${NP}_{CR}$ checks for $({\beta}_{j},{ID}_{j},{P}_{j})$ in the list ${L}_{PKQ}$, if it is available, it sends $({\beta}_{j},{P}_{j})$ to ${NP}_{{A}_{1}}.$ Otherwise, ${NP}_{CR}$ chooses ${\beta}_{j}\in {{Z}_{{q}_{KC}}}^{*},$ obtained ${P}_{j}$ from $PCGUQuery,$ sends $({\beta}_{j},{P}_{j})$ to ${NP}_{{A}_{1}}$, and add $({\beta}_{j},{ID}_{j},{P}_{j})$ into ${L}_{PKQ}$.

**Public Key Query**$:$ If ${NP}_{CR}$ receives $({X}_{j},{ID}_{j},{U}_{j})$ as a query from ${NP}_{{A}_{1}},$ ${NP}_{CR}$ checks for $({X}_{j},{ID}_{j},{U}_{j})$ in the list ${L}_{PBKQ}$, if it is available, it sends $({X}_{j},{U}_{j}))$ to ${NP}_{{A}_{1}}.$ Otherwise, ${NP}_{CR}$ searches and finds $({\beta}_{j},{X}_{j})$ from ${L}_{PKQ}$ and ${L}_{PCGU}$, and then computes ${U}_{j}={\beta}_{j}.{\gamma}_{KC}$, sends $({X}_{j},{U}_{j}))$ to ${NP}_{{A}_{1}}$, and adds $({X}_{j},{ID}_{j},{U}_{j})$ into ${L}_{PBKQ}$.

**Replace Public Key Query:**${NP}_{{A}_{1}}$ sends (${{X}_{j}}^{/},{{U}_{j}}^{/})$ to ${NP}_{CR}$ and can replace $({X}_{j},{U}_{j})$ on (${{X}_{j}}^{/},{{U}_{j}}^{/})$ for the identity ${ID}_{j}.$

**Delegation Generation Query:**${NP}_{{A}_{1}}$ sends two identity (${ID}_{\mathrm{C}\mathrm{C}},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}}$) and a warrant ${m}_{\mathrm{w}}$ to ${NP}_{CR},$ it then checks the tuple $({X}_{CC},{ID}_{\mathrm{C}\mathrm{C}})$ in ${L}_{{H}_{KC1}}$. If $\left(l=1\right)$, it can abort further processing. Otherwise, it extracts $({\beta}_{\mathrm{C}\mathrm{C}},{P}_{CC})$ from ${L}_{PKQ}$, ${H}_{2}$ from ${L}_{{H}_{KC2}}$, chooses ${A}_{\mathrm{C}\mathrm{C}}\in {{Z}_{{q}_{KC}}}^{*}$, generates ${(m}_{\mathrm{w}},{S}_{\mathrm{C}\mathrm{C}},{O}_{\mathrm{C}\mathrm{C}})$, and sends it to ${NP}_{{A}_{1}}$.

**CL-Proxy Signcryption Query:**${NP}_{{A}_{1}}$ sends two identities (${ID}_{drone},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}}$) and a message $\left(m\right)$ to ${NP}_{CR},$ it then checks the tuple $({X}_{GCS},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}})$ in ${L}_{{H}_{KC1}}$; if $\left(l=1\right)$, it can abort further processing. Otherwise, it extracts $({\beta}_{\mathrm{G}\mathrm{C}\mathrm{S}},{P}_{GCS})$ from ${L}_{PKQ}$, ${H}_{2}$ from ${L}_{{H}_{KC2}}$, chooses ${F}_{GCS}\in {{Z}_{{q}_{KC}}}^{*}$, generates (${C}_{GCS},{S}_{GCS},{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}),$ and sends it to ${NP}_{{A}_{1}}$.

**CL-Proxy Un-Signcryption Query:**${NP}_{{A}_{1}}$ sends two identities (${ID}_{drone},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}}$) and (${C}_{GCS},{S}_{GCS},{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}})$ to ${NP}_{CR},$ it then checks the tuple $({X}_{drone},{ID}_{drone})$ in ${L}_{{H}_{KC1}}$, the response is then provided in the subsequent methods.

- If $\left(l=0\right)$, ${NP}_{CR}$ can obtain $({X}_{GCS},{ID}_{GCS},{U}_{GCS})$ from ${L}_{PBKQ}$ according to identity ${ID}_{GCS}$, $({\beta}_{drone},{ID}_{drone},drone)$ from ${L}_{PKQ}$, performs the $ProxyUn-Signcryption$ algorithm and sends $\left(m\right)$ to ${NP}_{{A}_{1}}$.
- If $\left(l=1\right)$, ${NP}_{CR}$ can get $\left({H}_{4}\right)$ from ${L}_{{H}_{KC4}}$ and computes $m={H}_{KC4}\left(K\right)\oplus {C}_{GCS}$ perform the $ProxyUn-Signcryption$ algorithm. ${NP}_{CR}$ can further obtains $\left({X}_{GCS},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}},{{H}_{1}}^{\$},l\right)$ from ${L}_{{H}_{KC1}}$, $({X}_{GCS},{U}_{GCS})$ from ${L}_{PBKQ}$, $\left({H}_{3}\right)$ from the list ${L}_{{H}_{KC3}}$, and ${NP}_{CR}$ can verify the equation ${S}_{GCS}\left({X}_{GCS}+{U}_{GCS}+{{H}_{1}}^{\$}.{\delta}_{KC}+{H}_{3}.{\gamma}_{KC}\right)={U}_{GCS}+{Q}_{GCS}$. If the condition is met, the output is (m); otherwise, the procedure is repeated with new parameters.

**Challenge Stage:**Suppose ${m}_{KC1}$ and ${m}_{KC2}$ is adaptively generated two distinct messages by ${NP}_{{A}_{1}}$ and sends (${m}_{KC1},{m}_{KC2}$) and two challenged identities (${ID}_{\mathrm{G}\mathrm{C}\mathrm{S}},{ID}_{UAV}$) to ${NP}_{CR}$. Then, ${NP}_{CR}$ checks for the tuple $({X}_{drone},{ID}_{drone})$ in ${L}_{{H}_{KC1}}$, if $\left(l=0\right)$, ${NP}_{CR}$ stop; otherwise, it chooses (${{C}_{GCS}}^{/},{{S}_{GCS}}^{/},{{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}}^{/})\in {{Z}_{{q}_{KC}}}^{*}$ randomly and sends it to ${NP}_{{A}_{1}}$ as a challenge ciphertext.

**Guess Stage:**${NP}_{{A}_{1}}$ can make sure ${H}_{KC1}$ Query$,{H}_{KC2}$ Query, ${H}_{KC3}$ Query$,{H}_{KC4}$ Query, PCGU Query, Private Key Query, Public Key Query, Replace Public Key Query, Delegation Generation Query, CL-Proxy Signcryption Query, CL-Proxy Un-Signcryption Query is performed as same as above in Find Stage. So, ${NP}_{CR}$ returns ${l}^{/}$, ${NP}_{{A}_{1}}$ can made ${H}_{KC4}$ Query with ${K}^{/}={F}_{GCS}({U}_{drone}+{X}_{drone}+{\delta}_{KC}.{H}_{1})$. In this situation, the valid answer for ECDHP is included to ${L}_{{H}_{KC4}}$. The second situation is that ${NP}_{CR}$ can ignore the randomly selected/guessed value of ${NP}_{{A}_{1}}$, then ${NP}_{CR}$ randomly selects ${K}^{/}$ from ${L}_{{H}_{KC4}}$ and computes $\left(\frac{{K}^{/}-({\beta}_{drone}+{\alpha}_{drone}){{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}}^{/}}{{k}_{KC}}\right)={F}_{\mathrm{G}\mathrm{C}\mathrm{S}}.{{\Phi}_{KC}.\gamma}_{KC}$, where ${NP}_{CR}$ already knows the value ${\beta}_{drone},{\alpha}_{drone},{{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}}^{/}$, and ${K}^{/}$. Otherwise, ${NP}_{CR}$ failed to solve ECDHP.

**Theorem**

**2.**

**Proof.**

**Setup:**Here, ${NP}_{CR}$ selects ${\Phi}_{KC}\in {{Z}_{{q}_{KC}}}^{*},$ computes ${\delta}_{KC}$, make a param ${PAR}_{KC}$, and sends ${PAR}_{KC}$ and ${\Phi}_{KC}$ to ${NP}_{{A}_{2}}.$ Then, ${NP}_{{A}_{2}}$ can ask for the following queries.

**Find Stage:**Here, in this section, ${NP}_{{A}_{2}}$ can ask for the following polynomial bounded queries.

**PCGU Query**$:$ If ${NP}_{CR}$ receives $({X}_{j},{ID}_{j},{P}_{j})$ as a query from ${NP}_{{A}_{1}},$ ${NP}_{CR}$ checks for $({X}_{j},{ID}_{j},{P}_{j})$ in the list ${L}_{PCGU}$. If it is available, it sends $({X}_{j},{P}_{j})$ to ${NP}_{{A}_{2}}.$ Otherwise, ${NP}_{CR}$ chooses ${\alpha}_{j},{\Phi}_{j}\in {{Z}_{{q}_{KC}}}^{*},$ computes ${P}_{j}={\alpha}_{j}+{\Phi}_{j}{H}_{KC1}({X}_{j},{ID}_{\mathrm{j}})$, sends $({X}_{j},{P}_{j})$ to ${NP}_{{A}_{2}}$, and adds $({X}_{j},{ID}_{j},{P}_{j})$ into ${L}_{PCGU}$.

**Private Key Query**$:$ If ${NP}_{CR}$ receives $({\beta}_{j},{ID}_{j},{P}_{j})$ as a query from ${NP}_{{A}_{2}},$ ${NP}_{CR}$ checks for $({\beta}_{j},{ID}_{j},{P}_{j})$ in the list ${L}_{PKQ}$. If it is available, it sends $({\beta}_{j},{P}_{j})$ to ${NP}_{{A}_{2}}.$ Otherwise, ${NP}_{CR}$ chooses ${\beta}_{j}\in {{Z}_{{q}_{KC}}}^{*},$ obtained ${P}_{j}$ from $PCGUQuery,$ sends $({\beta}_{j},{P}_{j})$ to ${NP}_{{A}_{2}}$, and adds $({\beta}_{j},{ID}_{j},{P}_{j})$ into ${L}_{PKQ}$.

**Public Key Query**$:$ If ${NP}_{CR}$ receives $({X}_{j},{ID}_{j},{U}_{j})$ as a query from ${NP}_{{A}_{2}},$ ${NP}_{CR}$ checks for $({X}_{j},{ID}_{j},{U}_{j})$ in the list ${L}_{PBKQ}$, if it is available, it sends $({X}_{j},{U}_{j}))$ to ${NP}_{{A}_{1}}.$ Otherwise, ${NP}_{CR}$ searches and finds $({\beta}_{j},{X}_{j})$ from ${L}_{PKQ}$ and ${L}_{PCGU}$, and then computes ${X}_{j}={\Phi}_{KC}.{\gamma}_{KC}$, sends $({X}_{j},{U}_{j})$ to ${NP}_{{A}_{2}}$ and adds $({X}_{j},{ID}_{j},{U}_{j})$ into ${L}_{PBKQ}$.

**Delegation Generation Query:**${NP}_{{A}_{2}}$ sends two identity (${ID}_{\mathrm{C}\mathrm{C}},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}}$) and a warrant ${m}_{\mathrm{w}}$ to ${NP}_{CR},$ it then checks the tuple $({X}_{CC},{ID}_{\mathrm{C}\mathrm{C}})$ in ${L}_{{H}_{KC1}}$; if $\left(l=1\right)$, it can abort further processing. Otherwise, it extracts $({\beta}_{\mathrm{C}\mathrm{C}},{P}_{CC})$ from ${L}_{PKQ}$, ${H}_{2}$ from ${L}_{{H}_{KC2}}$, chooses ${A}_{\mathrm{C}\mathrm{C}}\in {{Z}_{{q}_{KC}}}^{*}$, generates ${(m}_{\mathrm{w}},{S}_{\mathrm{C}\mathrm{C}},{O}_{\mathrm{C}\mathrm{C}})$, and sends it to ${NP}_{{A}_{2}}$.

**CL-Proxy Signcryption Query:**${NP}_{{A}_{2}}$ sends two identities (${ID}_{drone},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}}$) and a message $\left(m\right)$ to ${NP}_{CR},$ it then checks the tuple $({X}_{GCS},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}})$ in ${L}_{{H}_{KC1}}$; if $\left(l=1\right)$, it can abort further processing. Otherwise, it extracts $({\beta}_{\mathrm{G}\mathrm{C}\mathrm{S}},{P}_{GCS})$ from ${L}_{PKQ}$, ${H}_{2}$ from ${L}_{{H}_{KC2}}$, chooses ${F}_{GCS}\in {{Z}_{{q}_{KC}}}^{*}$, generates (${C}_{GCS},{S}_{GCS},{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}),$ and sends it to ${NP}_{{A}_{2}}$.

**CL-Proxy Un-Signcryption Query:**${NP}_{{A}_{2}}$ sends two identities (${ID}_{drone},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}}$) and (${C}_{GCS},{S}_{GCS},{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}})$ to ${NP}_{CR},$ it then checks the tuple $({X}_{drone},{ID}_{drone})$ in ${L}_{{H}_{KC1}}$, and it gives the response in the following ways.

- If $\left(l=0\right)$, ${NP}_{CR}$ can obtain $({X}_{GCS},{ID}_{GCS},{U}_{GCS})$ from ${L}_{PBKQ}$ according to identity ${ID}_{GCS}$, $({\beta}_{drone},{ID}_{drone},{P}_{drone})$ from ${L}_{PKQ}$, perform the $ProxyUn-Signcryption$ algorithm, and sends $\left(m\right)$ to ${NP}_{{A}_{2}}$.
- If $\left(l=1\right)$, ${NP}_{CR}$ can obtain $\left({H}_{4}\right)$ from ${L}_{{H}_{KC4}}$, compute $m={H}_{KC4}\left(K\right)\oplus {C}_{GCS}$, and perform the $ProxyUn-Signcryption$ algorithm. ${NP}_{CR}$ further can get $\left({X}_{GCS},{ID}_{\mathrm{G}\mathrm{C}\mathrm{S}},{{H}_{1}}^{\$},l\right)$ from ${L}_{{H}_{KC1}}$, $({X}_{GCS},{U}_{GCS})$ from ${L}_{PBKQ}$, $\left({H}_{3}\right)$ from the list ${L}_{{H}_{KC3}}$, and ${NP}_{CR}$ can verify the equation ${S}_{GCS}\left({X}_{GCS}+{U}_{GCS}+{{H}_{1}}^{\$}.{\delta}_{KC}+{H}_{3}.{\gamma}_{KC}\right)={U}_{GCS}+{Q}_{GCS}$; if it is satisfied, its output will be $\left(m\right),$ otherwise, it repeats this process again with new parameters.

**Challenge Stage:**Suppose ${m}_{KC1}$ and ${m}_{KC2}$ adaptively generated two distinct messages by ${NP}_{{A}_{2}}$ and send (${m}_{KC1},{m}_{KC2}$) and two challenged identities (${ID}_{\mathrm{G}\mathrm{C}\mathrm{S}},{ID}_{drone}$) to ${NP}_{CR}$. Then, ${NP}_{CR}$ checks for the tuple $({X}_{drone},{ID}_{drone})$ in ${L}_{{H}_{KC1}}$, if $\left(l=0\right)$, ${NP}_{CR}$ stop; otherwise, it chooses (${{C}_{GCS}}^{/},{{S}_{GCS}}^{/},{{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}}^{/})\in {{Z}_{{q}_{KC}}}^{*}$ randomly and sends it to ${NP}_{{A}_{2}}$ as a challenge ciphertext.

**Guess Stage**: ${NP}_{{A}_{2}}$ can ensure ${H}_{KC1}$ Query$,{H}_{KC2}$ Query, ${H}_{KC3}$ Query$,{H}_{KC4}$ Query, PCGU Query, Private Key Query, Delegation Generation Query, CL-Proxy Signcryption Query, CL-Proxy Un-Signcryption Query is performed as same as above in Find Stage of Theorem 1 and Public Key Query of Theorem 2. So, ${NP}_{CR}$ returns ${l}^{/}$, ${NP}_{{A}_{2}}$ can make ${H}_{KC4}$ Query with ${K}^{/}={F}_{GCS}({U}_{drone}+{X}_{drone}+{\delta}_{KC}.{H}_{1})$; in this situation, the valid answer for ECDHP includes ${L}_{{H}_{KC4}}$. The second situation is that ${NP}_{CR}$ can ignore the randomly selected/guessed value of ${NP}_{{A}_{1}}$, ${NP}_{CR}$ then randomly selects ${K}^{/}$ from ${L}_{{H}_{KC4}}$ and computes $\left({K}^{/}-({\beta}_{\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{e}}+{\Phi}_{KC}{k}_{KC}){{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}}^{/}\right)={F}_{\mathrm{G}\mathrm{C}\mathrm{S}}.{{\Phi}_{KC}.\gamma}_{KC}$, where ${NP}_{CR}$ already knows the value ${\beta}_{\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{e}},{\alpha}_{drone},{{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}}^{/}$, and ${K}^{/}$. Otherwise, ${NP}_{CR}$ failed to solve ECDHP.

**Theorem**

**3.**

**Proof.**

**Setup:**Here, ${NP}_{CR}$ selects ${{\Phi}_{KC}}^{*}\in {{Z}_{{q}_{KC}}}^{*},$ computes ${\delta}_{KC}$, makes a param ${PAR}_{KC}$, and sends ${PAR}_{KC}$ to ${NP}_{{A}_{1}}.$ Then, ${NP}_{{A}_{1}}$ can ask for the following queries.

**Find Stage:**Here, in this section, ${NP}_{{A}_{1}}$ can ask for the following polynomial bounded queries.

**Forgery:**As ${NP}_{{A}_{1}}$ can ask for the following polynomial-bounded queries: ${H}_{KC1}$ Query$,{H}_{KC2}$ Query, ${H}_{KC3}$ Query$,{H}_{KC4}$ Query, PCGU Query, Private Key Query, Public Key Query, Replace Public Key Query, Delegation Generation Query, CL-Proxy Signcryption Query, CL-Proxy Un-Signcryption Query is performed as same as above in Find Stage of Theorem 1 and generates a forged proxy signcryption triple (${{C}_{GCS}}^{/},{{S}_{GCS}}^{/},{{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}}^{/})$ with the help of ${NP}_{CR}$. Note that ${NP}_{CR}$ can only solve the ECDLP if it accessed the actual value for ${\beta}_{\mathrm{C}\mathrm{C}}$ and ${A}_{\mathrm{C}\mathrm{C}}$ from ${U}_{\mathrm{C}\mathrm{C}}={\beta}_{\mathrm{C}\mathrm{C}}.{\gamma}_{KC}=a.{\gamma}_{KC}{andO}_{\mathrm{C}\mathrm{C}}={A}_{\mathrm{C}\mathrm{C}}.{\gamma}_{KC}=a.{\gamma}_{KC}$.

**Theorem**

**4.**

**Proof.**

**Setup:**Here, ${NP}_{CR}$ selects ${{\Phi}_{KC}}^{*}\in {{Z}_{{q}_{KC}}}^{*},$ computes ${\delta}_{KC}$, makes a param ${PAR}_{KC}$, and sends ${PAR}_{KC}$ and ${\Phi}_{KC}$ to ${NP}_{{A}_{1}}.$ Then, ${NP}_{{A}_{2}}$ can ask for the following queries.

**Find Stage:**Here, in this section, ${NP}_{{A}_{2}}$ can ask for the following polynomial-bounded queries: ${H}_{KC1}$ Query$,{H}_{KC2}$ Query, ${H}_{KC3}$ Query$,{H}_{KC4}$ Query, PCGU Query, Private Key Query, Delegation Generation Query, CL-Proxy Signcryption Query, CL-Proxy Un-Signcryption Query is performed in the same way as above in Find Stage of Theorem 1 and Public Key Query of Theorem 2.

**Forgery:**As ${NP}_{{A}_{2}}$ can ask for the following polynomial-bounded queries: ${H}_{KC1}$ Query$,{H}_{KC2}$ Query, ${H}_{KC3}$ Query$,{H}_{KC4}$ Query, PCGU Query, Private Key Query, Delegation Generation Query, CL-Proxy Signcryption Query, CL-Proxy Un-Signcryption Query is performed as same as above in Find Stage of Theorem 1 and Public Key Query of Theorem 2. Furthermore, it generates a forged proxy signcryption triple (${{C}_{GCS}}^{/},{{S}_{GCS}}^{/},{{Q}_{\mathrm{G}\mathrm{C}\mathrm{S}}}^{/})$ with the help of ${NP}_{CR}$. Note that ${NP}_{CR}$ can only solve the ECDLP if it accessed the actual value for ${\beta}_{\mathrm{C}\mathrm{C}}$ and ${A}_{\mathrm{C}\mathrm{C}}$ from ${U}_{\mathrm{C}\mathrm{C}}={\beta}_{\mathrm{C}\mathrm{C}}.{\gamma}_{KC}=a.{\gamma}_{KC}{andO}_{\mathrm{C}\mathrm{C}}={A}_{\mathrm{C}\mathrm{C}}.{\gamma}_{KC}=a.{\gamma}_{KC}$.

## 6. Performance Comparison

#### 6.1. Computational Cost

#### 6.2. Communication Cost

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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S. No | Symbol | Descriptions |
---|---|---|

1 | ${NP}_{KC}$ | The network provider that serves as KGC |

2 | ${k}_{KC}$ | The given security parameter to ${NP}_{KC}$ based on elliptic curve |

3 | ${G}_{KC}$ | A cyclic group of elliptic curve selected by the network provider |

4 | ${\gamma}_{KC}$ | It is the generator of a cyclic group ${G}_{KC}$ |

5 | ${{\rm Y}}_{m}$ | Indicates the length of plaintext |

6 | $\left|{{Z}_{{q}_{KC}}}^{\mathrm{*}}\right|$ | Indicates the length of selected parameter |

7 | ${\Phi}_{KC}$ | Indicates the master secret/master private key of ${NP}_{KC}$ |

8 | ${\delta}_{KC}$ | Indicates the master Public/Public key of ${NP}_{KC}$ |

9 | ${PAR}_{KC}$ | Represents the public parameter param that is distributed in a network |

10 | ${PP}_{i}$ | Represents the participated users, i.e., ($CC,GCS,drone$) |

11 | ${P}_{i}$ | Represents the partial private key of participated users, i.e., ($CC,GCS,drone$) |

12 | ${V}_{{PP}_{i}}$ | Represents the shared secret key between the participated users, i.e., ($CC,GCS,drone$) and ${NP}_{KC}$ |

13 | ${EID}_{{PP}_{i}}$ | Represents the encrypted identity of participated users, i.e., ($CC,GCS,drone$) |

14 | ${ID}_{{PP}_{i}}$ | Represents the identity of participated users, i.e., ($CC,GCS,drone$) |

15 | ${U}_{{PP}_{i}},{X}_{KC}$ | Represents the public key pair of participated users, i.e., ($CC,GCS,drone$) |

16 | ${P}_{i},{\beta}_{{PP}_{i}}$ | Represents the private key pair of participated users, i.e., ($CC,GCS,drone$) |

17 | ${U}_{\mathrm{C}\mathrm{C}},{X}_{CC}$ | Represents the public key pair of Control Centre ($CC$) |

18 | ${P}_{CC},{\beta}_{\mathrm{C}\mathrm{C}}$ | Represents the private key pair of Control Centre ($CC$) |

19 | ${U}_{GCS},{X}_{GCS}$ | Represents the public key pair of Ground Control Station ($GCS$) |

20 | ${U}_{drone},{X}_{drone}$ | Represents the public key pair of drone |

21 | ${P}_{drone},{\beta}_{drone}$ | Represents the private key pair of drone |

22 | ${P}_{GCS},{\beta}_{GCS}$ | Represenst the private key pair of Ground Control Station ($\mathrm{G}\mathrm{C}\mathrm{S}$) |

23 | ${H}_{KC1},{H}_{KC2},{H}_{KC3},{H}_{KC4}$ | Represents secure cryptographic hash functions |

24 | $\oplus $ | It is used for encryption and decryptions |

25 | ${S}_{{m}_{\mathrm{w}}}$ | It represents the signature generated on warrant message |

26 | ${m}_{\mathrm{w}}$ | Represents the warrant message that contains the delegation durations |

27 | ${C}_{GCS}$ | Represents the ciphertext, which is generated by GCS |

28 | $K$ | Represents the shared secret key between GCS and drone |

29 | $m$ | Represents the plaintext, which is chosen by GCS |

30 | ${S}_{GCS}$ | Represents the signature generated on message |

Schemes | Original User/Delegation | Delegation Verification and Signcryption | Un-Signcryption | Total |
---|---|---|---|---|

Yanfeng et al. [25] | - | $5{NP}_{ESM}$ | $10{NP}_{ESM}$ | $15{NP}_{ESM}$ |

Bhatia and Verma [26] | $2{NP}_{ESM}$ | $5{NP}_{ESM}$ | $4{NP}_{ESM}$ | $11{NP}_{ESM}$ |

Li et al. [27] | $2{NP}_{ESM}$ | $7{NP}_{ESM}$ | $5{NP}_{ESM}$ | $14{NP}_{ESM}$ |

Qu and Zeng [28] | $-$ | $3{NP}_{POP}$ | $3{NP}_{POP}$ | $6{NP}_{POP}$ |

Proposed Scheme | $1{NP}_{ESM}$ | $6{NP}_{ESM}$ | $4{NP}_{ESM}$ | $11{NP}_{ESM}$ |

Schemes | Original User/Delegation | Delegation Verification and Signcryption | Un-Signcryption | Total |
---|---|---|---|---|

Yanfeng et al. [25] | 0 | $14.24$ | $28.48$ | $42.72$ |

Bhatia and Verma [26] | $5.696$ | $14.24$ | 11.392 | $31.328$ |

Li et al. [27] | $5.696$ | $6.79$ | $14.24$ | 26.726 |

Qu and Zeng [28] | $0$ | $54.882$ | $54.882$ | $109.764$ |

Proposed Scheme | $2.848$ | $17.088$ | $11.392$ | $31.328$ |

Schemes | Signcryption Size | Signcryption Size in Bits |
---|---|---|

Yanfeng et al. [25] | $3\left|{NP}_{m}\right|+4\left|{NP}_{ID}\right|+12\left|{NP}_{q}\right|$ | 8704 |

Bhatia and Verma [26] | $3\left|{NP}_{m}\right|+4\left|{NP}_{ID}\right|+11\left|{NP}_{q}\right|$ | 8544 |

Li et al. [27] | $3\left|{NP}_{m}\right|+6\left|{NP}_{ID}\right|+10\left|{NP}_{q}\right|$ | 8704 |

Qu and Zeng [28] | $3\left|{NP}_{m}\right|+6\left|{NP}_{G}\right|$ | $\mathrm{12,288}$ |

Proposed Scheme | $3\left|{NP}_{m}\right|+5\left|{NP}_{q}\right|$ | $6944$ |

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## Share and Cite

**MDPI and ACS Style**

Khan, M.A.; Alhakami, H.; Ullah, I.; Alhakami, W.; Mohsan, S.A.H.; Tariq, U.; Innab, N.
A Resource-Friendly Certificateless Proxy Signcryption Scheme for Drones in Networks beyond 5G. *Drones* **2023**, *7*, 321.
https://doi.org/10.3390/drones7050321

**AMA Style**

Khan MA, Alhakami H, Ullah I, Alhakami W, Mohsan SAH, Tariq U, Innab N.
A Resource-Friendly Certificateless Proxy Signcryption Scheme for Drones in Networks beyond 5G. *Drones*. 2023; 7(5):321.
https://doi.org/10.3390/drones7050321

**Chicago/Turabian Style**

Khan, Muhammad Asghar, Hosam Alhakami, Insaf Ullah, Wajdi Alhakami, Syed Agha Hassnain Mohsan, Usman Tariq, and Nisreen Innab.
2023. "A Resource-Friendly Certificateless Proxy Signcryption Scheme for Drones in Networks beyond 5G" *Drones* 7, no. 5: 321.
https://doi.org/10.3390/drones7050321