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# Quantum-safe identity-based broadcast encryption with provable security from multivariate cryptography

Ψ The author is currently affiliated at the Department of Computer Science and Engineering, National Sun Yat-sen University

This work is supported by the University Grants Commission, Government of India under Grant No. 1228/(CSIRNETJUNE-2019); Ministry of Science and Technology of Taiwan under Grant No. MOST 111-2811-E-110-001; and Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India under Grant No. EMR/2017/002214

• Identity-Based Broadcast Encryption ($\textsf{IBBE}$) is a novel concept that can efficiently and securely transmit confidential content to a group of authorized users without the traditional Public-Key Infrastructure ($\textsf{PKI}$). After carefully exploring these areas, we have observed that none of the existing works have adopted the quantum-attack resistant cryptographic machinery Multivariate Public-Key Cryptography ($\textsf{MPKC}$) with provable security. We are the first to design a quantum-safe $\textsf{IBBE}$ that solely relies on the $\textsf{MPKC}$ framework. Our proposed protocol has achieved $\mathcal{O}(n)$-size communication bandwidth and ${n^3}\cdot\mathcal{O}\big(\max\big\{N, {\delta}^4\big\}\big)$-size overhead storage without any security breach. Here, $n$ is the number of variables for each multivariate polynomial, $N$ represents the total number of system users, and $\delta$ denotes a positive fixed-length. More positively, our design has achieved the adaptive INDistinguishable Chosen-Ciphertext Attack ($\textsf{IND-CCA}$) security in the Random Oracle Model ($\textsf{ROM}$) under the hardness of standard Multivariate Quadratic ($\textsf{MQ}$) problem. We emphasize that our system can also be immune against collusion attacks where several users come together to create an illicit decryption box.

Mathematics Subject Classification: Primary: 94A60, 68P25, 68P30, 68M12.

 Citation:

• Figure 1.  System model of $\textsf{IBBE-MPKC}$

Figure 2.  Flow diagram of our proposed $\textsf{IBBE-MPKC}$

Table 1.  Notations

 Symbol Description $\eta$ Security parameter of the system $\perp$ Null string $ID_u$ Identity for the user $u$ $|M|$ Length of a message $M$ $I_S$ The index set for a set $S$ $|S|$ Cardinality of a set $S$ $[N]$ $\{1, 2, \ldots, N\}$ $\delta$$\in_R$ $\{0, 1\}$ Bit $\delta$ is chosen randomly from the set $\{0, 1\}$ $\textsf{IND-ID-CCA}$ Indistinguishability under identity chosen-ciphertext attack

Table 2.  Comparison of communication bandwidth and storage overhead

 $\textsf{Schemes}$ $\textsf{Communication Bandwidth}$ $\textsf{Storage Overhead}$ $\textsf{MPK}$ $\textsf{MSK}$ $\textsf{SKey}_\textsf{u}$ Srivastava et al. [23] $n{{N+9}\choose 9}+1$ $n{n+2 \choose 2}{N+8 \choose 8}$ $\big(2n(n+1)$ +$n{n+2 \choose 2}\big){N+2 \choose 2}$ $2n(n+1)$ $+n{n+2 \choose 2}$ Our Scheme $n$ $n{n+2 \choose 2}{\delta+4 \choose 4}$ $(n+4)\delta{n+1 \choose 2}$ $(n+4)N{n+1 \choose 2}$ $\textsf{MPK}$= master-public key, $\textsf{MSK}$= master-secret key, $\textsf{SKey}_\textsf{u}$= secret-key of a user, $n$= the number of variables, $N$= the total number of users in the system, $\delta \le N$= a positive integer

Table 3.  Comparison of security and other functionalities

 $\textsf{Scheme}$ $\textsf{Encryption Technique}$ $\textsf{Security}$ $\textsf{IBE}$ $\textsf{Provable Security Model}$ $\textsf{Assumption}$ Srivastava et al. [23] $\textsf{KEM}$ × $\textsf{MQ Problem}$ √ Our scheme $\textsf{DEM}$ Adaptive $\textsf{IND-CCA}$ $\textsf{MQ Problem}$ √ $\textsf{KEM}$= Key Encapsulation Mechanism, $\textsf{DEM}$= Data Encapsulation Mechanism, $\textsf{IBE}$= Identity-based Encryption, $\textsf{MQ}$= Multivariate Quadratic, $\textsf{IND-CCA}$= Indistinguishable Chosen-Ciphertext Attack

Table 4.  Comparison of computation costs

 $\textsf{Scheme}$ $\textsf{Encryption Cost}$ $\textsf{Decryption Cost}$ $\textsf{#M}$ $\textsf{#A}$ $\textsf{#M}$ $\textsf{#I}$ Srivastava et al. [23] $nN{N+8 \choose 8}\big[n+N{n+2 \choose 2}\big]$ $n$ n$\sum_{i=1}^{9}i{N+i-1 \choose i}$ 0 Our Scheme $Nn\big[{n+2 \choose 2 }\sum_{i=1}^{4}{\delta+i-1 \choose i}+ {n+1 \choose 2}\big]$ $0$ $2nN^2$ $N$ $\textsf{#M}$ = cost of modular multiplication over finite field, $\textsf{#A}$= cost of modular addition over finite field, $\textsf{#I}$= cost to invert a function over a finite field.
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