基于多线性映射的可验证多接收者加密方案：安全性与验证设计

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本文探讨的是"来自多线性映射的可验证多收件人加密方案"，这是一项重要的研究论文，针对量子计算时代的信息安全需求而设计。在传统的公钥加密系统中，多收件人加密（MRPKE）是一个关键的应用，特别是在广播数据等场景中具有显著优势。作者Zhengjun Jing、Guoping Jiang 和 Chunsheng Gu 来自南京邮电大学计算机学院和江苏科技大学计算机工程系，他们共同提出了一个新颖的MRPKE体系。该方案构建在Garg-Gentry-Halevi (GGH) 架构之上，GGH框架是一种基于理想格的多线性映射的等距类比，类似于在密码学中的阶梯结构。在GDDH（分级决策 Diffie-Hellman）假设的保护下，这个新设计的加密方案能够抵抗选择性plaintext攻击（CPA），确保了消息的安全性。与传统MRPKE不同，这个方案允许每个接收者在无需先解密的情况下验证消息的真实性，提高了系统的效率和安全性。加密和解密过程仅涉及多项式模运算，这使得它在性能上具有高效性，对于大规模数据传输尤其适用。此外，该论文不仅关注加密技术本身，还可能讨论了如何将多线性映射的特性巧妙地融入到MRPKE设计中，以及如何通过GDDH假设来保障系统的安全性。这对于理解后量子密码学的进展以及如何利用新型数学构造来提升公钥加密系统的实用性和安全性具有重要意义。这篇研究为多线性映射在现代密码学中的实际应用提供了新的视角，并对未来的多收件人加密和信息共享协议提供了理论支持。通过阅读这篇文章，读者可以深入理解如何在量子计算时代设计出既安全又高效的公共密钥加密方案。

A Verifiable Multi-Recipient Encryption Scheme

from Multilinear Maps

Zhengjun Jing, Guoping Jiang

College of Computer

Nanjing University of Posts and Telecommunications

Nanjing, China

jzjing@jsut.edu.cn, jianggp@njupt.edu.cn

Chunsheng Gu

Department of Computer Engineering

Jiangsu University of Technology

Changzhou, China

guchunsheng@gmail.com

Abstract—Multi-recipient encryption is an important

public key cryptosystem, which can be applied for a

variety of purposes, such as broadcasting data. In order to

design an secure multi-recipient public key encryption

(MRPKE) in post-quantum era, in this paper, we construct

a novel MRPKE scheme base on Garg-Gentry-Halevi

(GGH) framework which is a graded algebras analogue of

multilinear maps from ideal lattice. Under the grade

decisional Diffie-Hellman (GDDH) assumption of GGH,

the proposed scheme has semantically safety against

chosen plaintext attack (CPA). At the same time, each

recipient, without first decrypting, can verify whether the

message to be received is from a legitimate sender.

Furthermore, the encryption and decryption only involves

the polynomial modular addition and multiplication in

polynomial ring, so the efficiency of the proposed scheme

is higher.

Keywords—

public key encryption; multi-recipient

verifiable; multilinear maps

INTRODUCTION

In 2000, Bellare et al.[1] and Baudron et al.[2]

independently proposed the concept of multi-recipient public

key encryption and its security definitions. A multi-recipient

public key encryption system generally includes a sender and

n receivers, in which both the sender and receivers have their

own private and public key pairs denoted by

(,)

pk sk ,

where

0 in and

(,)pk sk is for the sender. The sender can

use his (or her) private key and the

n public keys of recipients

to encrypt the message

M , while any authorized recipient can

use own private key to decrypt the received ciphertext, and

obtain the correct plaintext message. Such properties of multi-

recipient public key encryption can be applied for a variety of

purposes, such as broadcasting encryption and multicast

security protocols in wireless sensor network.

The initial MRPKE schemes were just an extension of the

general public key encryption (such as 2-party) [1-3]. For

receivers, they needed to encrypt

n times to generate the

ciphertext

(, )

CCC " . Obviously, such MRPKE schemes

were less efficient and high bandwidth requirements in

communication. To improve the efficiency, Lu et al. [4]and

Pang et al. [5] respectively proposed new schemes based on the

Shamir’s secrets sharing protocol and bilinear paring on elliptic

curve, in which the message to be sent was encrypted only

once while the corresponding ciphertext could be decrypted by

multiple private keys. After the invention of identity-based

public key encryption system[6], many outstanding identity-

based multi-recipient public key encryption schemes had also

been proposed. Furthermore, the formal definition of security

of MRPKE was described[7]. In order to protect the privacy of

users, several MRPKE schemes with anonymity of receipts (or

the sender) were proposed [8-10]. Recently, Pang et al.

designed a new identity-based multi-recipient public key

encryption scheme[11] , which can not only ensure the

anonymity of the recipients, but also satisfy the fairness and

make the identity of the sender verifiable. However, with the

increase in the number of recipients, the ciphertext length of

schemes described above becomes longer. More importantly,

the security of all these schemes is based on the elliptic curve

bilinear paring, which is threatened by the power of quantum

computers.

In order to design a secure multi-recipient public key

encryption in post-quantum era, in this paper, we construct a

novel MRPKE scheme from GGH multilinear maps. Its

security can be reduced to the assumption that the GDDH

problem is difficult to solve in GGH framework, which is

similar to the NTRU hard assumption[12]. Although GDDH

hard assumption can not be directly reduced to SVP (shortest

vector problem), LWE (Learning with error) and other general

problems about lattice, there is no effective algorithm in

polynomial time to solve this problem, including quantum

algorithms. Under the grade decisional Diffie-Hellman

(GDDH) assumption of GGH, the proposed scheme has

semantically safety against chosen plaintext attack (CPA). At

the same time, each recipient, without first decrypting, can

verify whether the message to be received is from a legitimate

sender. Furthermore, the encryption and decryption only

involves the polynomial modular addition and multiplication in

polynomial ring, so the efficiency of the proposed scheme is

higher.

The rest of this paper is organized as follows. In section 2,

we introduce the background about multilinear map and the

algorithms about GGH framework, while the definition of

MRPKE and its security model is described in section 3. In

2014 Ninth International Conference on P2P, Parallel, Grid, Cloud and Internet Computing

DOI 10.1109/3PGCIC.2014.49

151

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