增强Cramer-Shoup公钥加密系统抵抗线性相关密钥攻击的方法

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"这篇研究论文探讨了如何增强Cramer-Shoup公钥加密系统对线性相关密钥攻击的防御能力。作者包括来自不同科研机构的学者，他们分析了相关密钥攻击的威胁，并提出了解决策略，以确保Cramer-Shoup加密方案在遭受这种攻击时仍能保持安全性。" Cramer-Shoup加密系统是一种基于离散对数问题的安全公钥加密算法，由丹麦密码学家Daniele Cramer和Shafi Goldwasser于1998年在国际密码学会议CRYPTO上首次提出。该系统以其高效性和安全性而受到关注，能够抵抗已知的多种密码攻击，包括适应性选择密文攻击。然而，随着密码学研究的深入，新的攻击方式如相关密钥攻击（Related-Key Attacks）被发现，这些攻击方法允许对手通过篡改设备内存中的密钥来改变密钥，并观察使用修改后密钥的加密结果。相关密钥攻击是针对物理设备的安全威胁，攻击者可以通过物理手段或恶意软件篡改存储在设备中的密钥，然后利用这些修改后的密钥进行加密或解密操作，以此来寻找系统的漏洞。对于Cramer-Shoup这样的公钥加密系统来说，这类攻击可能破坏其安全性保证。论文中，作者Baodong Qin、Shuai Han、Yu Chen、Shengli Liu和Zhuo Wei分析了Cramer-Shoup系统在面对线性相关密钥攻击时的弱点，并提出了强化措施。他们可能采用了额外的随机化技术，如密钥派生函数（Key Derivation Functions, KDFs）和密钥伪装技术，以增加攻击者篡改密钥后的不确定性。此外，可能还讨论了如何通过增加加密过程的复杂性，比如使用多轮加密或者结合其他密码学构造，来提高系统对相关密钥攻击的抵抗力。这篇论文致力于填补Cramer-Shoup加密系统在应对相关密钥攻击方面的安全空白，提供了一套理论和实践上的解决方案，对于加强实际应用中的公钥加密安全具有重要意义。研究人员和安全工程师可以参考这些方法来改进现有的加密系统，提高其抵抗新型攻击的能力。

4 B. Qin et al.

Related Work. There are some basic RKA-secure primitives available for

achieving RKA-secure PKE schemes. They include RKA-PRFs [2, 3, 1], RKA-

IBE [4, 20], algebraic manipulation detection (AMD) codes [12] and (continu-

ous) non-malleable codes/functions [19, 21, 29, 11]. The framework of RKA-IBE

relies on a normal one-time signature scheme. Compared to the underlying IBE

scheme, it usually involves additional computational operations, including sign-

ing and veriﬁcation, during the encryption and decryption phases. Moreover,

it also extends the ciphertext sizes with the signing key and the signature. The

other methods usually require to change the key generation algorithm of a normal

CCA-secure PKE scheme and blows up the key sizes. There are also numerous

concrete and eﬃcient RKA-PKE schemes [20, 32, 23, 16, 17, 22] based on speciﬁc

number-theoretic assumptions, e.g., DDH and factoring. Speciﬁcally, to avoid the

usage of one-time signature and pairing, Cui et al. [16] proposed an eﬃcient PKE

scheme based on the CS-PKE scheme. Jia et al. proposed two eﬃcient RKA-

secure PKE schemes [23] based on Kiltz et al.’s (normal) CCA-secure hybrid

PKE schemes [24]. The two schemes almost have the same eﬃciency as that

of the underlying CCA-PKE schemes. Later, Jia et al. [22] showed that their

schemes can be generalized to constructing RKA secure hybrid PKE schemes

from 1-universal hash proof systems (with some special properties).

Organization. The rest of this paper is organized as follows. Section 2 reviews

some basic cryptographic notations that will be used. We present our construc-

tion and show its security in Section 3. In Section 4, we compare the performance

of our scheme with known RKA-PKE schemes. Section 5 is the summary of this

paper.

2 Preliminary

Notations. Throughout this paper, κ ∈ N denotes the security parameter. If S

is a ﬁnite set, then s ←

S denotes the operation of picking s from S uniformly

at random. If s ∈ S is an element, |s| denotes its bit length. By y ← A(x), we

denote the operation of running algorithm A on input x, and letting y denote

its output.

2.1 Public-Key Encryption and Related-Key Security

Public-Key Encryption. A public-key encryption scheme PKE=(Sys, Gen,

Enc, Dec) consists of four probabilistic polynomial-time (PPT) algorithms. Sys(1

)

is the randomized public parameter generation algorithm, which takes as in-

put a security parameter 1

, and outputs a global parameter sp. Gen(sp) is

the randomized public key generation algorithm that takes as input the system

parameter sp, and outputs a pair of public/secret keys (pk, sk). Enc(pk, m) is

the randomized encryption algorithm, which takes as input a public pk and a

message m ∈ M, and outputs a ciphertext C. Dec(sk, C) is the deterministic

decryption algorithm, which takes as input a secret key sk and a ciphertext C,

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