LFSR流密码与Stream Ciphers概述

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本章节来自《应用密码学手册》一书，作者为A.Menezes、P.van Oorschot和S.Vanstone，该书由CRC Press于1996年出版。本书主要探讨了流密码的相关内容，特别是基于线性反馈移位寄存器（Linear Feedback Shift Registers, LFSRs）的流密码。流密码是一种重要的加密技术，它们的工作原理是生成连续的无限序列比特流，这个比特流可以与明文进行异或操作来加密数据。LFSRs作为流密码的基础，是由一组可配置的寄存器组成，通过循环移位和反馈函数生成密钥流。由于每个阶段的输出仅依赖于前几个状态，LFSRs能提供高度的并行性和效率，适合在硬件实现上使用。章节讨论的重点可能包括以下几个方面： 1. **LFSR结构与工作原理**：详细介绍了如何构建和配置LFSR，以及其内部机制如何决定密钥流的生成。这包括描述不同的反馈函数，如何选择初始状态以确保安全性，以及如何利用多项式描述LFSR的行为。 2. **流密码算法**：可能介绍了如移位寄存器反馈（Shift Register Feedback, SRF）流密码，这是一种常见的基于LFSR的流密码设计，如Trivium、Salsa20或ChaCha等，它们的安全性和性能特点。 3. **随机性和伪随机性**：流密码依赖于随机性和伪随机性来确保加密的不可预测性，章节可能会解释这些概念在流密码设计中的重要性，以及如何通过LFSR生成高质量的随机数。 4. **安全分析**：针对流密码的潜在威胁，如预测攻击、线性分析和差分分析，以及如何通过设计策略来对抗这些攻击。 5. **实际应用与局限性**：可能讨论流密码在实际通信、网络安全和密码学协议中的应用，同时指出它们可能面临的挑战，如数据传输时的同步问题以及长期密钥管理的需求。 6. **版权声明与使用规定**：这部分提醒读者，电子版的《应用密码学手册》的使用受到严格的版权保护，只有在个人学习用途下才能打印、存储和复制单章，并且必须获得CRC Press的书面许可才能进行任何形式的传播。本章节深入探讨了基于LFSR的流密码在密码学领域的理论基础和实践应用，对于了解这类加密技术的开发者、研究者以及对密码学感兴趣的读者来说具有很高的价值。

6.2 Feedback shift registers 195

6.6 Note (properties of self-synchronizing stream ciphers)

(i) self-synchronization. Self-synchronization is possible if ciphertext digits are deleted

or inserted, because the decryption mapping depends only on a ﬁxed number of pre-

ceding ciphertext characters. Such ciphers are capable of re-establishing proper de-

cryption automatically after loss of synchronization, with only a ﬁxed number of

plaintext characters unrecoverable.

(ii) limited error propagation. Suppose that the state of a self-synchronizationstream ci-

pher depends on t previous ciphertext digits. If a single ciphertext digit is modiﬁed

(or even deleted or inserted) during transmission, then decryption of up to t subse-

quent ciphertext digits may be incorrect, after which correct decryption resumes.

(iii) active attacks. Property (ii) implies that any modiﬁcation of ciphertext digits by an

active adversary causes several other ciphertext digits to be decrypted incorrectly,

thereby improving(compared to synchronousstream ciphers) the likelihood of being

detected by the decryptor. As a consequence of property (i), it is more difﬁcult (than

for synchronous stream ciphers) to detect insertion, deletion, or replay of ciphertext

digits by an active adversary. This illustrates that additional mechanisms must be

employed in order to provide data origin authentication and data integrity guarantees

(see §9.5.4).

(iv) diffusion of plaintext statistics. Since each plaintext digit inﬂuences the entire fol-

lowing ciphertext, the statistical properties of the plaintext are dispersed through the

ciphertext. Hence, self-synchronizingstream ciphers may be more resistant than syn-

chronous stream ciphers against attacks based on plaintext redundancy.

6.2 Feedback shift registers

Feedback shift registers, in particular linear feedback shift registers, are the basic compo-

nents of many keystream generators. §6.2.1 introduces linear feedback shift registers. The

linear complexity of binary sequences is studied in §6.2.2, while the Berlekamp-Massey al-

gorithm for computing it is presented in §6.2.3. Finally, nonlinear feedback shift registers

are discussed in §6.2.4.

6.2.1 Linear feedback shift registers

Linear feedback shift registers (LFSRs) are used in many of the keystream generators that

have been proposed in the literature. There are several reasons for this:

1. LFSRs are well-suited to hardware implementation;

2. they can produce sequences of large period (Fact 6.12);

3. they can produce sequences with good statistical properties (Fact 6.14); and

4. because of their structure, they can be readily analyzed using algebraic techniques.

6.7 Deﬁnition A linear feedback shift register (LFSR)oflengthL consists of L stages (or

delay elements) numbered 0, 1,... ,L− 1, each capable of storing one bit and having one

input and one output; and a clock which controls the movement of data. During each unit

of time the following operations are performed:

(i) the content of stage 0 is output and forms part of the output sequence;

Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and S. Vanstone.

196 Ch. 6 Stream Ciphers

(ii) the content of stage i is moved to stage i − 1 for each i, 1 ≤ i ≤ L − 1;and

(iii) the new content of stage L − 1 is the feedback bit s

which is calculated by adding

together modulo 2 the previous contents of a ﬁxed subset of stages 0, 1,... ,L− 1.

Figure 6.4 depicts an LFSR. Referring to the ﬁgure, each c

is either 0 or 1; the closed

semi-circles are AND gates; and the feedback bit s

is the modulo 2 sum of the contents of

those stages i, 0 ≤ i ≤ L − 1, for which c

L−i

=1.

Stage Stage

L-2

L-1

L−1

output

StageStage

Figure 6.4:

A linear feedback shift register (LFSR) of length L.

6.8 Deﬁnition The LFSR of Figure 6.4 is denoted hL, C(D)i,whereC(D)=1+c

D +

+ ···+ c

∈ Z

[D] is the connection polynomial. The LFSR is said to be non-

singular if the degree of C(D) is L (that is, c

=1). If the initial content of stage i is

∈{0, 1} for each i, 0 ≤ i ≤ L − 1,then[s

L−1

,... ,s

] is called the initial state of

the LFSR.

6.9 Fact If the initial state of the LFSR in Figure 6.4 is [s

L−1

,... ,s

], then the output

sequence s = s

,... is uniquely determined by the following recursion:

=(c

j−1

+ c

j−2

+ ···+ c

j−L

)mod2 for j ≥ L.

6.10 Example (output sequence of an LFSR) Consider the LFSR h4, 1+D + D

i depicted

in Figure 6.5. If the initial state of the LFSR is [0, 0, 0, 0], the output sequence is the zero

sequence. The following tables show the contents of the stages D

, D

at the end

of each unit of time t when the initial state is [0, 1, 1, 0].

t D

0 0 1 1 0

1 0 0 1 1

2 1 0 0 1

3 0 1 0 0

4 0 0 1 0

5 0 0 0 1

6 1 0 0 0

7 1 1 0 0

t D

8 1 1 1 0

9 1 1 1 1

10 0 1 1 1

11 1 0 1 1

12 0 1 0 1

13 1 0 1 0

14 1 1 0 1

15 0 1 1 0

The output sequence is s =0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1,..., and is periodic with

period 15 (see Deﬁnition 5.25). 

The signiﬁcance of an LFSR being non-singular is explained by Fact 6.11.

1997 by CRC Press, Inc. — See accompanying notice at front of chapter.

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