What are correlation attacks and algebraic attacks, and how do they exploit the vulnerabilities of single LFSRs?

/ / Published in Cybersecurity, EITC/IS/CCF Classical Cryptography Fundamentals, Stream ciphers, Stream ciphers and linear feedback shift registers, Examination review

Linear Feedback Shift Registers (LFSRs) are critical components in the design of stream ciphers used in classical cryptography. Their simplicity and efficiency make them attractive for generating pseudo-random sequences. However, despite these advantages, LFSRs are susceptible to various forms of cryptanalysis, including correlation attacks and algebraic attacks. These attacks exploit inherent vulnerabilities in LFSRs, compromising the security of the cryptographic systems that rely on them.

Correlation Attacks:

Correlation attacks are a form of cryptanalysis that exploits statistical correlations between the output of an LFSR-based stream cipher and the known plaintext or other parts of the cipher. The basic premise of a correlation attack is that the pseudo-random sequence generated by an LFSR is not entirely random; instead, it exhibits certain patterns or correlations that can be detected and exploited.

An LFSR generates a sequence by shifting its bits and applying a linear feedback function. For example, consider an LFSR of length n with feedback polynomial P(x) = x^n + c_{n-1}x^{n-1} + \cdots + c_1x + c_0. The state of the LFSR at time t is represented by the vector \mathbf{s}(t) = (s_0(t), s_1(t), \ldots, s_{n-1}(t)), and the feedback function updates the state as follows:

    \[ s_0(t+1) = \sum_{i=0}^{n-1} c_i s_i(t) \mod 2 \]

In a correlation attack, the attacker seeks to identify a correlation between the output sequence of the LFSR and a known sequence. If such a correlation exists, the attacker can use it to deduce information about the internal state of the LFSR, ultimately compromising the security of the cipher.

For example, suppose an LFSR-based stream cipher generates a keystream \mathbf{k} = (k_0, k_1, \ldots) that is XORed with the plaintext \mathbf{p} = (p_0, p_1, \ldots) to produce the ciphertext \mathbf{c} = (c_0, c_1, \ldots). If the attacker knows or can guess part of the plaintext, they can compute the corresponding part of the keystream:

    \[ k_i = c_i \oplus p_i \]

The attacker then analyzes the keystream to detect any statistical correlations with the output of the LFSR. If such correlations are found, the attacker can use them to construct a system of linear equations that describe the internal state of the LFSR. Solving these equations allows the attacker to recover the initial state of the LFSR, and thus predict future keystream bits.

To illustrate this, consider a simple example where the LFSR has a feedback polynomial P(x) = x^3 + x + 1 and an initial state \mathbf{s}(0) = (1, 0, 1). The output sequence generated by this LFSR is:

    \[ \mathbf{k} = (1, 0, 1, 1, 0, 1, \ldots) \]

Suppose the attacker has access to a segment of the keystream and the corresponding plaintext:

    \[ \mathbf{k} = (1, 0, 1, 1, 0, 1) \]

    \[ \mathbf{p} = (0, 1, 1, 0, 1, 0) \]

The attacker computes the ciphertext:

    \[ \mathbf{c} = \mathbf{k} \oplus \mathbf{p} = (1, 1, 0, 1, 1, 1) \]

By analyzing the keystream, the attacker may detect a correlation with the output of the LFSR. For instance, they might observe that certain patterns in the keystream repeat with a period that matches the length of the LFSR. This information can be used to construct a system of linear equations that describe the internal state of the LFSR. Solving these equations reveals the initial state, allowing the attacker to predict future keystream bits and decrypt the ciphertext.

Algebraic Attacks:

Algebraic attacks are another form of cryptanalysis that targets the algebraic structure of the LFSR and its feedback function. These attacks exploit the fact that the output of an LFSR can be described by a system of linear equations, which can be manipulated to reveal the internal state of the LFSR.

The key idea behind algebraic attacks is to express the keystream bits as algebraic equations in terms of the internal state bits. Once these equations are obtained, the attacker can solve them to recover the initial state of the LFSR.

Consider an LFSR with a feedback polynomial P(x) = x^n + c_{n-1}x^{n-1} + \cdots + c_1x + c_0 and an initial state \mathbf{s}(0) = (s_0, s_1, \ldots, s_{n-1}). The output sequence \mathbf{k} = (k_0, k_1, \ldots) is generated by the LFSR as follows:

    \[ k_t = s_0(t) \]

    \[ s_0(t+1) = \sum_{i=0}^{n-1} c_i s_i(t) \mod 2 \]

The attacker constructs a system of linear equations that describe the relationship between the keystream bits and the internal state bits. For example, the first few equations might be:

    \[ k_0 = s_0 \]

    \[ k_1 = s_1 \]

    \[ k_2 = s_2 \]

    \[ k_3 = s_3 \]

    \[ k_4 = c_0 s_0 + c_1 s_1 + c_2 s_2 + c_3 s_3 \mod 2 \]

The attacker then solves this system of equations to recover the initial state \mathbf{s}(0). Once the initial state is known, the attacker can predict future keystream bits and decrypt the ciphertext.

To illustrate this, consider an LFSR with a feedback polynomial P(x) = x^4 + x + 1 and an initial state \mathbf{s}(0) = (1, 0, 1, 1). The output sequence generated by this LFSR is:

    \[ \mathbf{k} = (1, 0, 1, 1, 0, 1, 1, 0, \ldots) \]

The attacker constructs the following system of linear equations:

    \[ k_0 = s_0 \]

    \[ k_1 = s_1 \]

    \[ k_2 = s_2 \]

    \[ k_3 = s_3 \]

    \[ k_4 = s_0 + s_3 \mod 2 \]

    \[ k_5 = s_1 + s_0 \mod 2 \]

    \[ k_6 = s_2 + s_1 \mod 2 \]

    \[ k_7 = s_3 + s_2 \mod 2 \]

By solving this system of equations, the attacker recovers the initial state \mathbf{s}(0) = (1, 0, 1, 1), allowing them to predict future keystream bits and decrypt the ciphertext.

Both correlation attacks and algebraic attacks highlight the inherent vulnerabilities of single LFSRs in stream ciphers. These vulnerabilities arise from the linear nature of LFSRs, which makes their output sequences susceptible to statistical and algebraic analysis. To mitigate these vulnerabilities, cryptographic designers often use more complex constructions, such as combining multiple LFSRs with non-linear functions or using non-linear feedback shift registers (NLFSRs) that provide stronger security guarantees.

Correlation attacks and algebraic attacks are powerful cryptanalytic techniques that exploit the vulnerabilities of single LFSRs in stream ciphers. By detecting statistical correlations or constructing algebraic equations, attackers can recover the internal state of the LFSR, compromising the security of the cipher. Understanding these attacks is essential for designing more secure cryptographic systems that can withstand advanced cryptanalysis.

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