With an attack on a single LFSR is it possible to encounter combination of encrypted and decrypted part of the transmission of length 2m from which it is not possible to build solvable linear equations system?
In the field of classical cryptography, stream ciphers play a significant role in securing data transmission. One commonly used component in stream ciphers is the linear feedback shift register (LFSR), which generates a pseudorandom sequence of bits. However, it is important to analyze the security of stream ciphers to ensure that they are resistant to attacks.
The question asks whether it is possible to encounter a combination of encrypted and decrypted parts of a transmission, of length 2m, from which it is not possible to build a solvable linear equations system when attacking a single LFSR. To answer this question, we need to consider the properties of LFSRs and the nature of the attack.
An LFSR is a shift register whose input bit is a linear combination of its previous bits, determined by a feedback function. The output of an LFSR is the bit at the rightmost position of the shift register. The feedback function is defined by a polynomial, and the length of the LFSR is determined by the number of stages in the shift register.
When attacking an LFSR, the goal is to recover the internal state of the LFSR, which allows an attacker to predict the keystream generated by the LFSR and potentially decrypt the encrypted data. One common attack on LFSRs is the known-plaintext attack, where an attacker has access to both the plaintext and the corresponding ciphertext.
In the context of the question, if an attacker can obtain a combination of encrypted and decrypted parts of a transmission, it means that the attacker has access to both the plaintext and the corresponding ciphertext. In this scenario, it is possible for the attacker to analyze the keystream generated by the LFSR and attempt to recover the internal state of the LFSR.
To build a solvable linear equations system, the attacker needs to have a sufficient number of equations that are linearly independent. In the case of an LFSR, the internal state can be represented as a vector of bits. Each bit in the internal state is a linear combination of previous bits, determined by the feedback function of the LFSR.
If the attacker has access to a combination of encrypted and decrypted parts of the transmission, they can compare the corresponding bits in the plaintext and ciphertext. By XORing the plaintext and ciphertext bits, the attacker obtains the keystream bits generated by the LFSR. These keystream bits can be used to form linear equations that represent the feedback function of the LFSR.
However, it is important to note that the number of linearly independent equations that can be obtained from a combination of encrypted and decrypted parts of the transmission is limited. The length of the transmission is 2m, which means that there are m bits in the plaintext and m bits in the ciphertext. Therefore, the maximum number of linearly independent equations that can be formed is m.
If the length of the LFSR is greater than m, it is possible that the attacker may not have enough equations to build a solvable linear equations system. In this case, the attacker will not be able to recover the internal state of the LFSR and predict the keystream generated by the LFSR.
However, if the length of the LFSR is less than or equal to m, it is likely that the attacker will have enough equations to build a solvable linear equations system. In this case, the attacker can recover the internal state of the LFSR and predict the keystream generated by the LFSR.
When attacking a single LFSR, it is possible to encounter a combination of encrypted and decrypted parts of a transmission, of length 2m, from which it may or may not be possible to build a solvable linear equations system. The ability to build such a system depends on the length of the LFSR and the number of linearly independent equations that can be obtained from the available plaintext and ciphertext.
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