How does an LFSR generate a key stream, and what role does the feedback polynomial play in this process?

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

A Linear Feedback Shift Register (LFSR) is a key component in the generation of pseudorandom sequences, which are important in stream ciphers for cryptographic applications. The LFSR generates a key stream by shifting bits through a register and using a feedback mechanism defined by a polynomial. This process is deterministic and relies heavily on the properties of the feedback polynomial, which determines the sequence's period and randomness.

An LFSR consists of a shift register and a feedback function. The shift register is an array of bits, and the feedback function is typically a linear function of the bits in the register. The feedback polynomial, also known as the characteristic polynomial, defines which bits from the register are combined to form the feedback bit. This polynomial is of the form:

    \[ P(x) = x^n + c_{n-1}x^{n-1} + \cdots + c_2x^2 + c_1x + c_0 \]

where c_i are the coefficients that are either 0 or 1, and n is the length of the LFSR. The polynomial is used to determine which bits are tapped for feedback. For example, if the polynomial is P(x) = x^4 + x + 1, the feedback would be determined by the XOR of the first and fourth bits of the register.

The process of generating the key stream starts with an initial state, known as the seed, which is a non-zero vector of length n. The bits in the register are shifted to the right, and the feedback bit is computed and placed in the leftmost position of the register. This operation is repeated to produce a sequence of bits that form the key stream. The sequence generated by an LFSR is periodic, and the maximum period of the sequence is 2^n - 1, which occurs when the feedback polynomial is a primitive polynomial.

A primitive polynomial is a polynomial that cannot be factored into polynomials of lower degree over the field GF(2), and it generates the maximum-length sequence. For instance, the polynomial P(x) = x^4 + x + 1 is a primitive polynomial, and an LFSR with this polynomial will produce a sequence with a period of 2^4 - 1 = 15.

The role of the feedback polynomial is important in determining the properties of the key stream. The polynomial defines the taps, which are the positions in the register that are used to compute the feedback bit. The choice of taps affects the period, linear complexity, and statistical properties of the sequence. A well-chosen feedback polynomial ensures that the sequence has a long period and good randomness properties, making it suitable for cryptographic applications.

For example, consider an LFSR with a length of 4 and the feedback polynomial P(x) = x^4 + x^3 + x^2 + x + 1. The initial state is 1000. The LFSR operates as follows:

1. Initial state: 1000
2. Shift right: 0100, feedback bit: 1 (XOR of all bits)
3. New state: 1100
4. Shift right: 0110, feedback bit: 1
5. New state: 1110
6. Shift right: 0111, feedback bit: 1
7. New state: 1111
8. Shift right: 0111, feedback bit: 1
9. New state: 1110

This process continues, generating a sequence of bits. The period of the sequence is determined by the feedback polynomial and the length of the register. If the polynomial is not primitive, the sequence will have a shorter period and may exhibit poor randomness properties.

In cryptographic applications, the key stream generated by the LFSR is used to encrypt plaintext by performing an XOR operation between the key stream and the plaintext. The resulting ciphertext can be decrypted by performing the same XOR operation between the key stream and the ciphertext. The security of the stream cipher depends on the properties of the key stream, which are determined by the LFSR and the feedback polynomial.

To illustrate the importance of the feedback polynomial, consider two LFSRs with different polynomials. The first LFSR has a feedback polynomial P(x) = x^4 + x + 1, and the second LFSR has a feedback polynomial Q(x) = x^4 + x^3 + 1. The first LFSR generates a maximum-length sequence with a period of 15, while the second LFSR generates a sequence with a shorter period. The first LFSR's sequence has better randomness properties and is more suitable for cryptographic applications.

The LFSR generates a key stream by shifting bits through a register and using a feedback mechanism defined by a polynomial. The feedback polynomial determines the sequence's period, linear complexity, and randomness properties. A well-chosen feedback polynomial ensures that the sequence has a long period and good randomness properties, making it suitable for cryptographic applications. The key stream generated by the LFSR is used to encrypt plaintext by performing an XOR operation between the key stream and the plaintext. The security of the stream cipher depends on the properties of the key stream, which are determined by the LFSR and the feedback polynomial.

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