Determining Whether a Characteristic Polynomial Can Generate an m-Sequence

An m-sequence is a type of pseudorandom sequence widely used in communications and cryptography, with a maximum length of 2^n - 1 (where n is the number of shift register stages). The key to determining whether a characteristic polynomial can generate an m-sequence lies in verifying if the polynomial is primitive.

The step-by-step verification process for characteristic polynomials to generate m-sequences includes: Irreducibility check: First, confirm that the polynomial is irreducible over GF(2), meaning it cannot be factored into lower-degree polynomials. In code implementation, this can be tested using factorization algorithms like Berlekamp's algorithm or through trial division with irreducible polynomials of lower degrees. Order verification: Ensure the polynomial's order equals 2^n - 1, where n is the polynomial degree. This guarantees the generated sequence can achieve maximum length. The order can be computed by finding the smallest integer k such that the polynomial divides x^k - 1. Primitivity test: Verify that the polynomial can generate a maximum-length sequence by checking whether the minimal k value for which the polynomial divides x^k - 1 is exactly 2^n - 1. This typically involves modular polynomial exponentiation computations in GF(2).

Modeling verification process: Construct the corresponding linear feedback shift register (LFSR) structure where feedback connections are determined by the characteristic polynomial coefficients. In implementation, this can be represented using a state vector and feedback logic based on polynomial coefficients. Initialize the register to a non-all-zero state. Run sufficient clock cycles while monitoring the output sequence's periodicity. A typical implementation would track state repetition using cycle detection algorithms. Check whether the sequence period reaches the theoretical maximum of 2^n - 1. Additionally, perform statistical tests to verify the sequence's pseudorandom properties, such as autocorrelation tests or frequency tests commonly used in cryptography.

In practical applications, precomputed tables of known primitive polynomials are often stored and referenced, as the computational complexity of primitivity testing is significant. For a given n value, multiple primitive polynomials may exist, each capable of generating distinct m-sequences with different autocorrelation properties.