Likelihood that a pseudorandom sequence generator has optimal properties

The authors prove that the probability of choosing a nonlinear filter of m-sequences with optimal properties, that is, maximum period and maximum linear complexity, tends assymptotically to 1 as the linear feedback shift register length increases.

💡 Research Summary

The paper investigates the likelihood that a nonlinear filter applied to an m‑sequence generated by a linear feedback shift register (LFSR) will simultaneously exhibit the two most desirable properties for pseudorandom sequence generators: maximal period and maximal linear complexity. The authors start by modeling the LFSR output as a sequence of field elements in GF(2^L), where L is the register length and the feedback polynomial is primitive. A k‑th order nonlinear Boolean function F is then applied to a sliding window of L consecutive LFSR bits, producing a filtered output sequence {z_n}. By expanding F in terms of cosets of the underlying finite field, the filtered sequence can be expressed as a weighted sum of characteristic sequences S_{E_i,n} associated with each coset E_i, with coefficients C_i belonging to GF(2^{r_i}), where r_i is the cardinality of the coset. The key observations are: (i) each coefficient C_i lies in its appropriate subfield, (ii) a zero coefficient eliminates the corresponding coset from contributing to linear complexity, and (iii) the period of the filtered sequence is the least common multiple of the periods of the constituent characteristic sequences, which are divisors of 2^L−1.

Using these structural facts, the authors derive a closed‑form expression for the probability P_r that a randomly chosen k‑order nonlinear filter yields a sequence with maximal linear complexity. Let nf_k denote the total number of k‑order nonlinear filters and nf_m the number of those whose output has maximal linear complexity (i.e., all C_i ≠ 0). The probability is P_r = nf_m / nf_k = ∏{i=1}^{N} (2^{r_i}−2) / ∏{i=1}^{N} (2^{Lk_i}−1), where N is the number of relevant cosets and Lk_i are the exponents associated with each coset. When L is prime—a common situation—all r_i equal L, simplifying the expression to P_r = (2^L−1)^N / ∏_{i=1}^{N} (2^{Lk_i}−1).

Applying the limit (1−b^{-1}n)^b → e^{−1} as b → ∞, the authors obtain a lower bound P_r > e^{−N_k/2^L}, where N_k is the number of possible k‑order nonlinear functions. For k ≈ L/2, N_k ≈ 2^{L−1}, yielding P_r > e^{−1/2^L}. As L grows, e^{−1/2^L} approaches 1, implying that the probability of randomly selecting an optimal filter converges to certainty. Numerical evaluation for a typical communication‑system register length L = 25 gives P_r > 0.998, confirming the theoretical prediction for practical parameters.

The paper also argues that the maximal period property follows automatically because the filter includes the characteristic sequence of the coset whose period is 2^L−1; thus the filtered sequence inherits at least this period.

In conclusion, the authors demonstrate that nonlinear filters of m‑sequences are not only easy to implement in high‑speed hardware but also overwhelmingly likely to produce sequences with both maximal period and maximal linear complexity. This makes them highly attractive for a wide range of applications, including stream ciphers, radar, Monte‑Carlo simulations, error‑correcting codes, and spread‑spectrum communications.