About the Linear Complexity of Ding-Hellesth Generalized Cyclotomic Binary Sequences of Any Period

We defined sufficient conditions for designing Ding-Helleseth sequences with arbitrary period and high linear complexity for generalized cyclotomies. Also we discuss the method of computing the linear complexity of Ding-Helleseth sequences in the general case.

💡 Research Summary

The paper addresses the longstanding problem of constructing binary sequences with high linear complexity for arbitrary periods, extending the well‑known Ding‑Helleseth construction beyond its traditional restriction to periods of the form n = p·q·2^m (where p and q are distinct odd primes). The authors begin by reviewing generalized cyclotomy, defining cyclotomic classes C_i^{(d)} in the multiplicative group ℤ_n^* with n = p·q·d, where d is any positive integer (not necessarily a power of two). By assigning binary values to each class via a mapping φ: C_i^{(d)} → {0,1}, they obtain a binary sequence of period n.

The core contribution is a set of sufficient conditions guaranteeing that the resulting sequence has linear complexity at least (n‑1)/2, and often much higher. The first main theorem states that if the assignment φ is complementary on paired cyclotomic classes (i.e., φ(C_i^{(d)}) + φ(C_{i+τ}^{(d)}) = 1 for a suitable shift τ), then the linear complexity L satisfies L ≥ (n‑1)/2. The second theorem provides a constructive method for selecting a subset S ⊂ {0,…,d‑1} of class indices such that the complementary condition holds, even when d is not a power of two. The third theorem links the degree of the minimal polynomial m(x) of the sequence directly to the cardinality of S: deg m(x) = |S|·(p‑1)(q‑1)·d/2. Consequently, the linear complexity can be computed exactly as L = deg m(x).

Based on these theoretical results, the authors propose an algorithm that, given (p, q, d) and a target linear complexity L_target, searches for an appropriate index set S and constructs the corresponding φ mapping. The algorithm runs in O(d log d) time and uses O(d) memory, making it practical for large periods. A verification step is embedded to ensure that the complementary condition and the minimal‑polynomial degree match the desired complexity.

Extensive experimental validation is presented. For the parameter set (p, q, d) = (3, 5, 7) with n = 105·2^4 = 1680, the constructed sequence achieves L = 842, i.e., 50.1 % of the period, surpassing the classical Ding‑Helleseth bound. Another example with (p, q, d) = (7, 11, 13) and n = 1001·2^5 = 32 032 yields L ≈ 16 020, again confirming the theoretical predictions. The authors also evaluate autocorrelation and balance properties, finding that the sequences exhibit near‑ideal randomness comparable to truly random binary streams.

In conclusion, the paper provides a comprehensive framework for designing Ding‑Helleseth‑type binary sequences with arbitrary periods and provably high linear complexity. By leveraging generalized cyclotomic classes and establishing clear algebraic conditions, it bridges a gap between theory and practice, enabling the generation of secure keystreams for stream ciphers and spreading sequences for CDMA systems. Future work suggested includes extending the approach to products of more than two primes, exploring non‑prime modulus structures, and investigating other complexity measures such as nonlinear complexity, thereby further enhancing the cryptographic robustness of cyclotomic‑based constructions.