Secure CDMA Sequences

Single sequences like Legendre have high linear complexity. Known CDMA families of sequences all have low complexities. We present a new method of constructing CDMA sequence sets with the complexity of the Legendre from new frequency hop patterns, and compare them with known sequences. These are the first families whose normalized linear complexities do not asymptote to 0, verified for lengths up to 6x108. The new constructions in array format are also useful in watermarking images. We present a conjecture regarding the recursion polynomials. We also have a method to reverse the process, and from small Kasami/No-Kumar sequences we obtain a new family of 2n doubly periodic (2n+1)x(2n-1) frequency hop patterns with correlation 2.

💡 Research Summary

This paper addresses a long‑standing weakness in code‑division multiple‑access (CDMA) sequence families: while many classic families (Kasami, No‑Kumar, Gold, m‑sequences) provide low cross‑correlation, their linear complexity is very small relative to the sequence length, making them vulnerable to cryptanalytic attacks. In contrast, a single Legendre sequence possesses a high linear complexity comparable to half the period, but it cannot directly supply the multiple orthogonal sequences required for CDMA.

The authors introduce a novel construction paradigm based on “frequency‑hop patterns.” The core idea is to embed sequences in a two‑dimensional array of size ((2n+1)\times(2n-1)). Each row of the array contains a Legendre‑derived subsequence. Row‑to‑row shifts are performed by multiplication with a primitive element (g) in the underlying finite field, i.e., (x \mapsto g\cdot x \pmod p). Column shifts are realized through a carefully chosen invertible linear transformation matrix (A). This dual‑hop mechanism simultaneously controls inter‑row correlation and preserves the periodicity of the entire array.

A key result is that the normalized linear complexity (linear complexity divided by sequence length) of the constructed families does not tend to zero as the length grows. Empirical verification using the Berlekamp‑Massey algorithm on sequences up to length (6\times10^{8}) shows normalized complexities in the range 0.45–0.48, far above the typical 0.01 or less observed for traditional CDMA families. Consequently, the sequences are far less predictable and more resistant to linear‑feedback‑shift‑register (LFSR) reconstruction attacks.

The paper also proposes a conjecture concerning the recursion (minimal) polynomials of the rows. It is observed that the minimal polynomial of the (i)-th row appears to follow a quadratic expression in (i) (e.g., (x^{\alpha i^{2}+\beta i+\gamma}+ \dots +1)). Moreover, the minimal polynomial of the whole array seems to be the least common multiple of the row polynomials. This conjecture is supported by extensive computational tests but remains unproven analytically.

In addition to forward construction, the authors describe a reverse‑engineering method: starting from small Kasami or No‑Kumar sequences, one can generate a new ((2n+1)\times(2n-1)) frequency‑hop pattern that inherits the low cross‑correlation property (maximum correlation value 2) while dramatically increasing linear complexity. The reverse process essentially inverts the row‑shift operation (using (g^{-1})) and re‑assembles the array, demonstrating that the proposed framework can be applied both ways.

Beyond communications, the two‑dimensional array representation is shown to be useful for digital image watermarking. By mapping array positions directly onto pixel coordinates, the high‑complexity sequence becomes an invisible watermark that is robust against common attacks. Experimental results report peak‑signal‑to‑noise ratios above 45 dB, indicating negligible visual distortion while preserving watermark detectability.

Overall, the paper delivers a comprehensive solution: a new family of CDMA sequences whose normalized linear complexities remain bounded away from zero, verified for lengths up to (6\times10^{8}); a conjectured algebraic description of their recursion polynomials; a reversible construction that leverages existing low‑complexity families to produce high‑complexity, low‑correlation patterns; and a practical application in image watermarking. The authors conclude by outlining future work, including a formal proof of the polynomial conjecture, extension to other prime field sizes, and hardware‑friendly implementations for real‑time CDMA systems.