[Paper Review] Some New Results on the Cross Correlation of $m$-sequences

Tao Zhang, Shuxing Li+2 more|arXiv (Cornell University)|Sep 30, 2013

Coding theory and cryptography20 references4 citations

TL;DR

This paper determines the cross-correlation distribution of ternary $m$-sequences of period $3^{3r}-1$ with decimations $d = 3^r + 2$ and $d = 3^{2r} + 2$ under the condition $\gcd(r,3) = 1$, and investigates the cross-correlation properties of binary $m$-sequences of period $2^{2lm}-1$ with decimation $d = \frac{2^{2lm}-1}{2^m+1} + 2^s$, proving it takes at least four values and confirming two major conjectures by Sarwate et al. and Helleseth.

ABSTRACT

The determination of the cross correlation between an $m$-sequence and its decimated sequence has been a long-standing research problem. Considering a ternary $m$-sequence of period $3^{3r}-1$, we determine the cross correlation distribution for decimations $d=3^{r}+2$ and $d=3^{2r}+2$, where $\gcd(r,3)=1$. Meanwhile, for a binary $m$-sequence of period $2^{2lm}-1$, we make an initial investigation for the decimation $d=\frac{2^{2lm}-1}{2^{m}+1}+2^{s}$, where $l \ge 2$ is even and $0 \le s \le 2m-1$. It is shown that the cross correlation takes at least four values. Furthermore, we confirm the validity of two famous conjectures due to Sarwate et al. and Helleseth in this case.

Motivation & Objective

• To determine the cross-correlation distribution of ternary $m$-sequences with specific decimations $d = 3^r + 2$ and $d = 3^{2r} + 2$ for $\gcd(r,3) = 1$.
• To investigate the cross-correlation properties of binary $m$-sequences with period $2^{2lm}-1$ and decimation $d = \frac{2^{2lm}-1}{2^m+1} + 2^s$, where $l \geq 2$ is even and $0 \leq s \leq 2m-1$.
• To confirm the validity of two long-standing conjectures—by Sarwate et al. and Helleseth—within this new decimation framework.
• To establish that the cross-correlation function takes at least four distinct values in the binary case, advancing beyond known three-value bounds.

Proposed method

• The cross-correlation function is analyzed via Weil sums, expressed as $ C_d(z) = \sum_{x \in \text{GF}(p^n)^*} \chi(zx - x^d) $, where $ \chi(x) = \omega^{\text{Tr}(x)} $.
• For the ternary case, the authors use algebraic techniques inspired by Dobbertin and Feng et al., leveraging the structure of finite fields and character sums.
• For the binary case, the method involves decomposing the multiplicative group $ \text{GF}(2^{4m})^* $ into cosets of subgroups $ C_0 $ and $ D_0 $, and analyzing the distribution of $ zx + x^{d2^{-s}} $ across these cosets.
• The analysis uses the number of solutions to equations like $ (zx + x^{d2^{-s}})^{(2^{4m}-1)/(2^m+1)} = 1 $, relating them to the cross-correlation values.
• The authors define $ n_i(z) $ as the number of $ x $ such that $ zx + x^{d2^{-s}} \in C_i $, and use these to compute the sum $ S_d(z) $, which corresponds to the cross-correlation value.
• The proof relies on counting solutions under group action and using properties of cyclotomic cosets and Dickson polynomials in the binary case.

Experimental results

Research questions

• RQ1 What is the exact cross-correlation distribution for a ternary $m$-sequence of period $3^{3r}-1$ under decimation $d = 3^r + 2$ with $\gcd(r,3) = 1$?
• RQ2 Does the cross-correlation of a binary $m$-sequence of period $2^{2lm}-1$ with decimation $d = \frac{2^{2lm}-1}{2^m+1} + 2^s$ take at least four distinct values?
• RQ3 Are the conjectures by Sarwate et al. and Helleseth valid for the decimation $d = \frac{2^{2lm}-1}{2^m+1} + 2^s$ with even $l \geq 2$?
• RQ4 Can the cross-correlation distribution be fully determined in the binary case, or does it take more than four values?

Key findings

• The cross-correlation distribution for ternary $m$-sequences with $d = 3^r + 2$ and $d = 3^{2r} + 2$ is completely determined under $\gcd(r,3) = 1$, with the distribution matching that of the $\gcd(r,3) = 1$ case.
• For the binary $m$-sequence with $d = \frac{2^{2lm}-1}{2^m+1} + 2^s$, the cross-correlation function takes at least four distinct values, confirming a lower bound beyond the known three-value minimum.
• The cross-correlation sum $ S_d(z) $ takes at least two distinct positive values, and one of them is at least $ 2^{2m+1} $, indicating non-trivial distribution.
• When $ u = (d2^{-s} + 1, 2^m + 1) = 2^m + 1 $, $ S_d(z) = 2^{3m} $ for $ z \neq 1 $, and $ S_d(1) = 0 $, showing a clear separation of values.
• The authors confirm that the two famous conjectures by Sarwate et al. and Helleseth hold true for this decimation, despite the complexity of the distribution.
• Numerical experiments suggest the cross-correlation may take eight or more values, indicating that a full distribution determination remains a challenging open problem.