2

I want to see and mathematically verify how pseudonoise (PN) sequence generated by LFSR or maximum length codes would correlate (auto-correlation and cross correlation) for seeds of varying correlation. Say, I want to generate $512$ bits of PN sequence on two sides ($A$ and $B$). To hide the possible output from eavesdropper, I choose some seeds (which can be generated by both $A$ and $B$ simultaneously). The issue is the seeds (sequence of $512$ bits) generated at various instants of time are correlated with previous time instants.

I want to investigate how correlation in the input seeds affects correlation in output. Is there any paper or reference literature survey for the same?

My research efforts

As of now, I am studying the properties of PN sequence generators. In my research, I have a source that generates correlated sequences of various degrees. I want to analyze what happens to the PN sequence generated using those seeds.

kelalaka
  • 49,797
  • 12
  • 123
  • 211
Jay
  • 195
  • 11

1 Answers1

3

Caveat: LFSRs outputs are vulnerable to known plaintext attacks using the Berlekamp Massey algorithm.

Given a maximal length sequence $s(t)$ of period $2^n-1$ generated by a primitive LFSR, all its windows of length $\ell \geq n$ are unique so partial period correlation of such lengths can uniquely determine the phase, thus the seed.

All primitive LFSR sequences of this length are obtainable by regular decimations of this sequence, i.e., $s(dt)$ where $gcd(2^n-1,d)=1.$

So this is an exploitable correlation.

Another relevant property is the shift and add property

$$s(t+k)\oplus s(t)=s(t+z(k))$$

that comes from the fact that the maximal length sequence and its cyclic shifts together with the all zero vector form a linear code. This is another source of correlation.Here $$a^{z(k)} =1+ a^k$$ defines the so called Zech logarithm $z(\cdot)$, if we use the trace representation $$s(t)=Tr(a^t)$$ with $a \in GF(2^n).$

The book by Golomb and Gong and a survey by Sarwate and Pursley in the Proceedings of the IEEE are good places to start.

kodlu
  • 25,146
  • 2
  • 30
  • 63