3
$\begingroup$

I am reading a paper and say this

"The idea is to load $f(X)$ into LFSR to multiply by $X$ mod $g(X)$(primitive polynomial deg $g=n$). We next compute a polynomial h(X) whose coefficients are given by successive values of a particular cell of register".

and say "$h(Y)=\sum_{i=0}^{n-1}{a_iY^i}$, where $a_i$ is a coefficient of $X^{n-1}$ in $X^if(X)$ mod $g(X)$"

My question, Please help me How design this LFSR? and The paper too say :

$f(X)=(\sum_{i=0}^{n}{a_iX^{-(i+1)}}+\sum_{i=n}^{\infty}{b_iX^{-(i+1)}})g(X)$,

Why this last expression?

  • 1
    Do you want to know how to _design_ the LFSR, or how to compute the coefficients of the polynomial $h(X)$? Also, please edit your question a little. Did you write "$x \bmod g(X)$" instead of $X \bmod g(X)$"? Is $h(X)$ a polynomial in $X$ as the name suggests, or is it a polynomial in $y$ as $\sum a_i y^i$ seems to indicate?2011-11-23
  • 0
    @Dilip Sarwate thanks by your response, I edit the question, ... with your suggestions, I like too a design to general case to this problem please ....2011-11-24
  • 0
    @Dilip Sarwate thanks by your response, I edit the question, ... with your suggestions, I like too a design to general case to this problem please ...., about your response F(X) = f(X)?, where are $a_i$ values?2011-11-24
  • 0
    Hi thanks by your response,here a an example For example: $K_2[a] = GF(2^4),F[X] = {\rm Polynomial Ring}(K_2[a])$ Give: $g(X) = (X + 1) * (X + a^2 + a + 1) * (X + a^3 + a + 1) * (X + a^3 + a^2)$ $\eqalign{f(X) &= a^2*X^3 + (a^3 + a)*X^2 + (a^3 + 1)*X\cr}$2011-11-23
  • 0
    What's the minimal polynomial of $a$? I will return to this later, but will be off-air for a few hours. If this is urgent, hopefully somebody else can answer.2011-11-23
  • 0
    Hi Jyrki, the minimal polynomial is $x^4 + x + 1$2011-11-23
  • 0
    @Juan $x^4 + x + 1$ is _not_ irreducible over $GF(2^4)$. Again I ask you, _please_ edit your question so that it is less ambiguous.2011-11-24
  • 0
    Hi Dilip thanks by your response, I test if $x^4+x+1$, is irreducible in SAGE, that program say yes is irreducible, ... you can me which factor have $x^4+x+1$ ?2011-11-24
  • 0
    @Juan: Over $GF(16)$ we have the factorization $$x^4+x+1=(x+a)(x+a^2)(x+a^4)(x+a^8)=(x+a)(x+a^2)(x+a+1)(x+a^2+1).$$ This polynomial is irreducible over $GF(2)$ though. Your $g(x)$ has $1$, $a^{10}$, $a^7$ and $a^6$ as zeros, so I'm a bit curious as to why are you doing this? For example, it doesn't look like a generator polynomial of an RS-code. Meanwhile, Dilip has given a good answer. I seem to have missed the boat more or less :-)2011-11-24
  • 0
    @Jyrki: Is this what you had in mind?2011-11-24
  • 0
    sorry the g(X) = X^4 + (a + 1)*X^3 + (a^3 + a)*X^2 + a*X + 1 and R= (a^3 + a^2 + 1)*X + a^2 + a + 1 please I need a design for this LFSR2011-11-24

1 Answers 1