Question

Fibonacci LFSR is described on wiki, it’s pretty simple.

I’d like to calucate the period of some Fibonacci’s LFSR and use generated sequence for ciphering later.

Let’s take and example from wiki:

x¹⁶ + x¹⁴ + x¹³ + x¹¹ + 1;

//code from wiki:
include <stdint.h>
uint16_t lfsr = 0xACE1u;
unsigned bit;
unsigned period = 0;
do {
  /* taps: 16 14 13 11; characteristic polynomial: x^16 + x^14 + x^13 + x^11 + 1 */
  bit  = ((lfsr >> 0) ^ (lfsr >> 2) ^ (lfsr >> 3) ^ (lfsr >> 5) ) & 1;
  lfsr =  (lfsr >> 1) | (bit << 15);
  ++period;
} while(lfsr != 0xACE1u);

My weakly try so far in php:

function getPeriod(){
    $polynoms = array(16, 14, 13, 11);
    $input = $polynoms[0] - 1;
    $n = sizeof($polynoms);
    for ($i = 1; $i < $n; $i++)
        $polynoms[$i] = $polynoms[0] - $polynoms[$i];
    $polynoms[0] = 0;
    //reversed polynoms == array(0, 2, 3, 5);
    $lfsr = 0x1; //begining state       
    $period = 0;
    //gmp -- php library for long numbers;
    $lfsr = gmp_init($lfsr, 16);
    do {            
        $bit = $lfsr; //bit = x^16 >> 0;

        for($i = 1; $i < $n; $i++) {
            //bit ^= lfsr >> 2 ^ lfst >> 3 ^ lfst >> 5;
            $bit = gmp_xor($bit, ( gmp_div_q($lfsr, gmp_pow(2, $polynoms[$i])) ));                  
        }
        //bit &= 1;
        $bit = gmp_and($bit, 1);            
        //lfsr = $lfsr >> 1 | $lsfr << (16 - 1);
        $lfsr = gmp_or( (gmp_div_q($lfsr, 2)), (gmp_mul($bit, gmp_pow(2, $input))) );

        $period++;
    } while (gmp_cmp($lfsr, 0x1) != 0);
    echo '<br />period = '.$period;
    //period == 65535 == 2^16 - 1; -- and that's correct;
            //                        I hope, at least;
    return $period;
}

Problem:

If i try to modulate work of i.e.

x³²¹ + x¹⁴ + x¹³ + x¹¹ + 1;

i got an error:”Fatal error: Maximum execution time of 30 seconds exceeded in /var/www/Dx02/test.php“;

Can i somehow optimize (accelerate 🙂 ) the calculation?

Any help is appreciated. Thank you and excuse me for my English.

Answer 1

You simply can’t do it this way with a polynomial like x^321 + …;

If the polynomial is chosen well, you get a period length of 2^231 -1,
and this is approximately 4.27 *10^96. If I’m not mistaken, this number is
believed to exceed the number of atoms in the universe…
(Strictly speaking, I’m referring to the posted C-code since I do not know php, but that certainly makes no difference.)

However, there is a mathematical method to calculate the length of the period without doing a brute-force attack. Unfortunately, this can’t be explained in a few lines. If you have a solid background in math (especially calculations in finite fields), I’ll be glad to look for a helpful reference for you.

EDIT:
The first step in calculating the period of the LFSR obtained by using a polynomial p(x) is to obtain a factorization of p(x) mod 2, i.e. in GF(2). To do this, I recommend using software like Mathematica or Maple if available. You could also try the freely available Sage, see e.g. http://www.sagemath.org/doc/constructions/polynomials.html for usage details.

The period of p(x) is given by its order e, that means the smallest number such that p(x) divedes x^e+1. Unfortunately, I can’t provide more information at the moment, it will take me several days to look for the lecture notes of a course I took several years ago…

A small example: p(x) = (x^5+x^4+1) = (x^3+x+1)*(x^2+x+1), the individual periods are 2^3-1=7 and 2^2-1=3, and since all polynomial factors are different, the period of p(x) is 3*7=21, which I also verified in C++.