Is it possible to reverse the sequence of an LFSR if the GF(2) tap polynomial has the x^0 term set to zero?

Question

While exploring how to generate maximum period forward and reverse sequences using an LFSR, I ran into some confusion regarding the definition of a GF(2) primitive polynomial.

When I chose a primitive GF(2) polynomial from this table, all polynomials have the x^0 term set to 1 and everything works as expected (see below).

#include <cstdio>
#include <cstdint>

uint8_t rev (uint8_t xi)
{
  uint8_t xo ;

  for (auto n = 8 ; n ; -- n)
    {
      xo = xo << 1 | xi & 1 ;

      xi >>= 1 ;
    }

  return xo ;
}

int main (void)
{
  uint8_t const POLY_F = 0x85 ; // x^8 + x^7 + x^2 + 1 : Normal
  uint8_t const POLY_R = 0x43 ; // x^8 + x^6 + x   + 1 : Reversed

  uint8_t lfsr = 0xFF ;

  for (auto n = 1 ; n <= 5 ; ++ n)                    // 1 -> FF
    {                                                 // 2 -> 7B
      printf ("%u -> %02X\n" , n , lfsr) ;            // 3 -> F6
                                                      // 4 -> 69
      lfsr = - (lfsr >> 7) & POLY_F ^ lfsr << 1 ;     // 5 -> D2
    }

  printf ("\n") ;

  for (auto n = 5 ; n >= 1 ; -- n)                    // 5 -> D2
    {                                                 // 4 -> 69
      lfsr = rev (lfsr) ;                             // 3 -> F6
                                                      // 2 -> 7B
      lfsr = - (lfsr >> 7) & POLY_R ^ lfsr << 1 ;     // 1 -> FF

      lfsr = rev (lfsr) ;

      printf ("%u -> %02X\n" , n ,lfsr) ;
    }
}

When I generate a maximal length GF(2) polynomial using this program, it sometimes outputs a polynomial with the x^0 term set to 0.

I cannot figure out how to reverse the sequence when the x^0 term is 0.

Is a “maximal length” polynomial the same thing as a “primitive polynomial”?

Can a primitive polynomial have the x^0 term set to 0?

Is it possible to reverse a sequence if the x^0 term is 0?

Answer 1

Code has at least these problems:

Uninitialized value

it sometimes outputs a polynomial with the x^0 term set to 0.

The first time xo << 1 | xi & 1 ; is executed, xo is not initialize nor assigned.

Instead:

// uint8_t xo ;
uint8_t xo = 0;

Language?

Post is tagged C yet uses #include <cstdio>. Best to use a C compiler for C code.