class sdr.GLFSR

Implements a Galois linear-feedback shift register (LFSR).

Notes

A Galois LFSR is defined by its feedback polynomial \(f(x)\).

\[ f(x) = -c_{0} \cdot x^{n} - c_{1} \cdot x^{n-1} - \dots - c_{n-2} \cdot x^{2} - c_{n-1} \cdot x + 1 = x^n \cdot c(x^{-1}) \]

The feedback polynomial is the reciprocal of the characteristic polynomial \(c(x)\) of the linear recurrent sequence \(y\) produced by the Galois LFSR.

\[c(x) = x^{n} - c_{n-1} \cdot x^{n-1} - c_{n-2} \cdot x^{n-2} - \dots - c_{1} \cdot x - c_{0}\]

\[y_t = c_{n-1} \cdot y_{t-1} + c_{n-2} \cdot y_{t-2} + \dots + c_{1} \cdot y_{t-n+2} + c_{0} \cdot y_{t-n+1}\]

Galois LFSR Configuration
 +--------------+<-------------+<-------------+<-------------+
 |              |              |              |              |
 | c_0          | c_1          | c_2          | c_n-1        |
 | T[0]         | T[1]         | T[2]         | T[n-1]       |
 |              |              |              |              |
 |  +--------+  v  +--------+  v              v  +--------+  |
 +->|  S[0]  |--@->|  S[1]  |--@---  ...   ---@->| S[n-1] |--+--> y[t]
    +--------+     +--------+                    +--------+
      y[t-1]                                       y[t+1]

 S[k] = State vector
 T[k] = Taps vector
 y[t] = Output sequence
 @ = Finite field adder

The shift register taps \(T\) are defined left-to-right as \(T = [T_0, T_1, \dots, T_{n-2}, T_{n-1}]\). The state vector \(S\) is also defined left-to-right as \(S = [S_0, S_1, \dots, S_{n-2}, S_{n-1}]\).

In the Galois configuration, the shift register taps are \(T = [c_0, c_1, \dots, c_{n-2}, c_{n-1}]\).

References

See also

berlekamp_massey

Examples

Create a Galois LFSR from a degree-4 primitive characteristic polynomial over \(\mathrm{GF}(2)\).

In [1]: c = galois.primitive_poly(2, 4); c
Out[1]: Poly(x^4 + x + 1, GF(2))

In [2]: lfsr = sdr.GLFSR(c)

In [3]: print(lfsr)
Galois LFSR:
  field: GF(2)
  characteristic_poly: x^4 + x + 1
  feedback_poly: x^4 + x^3 + 1
  taps: [1 1 0 0]
  order: 4
  state: [1 1 1 1]
  initial_state: [1 1 1 1]

Step the Galois LFSR and produce 10 output symbols.

In [4]: lfsr.state
Out[4]: GF([1, 1, 1, 1], order=2)

In [5]: lfsr.step(10)
Out[5]: GF([1, 1, 1, 0, 0, 0, 1, 0, 0, 1], order=2)

In [6]: lfsr.state
Out[6]: GF([1, 1, 0, 1], order=2)

Create a Galois LFSR from a degree-4 primitive characteristic polynomial over \(\mathrm{GF}(7)\).

In [7]: c = galois.primitive_poly(7, 4); c
Out[7]: Poly(x^4 + x^2 + 3x + 5, GF(7))

In [8]: lfsr = sdr.GLFSR(c)

In [9]: print(lfsr)
Galois LFSR:
  field: GF(7)
  characteristic_poly: x^4 + x^2 + 3x + 5
  feedback_poly: 5x^4 + 3x^3 + x^2 + 1
  taps: [2 4 6 0]
  order: 4
  state: [1 1 1 1]
  initial_state: [1 1 1 1]

Step the Galois LFSR and produce 10 output symbols.

In [10]: lfsr.state
Out[10]: GF([1, 1, 1, 1], order=7)

In [11]: lfsr.step(10)
Out[11]: GF([1, 1, 0, 4, 6, 5, 3, 6, 1, 2], order=7)

In [12]: lfsr.state
Out[12]: GF([4, 3, 0, 1], order=7)

Create a Galois LFSR from a degree-4 primitive characteristic polynomial over \(\mathrm{GF}(2^3)\).

In [13]: c = galois.primitive_poly(2**3, 4); c
Out[13]: Poly(x^4 + x + 3, GF(2^3))

In [14]: lfsr = sdr.GLFSR(c)

In [15]: print(lfsr)
Galois LFSR:
  field: GF(2^3)
  characteristic_poly: x^4 + x + 3
  feedback_poly: 3x^4 + x^3 + 1
  taps: [3 1 0 0]
  order: 4
  state: [1 1 1 1]
  initial_state: [1 1 1 1]

Step the Galois LFSR and produce 10 output symbols.

In [16]: lfsr.state
Out[16]: GF([1, 1, 1, 1], order=2^3)

In [17]: lfsr.step(10)
Out[17]: GF([1, 1, 1, 0, 2, 2, 3, 2, 4, 5], order=2^3)

In [18]: lfsr.state
Out[18]: GF([4, 2, 2, 7], order=2^3)

Create a Galois LFSR from a degree-4 primitive characteristic polynomial over \(\mathrm{GF}(3^3)\).

In [19]: c = galois.primitive_poly(3**3, 4); c
Out[19]: Poly(x^4 + x + 10, GF(3^3))

In [20]: lfsr = sdr.GLFSR(c)

In [21]: print(lfsr)
Galois LFSR:
  field: GF(3^3)
  characteristic_poly: x^4 + x + 10
  feedback_poly: 10x^4 + x^3 + 1
  taps: [20  2  0  0]
  order: 4
  state: [1 1 1 1]
  initial_state: [1 1 1 1]

Step the Galois LFSR and produce 10 output symbols.

In [22]: lfsr.state
Out[22]: GF([1, 1, 1, 1], order=3^3)

In [23]: lfsr.step(10)
Out[23]: GF([ 1,  1,  1,  0, 19, 19, 20, 11, 25, 24], order=3^3)

In [24]: lfsr.state
Out[24]: GF([23, 25,  6, 26], order=3^3)

Constructors

GLFSR(characteristic_poly: PolyLike | None = None, ...)

Creates a new Galois LFSR.

classmethod Taps(taps: FieldArray, ...) → Self

Creates a Galois LFSR from its taps.

Methods

reset(state: ArrayLike | None = None)

Resets the Galois LFSR state to the specified state.

step(steps: int = 1) → FieldArray

Produces the next steps output symbols.

to_fibonacci_lfsr() → FLFSR

Converts the Galois LFSR to a Fibonacci LFSR that produces the same output.

Properties

property field : type[FieldArray]

The FieldArray subclass for the finite field that defines the linear arithmetic.

property taps : FieldArray

The shift register taps \(T = [c_0, c_1, \dots, c_{n-2}, c_{n-1}]\).

property order : int

The order of the linear recurrence/linear recurrent sequence.

Polynomials

property characteristic_poly : Poly

The characteristic polynomial \(c(x)\) that defines the linear recurrent sequence.

property feedback_poly : Poly

The feedback polynomial \(f(x)\) that defines the feedback arithmetic.

State

property initial_state : FieldArray

The initial state vector \(S = [S_0, S_1, \dots, S_{n-2}, S_{n-1}]\).

property state : FieldArray

The current state vector \(S = [S_0, S_1, \dots, S_{n-2}, S_{n-1}]\).