class galois.GLFSR

A Galois linear-feedback shift register (LFSR).

Notes

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

f(x)=−c0xn−c1xn−1−⋯−cn−2x2−cn−1x+1=xnc(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)=xn−cn−1xn−1−cn−2xn−2−⋯−c1x−c0

yt=cn−1yt−1+cn−2yt−2+⋯+c1yt−n+2+c0yt−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]

The shift register taps T are defined left-to-right as T=[T0,T1,…,Tn−2,Tn−1]. The state vector S is also defined left-to-right as S=[S0,S1,…,Sn−2,Sn−1].

In the Galois configuration, the shift register taps are T=[c0,c1,…,cn−2,cn−1].

References

See also

berlekamp_massey

Examples

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

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

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

In [3]: print(lfsr)
Galois LFSR:
  field: GF(2)
  feedback_poly: x^4 + x^3 + 1
  characteristic_poly: x^4 + x + 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 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 = galois.GLFSR(c.reverse())

In [9]: print(lfsr)
Galois LFSR:
  field: GF(7)
  feedback_poly: 5x^4 + 3x^3 + x^2 + 1
  characteristic_poly: x^4 + x^2 + 3x + 5
  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 GF(23).

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

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

In [15]: print(lfsr)
Galois LFSR:
  field: GF(2^3)
  feedback_poly: 3x^4 + x^3 + 1
  characteristic_poly: x^4 + x + 3
  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 GF(33).

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

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

In [21]: print(lfsr)
Galois LFSR:
  field: GF(3^3)
  feedback_poly: 10x^4 + x^3 + 1
  characteristic_poly: x^4 + x + 10
  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(feedback_poly: Poly, state: ArrayLike | None = None)

Constructs a Galois LFSR from its feedback polynomial f(x).

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

Constructs a Galois LFSR from its taps T=[c0,c1,…,cn−2,cn−1].

String representation

__repr__() → str

A terse representation of the Galois LFSR.

__str__() → str

A formatted string of relevant properties of the Galois LFSR.

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[galois._fields._array.FieldArray]

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

property order : int

The order of the linear recurrence/linear recurrent sequence. The order of a sequence is defined by the degree of the minimal polynomial that produces it.

property taps : FieldArray

The shift register taps T=[c0,c1,…,cn−2,cn−1]. The taps of the shift register define the linear recurrence relation.

Polynomials

property characteristic_poly : Poly

The characteristic polynomial c(x)=xn−cn−1xn−1−cn−2xn−2−⋯−c1x−c0 that defines the linear recurrent sequence.

property feedback_poly : Poly

The feedback polynomial f(x)=−c0xn−c1xn−1−⋯−cn−2x2−cn−1x+1 that defines the feedback arithmetic.

State

property initial_state : FieldArray

The initial state vector S=[S0,S1,…,Sn−2,Sn−1].

property state : FieldArray

The current state vector S=[S0,S1,…,Sn−2,Sn−1].