class galois.GLFSR

A Galois linear-feedback shift register (LFSR).

Notes

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

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

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

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

$y_t = c_{n-1}y_{t-1} + c_{n-2}y_{t-2} + \dots + c_{1}y_{t-n+2} + c_{0}y_{t-n+1}$

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]


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

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 = 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 $$\mathrm{GF}(7)$$.

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

In [8]: lfsr = 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 $$\mathrm{GF}(2^3)$$.

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

In [14]: lfsr = 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 $$\mathrm{GF}(3^3)$$.

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

In [20]: lfsr = 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(state: = None)

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

classmethod Taps(...) Self

Constructs a Galois LFSR from its taps $$T = [c_0, c_1, \dots, c_{n-2}, c_{n-1}]$$.

## 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: = None)

Resets the Galois LFSR state to the specified state.

step(steps: int = 1)

Produces the next steps output symbols.

to_fibonacci_lfsr()

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

## Properties¶

property field :

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 $$T = [c_0, c_1, \dots, c_{n-2}, c_{n-1}]$$. The taps of the shift register define the linear recurrence relation.

## Polynomials¶

property characteristic_poly : Poly

The characteristic polynomial $$c(x) = x^{n} - c_{n-1}x^{n-1} - c_{n-2}x^{n-2} - \dots - c_{1}x - c_{0}$$ that defines the linear recurrent sequence.

property feedback_poly : Poly

The feedback polynomial $$f(x) = -c_{0}x^{n} - c_{1}x^{n-1} - \dots - c_{n-2}x^{2} - c_{n-1}x + 1$$ 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 $$S = [S_0, S_1, \dots, S_{n-2}, S_{n-1}]$$.