galois.berlekamp_massey¶

galois.berlekamp_massey(sequence: FieldArray, output: 'minimal' = 'minimal') → Poly¶

galois.berlekamp_massey(sequence: FieldArray, output: 'fibonacci') → FLFSR

galois.berlekamp_massey(sequence: FieldArray, output: 'galois') → GLFSR

Finds the minimal polynomial \(c(x)\) that produces the linear recurrent sequence \(y\).

This function implements the Berlekamp-Massey algorithm.

Parameters

sequence: FieldArray¶

A linear recurrent sequence \(y\) in \(\mathrm{GF}(p^m)\).

output: 'minimal' = 'minimal'¶

output: 'fibonacci'

output: 'galois'

The output object type.

• "minimal" (default): Returns the minimal polynomial that generates the linear recurrent sequence. The minimal polynomial is the characteristic polynomial \(c(x)\) of minimal degree.

• "fibonacci": Returns a Fibonacci LFSR that produces \(y\).

• "galois": Returns a Galois LFSR that produces \(y\).

Returns

The minimal polynomial \(c(x)\), a Fibonacci LFSR, or a Galois LFSR, depending on the value of output.

Notes

The minimal polynomial is the characteristic polynomial \(c(x)\) of minimal degree that produces the linear recurrent sequence \(y\).

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

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

For a linear sequence with order \(n\), at least \(2n\) output symbols are required to determine the minimal polynomial.

References

• Gardner, D. 2019. “Applications of the Galois Model LFSR in Cryptography”. figshare. https://hdl.handle.net/2134/21932.

• Sachs, J. Linear Feedback Shift Registers for the Uninitiated, Part VI: Sing Along with the Berlekamp-Massey Algorithm. https://www.embeddedrelated.com/showarticle/1099.php

• https://crypto.stanford.edu/~mironov/cs359/massey.pdf

Examples

The sequence below is a degree-4 linear recurrent sequence over \(\mathrm{GF}(7)\).

In [1]: GF = galois.GF(7)

In [2]: y = GF([5, 5, 1, 3, 1, 4, 6, 6, 5, 5])

The characteristic polynomial is \(c(x) = x^4 + x^2 + 3x + 5\) over \(\mathrm{GF}(7)\).

In [3]: galois.berlekamp_massey(y)
Out[3]: Poly(x^4 + x^2 + 3x + 5, GF(7))

Use the Berlekamp-Massey algorithm to return equivalent Fibonacci and Galois LFSRs that reproduce the sequence.

Fibonacci LFSR

In [4]: lfsr = galois.berlekamp_massey(y, output="fibonacci")

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

In [6]: z = lfsr.step(y.size); z
Out[6]: GF([5, 5, 1, 3, 1, 4, 6, 6, 5, 5], order=7)

In [7]: np.array_equal(y, z)
Out[7]: True

Galois LFSR

In [8]: lfsr = galois.berlekamp_massey(y, output="galois")

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: [2, 6, 5, 5]
  initial_state: [2, 6, 5, 5]

In [10]: z = lfsr.step(y.size); z
Out[10]: GF([5, 5, 1, 3, 1, 4, 6, 6, 5, 5], order=7)

In [11]: np.array_equal(y, z)
Out[11]: True