galois.Poly

class galois.Poly(coeffs, field=None, order='desc')

Bases: object

A polynomial class with coefficients in any Galois field.

Parameters

• coeffs (array_like) – List of polynomial coefficients of type Galois field array, np.ndarray, list, or tuple. The first element is the highest-degree element if order=”desc” or the first element is the 0-th degree element if order=”asc”.

• field (galois.GFBase, optional) – Optionally specify the field to which the coefficients belong. The default field is galois.GF2. If coeffs is a Galois field array, then that field is used and the field parameter is ignored.

• order (str, optional) – The interpretation of the coefficient degrees, either “desc” (default) or “asc”. For “desc”, the first element of coeffs is the highest degree coefficient (x^(N-1)) and the last element is the 0-th degree element (x^0).

Examples

Create polynomials over GF(2)

# Construct a polynominal over GF(2)
In [13]: a = galois.Poly([1,0,1,1]); a
Out[13]: Poly(x^3 + x + 1, GF2)

# Construct the same polynomial by only specifying its non-zero coefficients
In [14]: b = galois.Poly.NonZero([1,1,1], [3,1,0]); b
Out[14]: Poly(x^3 + x + 1, GF2)

Create polynomials over GF(7)

# Construct the GF(7) field
In [15]: GF = galois.GF_factory(7, 1)

# Construct a polynominal over GF(7)
In [16]: a = galois.Poly([4,0,3,0,0,2], field=GF); a
Out[16]: Poly(4x^5 + 3x^3 + 2, GF7)

# Construct the same polynomial by only specifying its non-zero coefficients
In [17]: b = galois.Poly.NonZero([4,3,2], [5,3,0], field=GF); b
Out[17]: Poly(4x^5 + 3x^3 + 2, GF7)

Polynomial arithmetic

In [18]: a = galois.Poly([1,0,6,3], field=GF); a
Out[18]: Poly(x^3 + 6x + 3, GF7)

In [19]: b = galois.Poly([2,0,2], field=GF); b
Out[19]: Poly(2x^2 + 2, GF7)

In [20]: a + b
Out[20]: Poly(x^3 + 2x^2 + 6x + 5, GF7)

In [21]: a - b
Out[21]: Poly(x^3 + 5x^2 + 6x + 1, GF7)

# Compute the quotient of the polynomial division
In [22]: a / b
Out[22]: Poly(4x, GF7)

# True division and floor division are equivalent
In [23]: a / b == a // b
Out[23]: True

# Compute the remainder of the polynomial division
In [24]: a % b
Out[24]: Poly(5x + 3, GF7)

__init__(coeffs, field=None, order='desc')

Initialize self. See help(type(self)) for accurate signature.

Methods

Decimal(decimal[, field, order])
NonZero(coeffs, degrees[, field])	Examples
__init__(coeffs[, field, order])	Initialize self.
divmod(dividend, divisor)

Attributes

coeffs	The polynomial coefficients as a Galois field array, in descending order if order=”desc” or ascending order if order=”asc”.
coeffs_asc	The polynomial coefficients as a Galois field array in exponent-ascending order, i.e. the first element corresponds to x^0 and the last element corresponds to x^N-1.
coeffs_desc	The polynomial coefficients as a Galois field array in exponent-descending order, i.e. the first element corresponds to x^N-1 and the last element corresponds to x^0.
decimal
degree	The degree of the polynomial, i.e. the highest degree with non-zero coefficient.
field	The finite field to which the coefficients belong.
order	The interpretation of the ordering of the polynomial coefficients.
str

classmethod Decimal(decimal, field=<class 'galois.gf2.GF2'>, order='desc')

classmethod NonZero(coeffs, degrees, field=<class 'galois.gf2.GF2'>)

Examples

# Construct a polynomial over GF2 only specifying the non-zero terms
In [25]: a = galois.Poly.NonZero([1,1,1], [3,1,0]); a
Out[25]: Poly(x^3 + x + 1, GF2)

static divmod(dividend, divisor)

property coeffs

The polynomial coefficients as a Galois field array, in descending order if order=”desc” or ascending order if order=”asc”.

Type

GFBase

property coeffs_asc

The polynomial coefficients as a Galois field array in exponent-ascending order, i.e. the first element corresponds to x^0 and the last element corresponds to x^N-1.

Type

GFBase

property coeffs_desc

The polynomial coefficients as a Galois field array in exponent-descending order, i.e. the first element corresponds to x^N-1 and the last element corresponds to x^0.

Type

GFBase

property decimal

property degree

The degree of the polynomial, i.e. the highest degree with non-zero coefficient.

Type

int

property field

The finite field to which the coefficients belong.

Type

galois.GF2Base or galois.GFpBase

property order

The interpretation of the ordering of the polynomial coefficients. coeffs are in exponent-descending order if order=”desc” and in exponent-ascending order if order=”asc”.

Type

str

property str