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komm.BinaryPolynomial

Binary polynomial. A binary polynomial is a polynomial whose coefficients are elements in the finite field $\mathbb{F}_2 = \{ 0, 1 \}$. This class supports addition, multiplication, division, and exponentiation.

Examples:

>>> poly1 = komm.BinaryPolynomial(0b10100)  # X^4 + X^2
>>> poly2 = komm.BinaryPolynomial(0b11010)  # X^4 + X^3 + X
>>> poly1 + poly2  # X^3 + X^2 + X
BinaryPolynomial(0b1110)
>>> poly1 * poly2  # X^8 + X^7 + X^6 + X^3
BinaryPolynomial(0b111001000)
>>> poly1**2  # X^8 + X^4
BinaryPolynomial(0b100010000)

__init__()

Default constructor for the class.

Parameters:

  • integer (int) –

    An integer whose binary digits represent the coefficients of the polynomial—the leftmost bit standing for the highest degree term. For example, the binary polynomial $X^4 + X^3 + X$ is represented by the integer 0b11010 = 0o32 = 26.

Examples:

>>> komm.BinaryPolynomial(0b11010)  # X^4 + X^3 + X
BinaryPolynomial(0b11010)

See also the class methods from_coefficients and from_exponents for alternative ways to construct a binary polynomial.

from_coefficients() classmethod

Constructs a binary polynomial from its coefficients.

Parameters:

  • coefficients (Array1D[int]) –

    The coefficients of the binary polynomial—the $i$-th element of the array standing for the coefficient of $X^i$. For example, [0, 1, 0, 1, 1] represents the binary polynomial $X^4 + X^3 + X$.

Examples:

>>> komm.BinaryPolynomial.from_coefficients([0, 1, 0, 1, 1])  # X^4 + X^3 + X
BinaryPolynomial(0b11010)

from_exponents() classmethod

Constructs a binary polynomial from its exponents.

Parameters:

  • exponents (Array1D[int]) –

    The exponents of the nonzero terms of the binary polynomial. For example, [1, 3, 4] represents the binary polynomial $X^4 + X^3 + X$.

Examples:

>>> komm.BinaryPolynomial.from_exponents([1, 3, 4])  # X^4 + X^3 + X
BinaryPolynomial(0b11010)

degree property

The degree of the polynomial.

Examples:

>>> poly = komm.BinaryPolynomial(0b11010)  # X^4 + X^3 + X
>>> poly.degree
4

coefficients()

Returns the coefficients of the binary polynomial.

Parameters:

  • width (Optional[int]) –

    If this parameter is specified, the output will be filled with zeros on the right so that the its length will be the specified value.

Returns:

  • coefficients (Array1D[int]) –

    Coefficients of the binary polynomial. The $i$-th element of the array stands for the coefficient of $X^i$.

Examples:

>>> poly = komm.BinaryPolynomial(0b11010)  # X^4 + X^3 + X
>>> poly.coefficients()
array([0, 1, 0, 1, 1])
>>> poly.coefficients(width=8)
array([0, 1, 0, 1, 1, 0, 0, 0])

exponents()

Returns the exponents of the binary polynomial.

Returns:

  • exponents (Array1D[int]) –

    Exponents of the nonzero terms of the binary polynomial. The exponents are returned in ascending order.

Examples:

>>> poly = komm.BinaryPolynomial(0b11010)  # X^4 + X^3 + X
>>> poly.exponents()
array([1, 3, 4])

evaluate()

Evaluates the polynomial at a given point. Uses Horner's method.

Parameters:

  • point (RingElement) –

    Any Python object supporting the operations of addition, subtraction, and multiplication.

Returns:

  • result (RingElement) –

    The result of evaluating the binary polynomial at point. It has the same type as point.

Examples:

>>> poly = komm.BinaryPolynomial(0b11010)  # X^4 + X^3 + X
>>> poly.evaluate(7)  # same as 7**4 + 7**3 + 7
2751
>>> point = np.array([[1, 2], [3, 4]])
>>> poly.evaluate(point)  # same as point**4 + point**3 + point
array([[  3,  26],
       [111, 324]])

xgcd() classmethod

Performs the extended Euclidean algorithm on two given binary polynomials.

gcd() classmethod

Computes the greatest common divisor (gcd) of the arguments.

lcm() classmethod

Computes the least common multiple (lcm) of the arguments.