# galois.GF¶

galois.GF(order: int, irreducible_poly: = None, primitive_element: = None, verify: bool = True, compile: Literal['auto', 'jit-lookup', 'jit-calculate', 'python-calculate'] | None = None, display: Literal['int', 'poly', 'power'] | None = None)

Creates a FieldArray subclass for $$\mathrm{GF}(p^m)$$.

Parameters
order

The order $$p^m$$ of the field $$\mathrm{GF}(p^m)$$. The order must be a prime power.

irreducible_poly

Optionally specify an irreducible polynomial of degree $$m$$ over $$\mathrm{GF}(p)$$ that defines the finite field arithmetic. The default is None which uses the Conway polynomial $$C_{p,m}$$, see conway_poly().

primitive_element

Optionally specify a primitive element of the field. This value is used when building the exponential and logarithm lookup tables and as the base of numpy.log. A primitive element is a generator of the multiplicative group of the field.

For prime fields $$\mathrm{GF}(p)$$, the primitive element must be an integer and is a primitive root modulo $$p$$. The default is None which uses primitive_root().

For extension fields $$\mathrm{GF}(p^m)$$, the primitive element is a polynomial of degree less than $$m$$ over $$\mathrm{GF}(p)$$. The default is None which uses primitive_element().

verify

Indicates whether to verify that the user-provided irreducible polynomial is in fact irreducible and that the user-provided primitive element is in fact a generator of the multiplicative group. The default is True.

For large fields and irreducible polynomials that are already known to be irreducible (which may take a while to verify), this argument may be set to False.

The default irreducible polynomial and primitive element are never verified because they are already known to be irreducible and a multiplicative generator, respectively.

compile

The ufunc calculation mode. This can be modified after class construction with the compile() method. See Compilation Modes for a further discussion.

• None (default): For a newly-created FieldArray subclass, None corresponds to "auto". If the FieldArray subclass already exists, None does not modify its current compilation mode.

• "auto": Selects "jit-lookup" for fields with order less than $$2^{20}$$, "jit-calculate" for larger fields, and "python-calculate" for fields whose elements cannot be represented with numpy.int64.

• "jit-lookup": JIT compiles arithmetic ufuncs to use Zech log, log, and anti-log lookup tables for efficient computation. In the few cases where explicit calculation is faster than table lookup, explicit calculation is used.

• "jit-calculate": JIT compiles arithmetic ufuncs to use explicit calculation. The "jit-calculate" mode is designed for large fields that cannot or should not store lookup tables in RAM. Generally, the "jit-calculate" mode is slower than "jit-lookup".

• "python-calculate": Uses pure-Python ufuncs with explicit calculation. This is reserved for fields whose elements cannot be represented with numpy.int64 and instead use numpy.object_ with Python int (which has arbitrary precision).

display

The field element display representation. This can be modified after class construction with the display() method. See Element Representation for a further discussion.

Returns

A FieldArray subclass for $$\mathrm{GF}(p^m)$$.

Notes

FieldArray subclasses of the same type (order, irreducible polynomial, and primitive element) are singletons. So, calling this class factory with arguments that correspond to the same subclass will return the same class object.

Examples

Create a FieldArray subclass for each type of finite field.

Construct the binary field.

In [1]: GF = galois.GF(2)

In [2]: print(GF.properties)
Galois Field:
name: GF(2)
characteristic: 2
degree: 1
order: 2
irreducible_poly: x + 1
is_primitive_poly: True
primitive_element: 1


Construct a prime field.

In [3]: GF = galois.GF(31)

In [4]: print(GF.properties)
Galois Field:
name: GF(31)
characteristic: 31
degree: 1
order: 31
irreducible_poly: x + 28
is_primitive_poly: True
primitive_element: 3


Construct a binary extension field. Notice the default irreducible polynomial is primitive and $$x$$ is a primitive element.

In [5]: GF = galois.GF(2**8)

In [6]: print(GF.properties)
Galois Field:
name: GF(2^8)
characteristic: 2
degree: 8
order: 256
irreducible_poly: x^8 + x^4 + x^3 + x^2 + 1
is_primitive_poly: True
primitive_element: x


Construct a prime extension field. Notice the default irreducible polynomial is primitive and $$x$$ is a primitive element.

In [7]: GF = galois.GF(3**5)

In [8]: print(GF.properties)
Galois Field:
name: GF(3^5)
characteristic: 3
degree: 5
order: 243
irreducible_poly: x^5 + 2x + 1
is_primitive_poly: True
primitive_element: x


Create a FieldArray subclass for extension fields and specify their irreducible polynomials.

Construct the $$\mathrm{GF}(2^8)$$ field that is used in AES. Notice the irreducible polynomial is not primitive and $$x$$ is not a primitive element.

In [9]: GF = galois.GF(2**8, irreducible_poly="x^8 + x^4 + x^3 + x + 1")

In [10]: print(GF.properties)
Galois Field:
name: GF(2^8)
characteristic: 2
degree: 8
order: 256
irreducible_poly: x^8 + x^4 + x^3 + x + 1
is_primitive_poly: False
primitive_element: x + 1


Construct $$\mathrm{GF}(3^5)$$ with an irreducible, but not primitive, polynomial. Notice that $$x$$ is not a primitive element.

In [11]: GF = galois.GF(3**5, irreducible_poly="x^5 + 2x + 2")

In [12]: print(GF.properties)
Galois Field:
name: GF(3^5)
characteristic: 3
degree: 5
order: 243
irreducible_poly: x^5 + 2x + 2
is_primitive_poly: False
primitive_element: 2x


Finite fields with arbitrarily-large orders are supported.

Construct a large prime field.

In [13]: GF = galois.GF(36893488147419103183)

In [14]: print(GF.properties)
Galois Field:
name: GF(36893488147419103183)
characteristic: 36893488147419103183
degree: 1
order: 36893488147419103183
irreducible_poly: x + 36893488147419103180
is_primitive_poly: True
primitive_element: 3


Construct a large binary extension field.

In [15]: GF = galois.GF(2**100)

In [16]: print(GF.properties)
Galois Field:
name: GF(2^100)
characteristic: 2
degree: 100
order: 1267650600228229401496703205376
irreducible_poly: x^100 + x^57 + x^56 + x^55 + x^52 + x^48 + x^47 + x^46 + x^45 + x^44 + x^43 + x^41 + x^37 + x^36 + x^35 + x^34 + x^31 + x^30 + x^27 + x^25 + x^24 + x^22 + x^20 + x^19 + x^16 + x^15 + x^11 + x^9 + x^8 + x^6 + x^5 + x^3 + 1
is_primitive_poly: True
primitive_element: x


Construct a large prime extension field.

In [17]: GF = galois.GF(109987**4)

In [18]: print(GF.properties)
Galois Field:
name: GF(109987^4)
characteristic: 109987
degree: 4
order: 146340800268433348561
irreducible_poly: x^4 + 3x^2 + 100525x + 3
is_primitive_poly: True
primitive_element: x


Last update: May 18, 2022