galois.FieldArray.arithmetic_table

Generates the specified arithmetic table for the finite field.

Parameters¶

operation: '+' | '-' | '*' | '/'¶: The arithmetic operation.
x: FieldArray | None = None¶: Optionally specify the \(x\) values for the arithmetic table. The default is None which represents \(\{0, \dots, p^m - 1\}\).
y: FieldArray | None = None¶: Optionally specify the \(y\) values for the arithmetic table. The default is None which represents \(\{0, \dots, p^m - 1\}\) for addition, subtraction, and multiplication and \(\{1, \dots, p^m - 1\}\) for division.

Returns¶

A string representation of the arithmetic table.

Examples¶

Arithmetic tables can be displayed using any element representation.

Integer Polynomial Power

In [1]: GF = galois.GF(3**2)

In [2]: print(GF.arithmetic_table("+"))
x + y | 0  1  2  3  4  5  6  7  8 
------|---------------------------
    0 | 0  1  2  3  4  5  6  7  8 
    1 | 1  2  0  4  5  3  7  8  6 
    2 | 2  0  1  5  3  4  8  6  7 
    3 | 3  4  5  6  7  8  0  1  2 
    4 | 4  5  3  7  8  6  1  2  0 
    5 | 5  3  4  8  6  7  2  0  1 
    6 | 6  7  8  0  1  2  3  4  5 
    7 | 7  8  6  1  2  0  4  5  3 
    8 | 8  6  7  2  0  1  5  3  4 

In [3]: GF = galois.GF(3**2, display="poly")

In [4]: print(GF.arithmetic_table("+"))
 x + y |      0       1       2       α   α + 1   α + 2      2α  2α + 1  2α + 2 
-------|------------------------------------------------------------------------
     0 |      0       1       2       α   α + 1   α + 2      2α  2α + 1  2α + 2 
     1 |      1       2       0   α + 1   α + 2       α  2α + 1  2α + 2      2α 
     2 |      2       0       1   α + 2       α   α + 1  2α + 2      2α  2α + 1 
     α |      α   α + 1   α + 2      2α  2α + 1  2α + 2       0       1       2 
 α + 1 |  α + 1   α + 2       α  2α + 1  2α + 2      2α       1       2       0 
 α + 2 |  α + 2       α   α + 1  2α + 2      2α  2α + 1       2       0       1 
    2α |     2α  2α + 1  2α + 2       0       1       2       α   α + 1   α + 2 
2α + 1 | 2α + 1  2α + 2      2α       1       2       0   α + 1   α + 2       α 
2α + 2 | 2α + 2      2α  2α + 1       2       0       1   α + 2       α   α + 1 

In [5]: GF = galois.GF(3**2, display="power")

In [6]: print(GF.arithmetic_table("+"))
x + y |   0    1    α  α^2  α^3  α^4  α^5  α^6  α^7 
------|---------------------------------------------
    0 |   0    1    α  α^2  α^3  α^4  α^5  α^6  α^7 
    1 |   1  α^4  α^2  α^7  α^6    0  α^3  α^5    α 
    α |   α  α^2  α^5  α^3    1  α^7    0  α^4  α^6 
  α^2 | α^2  α^7  α^3  α^6  α^4    α    1    0  α^5 
  α^3 | α^3  α^6    1  α^4  α^7  α^5  α^2    α    0 
  α^4 | α^4    0  α^7    α  α^5    1  α^6  α^3  α^2 
  α^5 | α^5  α^3    0    1  α^2  α^6    α  α^7  α^4 
  α^6 | α^6  α^5  α^4    0    α  α^3  α^7  α^2    1 
  α^7 | α^7    α  α^6  α^5    0  α^2  α^4    1  α^3 

An arithmetic table may also be constructed from arbitrary \(x\) and \(y\).

Integer Polynomial Power

In [7]: GF = galois.GF(3**2)

In [8]: x = GF([7, 2, 8]); x
Out[8]: GF([7, 2, 8], order=3^2)

In [9]: y = GF([1, 4, 5, 3]); y
Out[9]: GF([1, 4, 5, 3], order=3^2)

In [10]: print(GF.arithmetic_table("+", x=x, y=y))
x + y | 1  4  5  3 
------|------------
    7 | 8  2  0  1 
    2 | 0  3  4  5 
    8 | 6  0  1  2 

In [11]: GF = galois.GF(3**2, display="poly")

In [12]: x = GF([7, 2, 8]); x
Out[12]: GF([2α + 1,      2, 2α + 2], order=3^2)

In [13]: y = GF([1, 4, 5, 3]); y
Out[13]: GF([    1, α + 1, α + 2,     α], order=3^2)

In [14]: print(GF.arithmetic_table("+", x=x, y=y))
 x + y |      1   α + 1   α + 2       α 
-------|--------------------------------
2α + 1 | 2α + 2       2       0       1 
     2 |      0       α   α + 1   α + 2 
2α + 2 |     2α       0       1       2 

In [15]: GF = galois.GF(3**2, display="power")

In [16]: x = GF([7, 2, 8]); x
Out[16]: GF([α^3, α^4, α^6], order=3^2)

In [17]: y = GF([1, 4, 5, 3]); y
Out[17]: GF([  1, α^2, α^7,   α], order=3^2)

In [18]: print(GF.arithmetic_table("+", x=x, y=y))
x + y |   1  α^2  α^7    α 
------|--------------------
  α^3 | α^6  α^4    0    1 
  α^4 |   0    α  α^2  α^7 
  α^6 | α^5    0    1  α^4 