Extremely large fields

Arbitrarily-large \(\mathrm{GF}(2^m)\), \(\mathrm{GF}(p)\), \(\mathrm{GF}(p^m)\) fields are supported. Because field elements can’t be represented with numpy.int64, we use dtype=object in the numpy arrays. This enables use of native python int, which doesn’t overflow. It comes at a performance cost though. There are no JIT-compiled arithmetic ufuncs. All the arithmetic is done in pure python. All the same array operations, broadcasting, ufunc methods, etc are supported.

Large GF(p) fields

In [1]: prime = 36893488147419103183

In [2]: galois.is_prime(prime)
Out[2]: True

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

In [4]: print(GF)
<class 'numpy.ndarray over GF(36893488147419103183)'>

In [5]: a = GF.Random(10); a
Out[5]: 
GF([21583199482365953973, 27794698775310868400, 27841110648374942378,
    23723181890755893088, 3484569634501057330, 35794391764370692213,
    3286013591502093028, 33843931425453273476, 25644997368629684079,
    6929312102685306206], order=36893488147419103183)

In [6]: b = GF.Random(10); b
Out[6]: 
GF([23902395712233070946, 30794931141691657225, 33352760668865333264,
    22136588761414881519, 1011262890113393303, 18032744029749184234,
    16198366898132711393, 13234709251699927907, 20360536585663187311,
    32106199342017140228], order=36893488147419103183)

In [7]: a + b
Out[7]: 
GF([8592107047179921736, 21696141769583422442, 24300383169821172459,
    8966282504751671424, 4495832524614450633, 16933647646700773264,
    19484380489634804421, 10185152529734098200, 9112045806873768207,
    2142023297283343251], order=36893488147419103183)

Large GF(2^m) fields

In [8]: GF = galois.GF(2**100)

In [9]: print(GF)
<class 'numpy.ndarray over GF(2^100)'>

In [10]: a = GF([2**8, 2**21, 2**35, 2**98]); a
Out[10]: 
GF([256, 2097152, 34359738368, 316912650057057350374175801344],
   order=2^100)

In [11]: b = GF([2**91, 2**40, 2**40, 2**2]); b
Out[11]: 
GF([2475880078570760549798248448, 1099511627776, 1099511627776, 4],
   order=2^100)

In [12]: a + b
Out[12]: 
GF([2475880078570760549798248704, 1099513724928, 1133871366144,
    316912650057057350374175801348], order=2^100)

# Display elements as polynomials
In [13]: GF.display("poly")
Out[13]: <galois.field.meta.DisplayContext at 0x7f660894a128>

In [14]: a
Out[14]: GF([α^8, α^21, α^35, α^98], order=2^100)

In [15]: b
Out[15]: GF([α^91, α^40, α^40, α^2], order=2^100)

In [16]: a + b
Out[16]: GF([α^91 + α^8, α^40 + α^21, α^40 + α^35, α^98 + α^2], order=2^100)

In [17]: a * b
Out[17]: 
GF([α^99, α^61, α^75,
    α^57 + α^56 + α^55 + α^52 + α^48 + α^47 + α^46 + α^45 + α^44 + α^43 + α^41 + α^37 + α^36 + α^35 + α^34 + α^31 + α^30 + α^27 + α^25 + α^24 + α^22 + α^20 + α^19 + α^16 + α^15 + α^11 + α^9 + α^8 + α^6 + α^5 + α^3 + 1],
   order=2^100)

# Reset the display mode
In [18]: GF.display()
Out[18]: <galois.field.meta.DisplayContext at 0x7f6608e36b38>