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_factory(prime, 1)

In [4]: print(GF)
<Galois Field: GF(36893488147419103183^1), prim_poly = x + 36893488147419103180 (73786976294838206363 decimal)>

In [5]: a = GF.Random(10); a
Out[5]: 
GF([28845574414943853915, 26347834850281415418, 10673849876396859878,
    14852493195359374018, 5866716745713132679, 20543662619226348879,
    6585728557858407962, 29317841840853367709, 26944402151343387836,
    31631072034578594488], order=36893488147419103183)

In [6]: b = GF.Random(10); b
Out[6]: 
GF([25529742649274386766, 22089256298665024175, 32097986997008250367,
    27015246243677871272, 10522436178410799754, 2535421012183951534,
    17593346243178071557, 17083604394999870984, 7967117575706747426,
    22901880998493248233], order=36893488147419103183)

In [7]: a + b
Out[7]: 
GF([17481828916799137498, 11543603001527336410, 5878348725986007062,
    4974251291618142107, 16389152924123932433, 23079083631410300413,
    24179074801036479519, 9507958088434135510, 34911519727050135262,
    17639464885652739538], order=36893488147419103183)

Large GF(2^m) fields

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

In [9]: print(GF)
<Galois Field: GF(2^100), prim_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 (1267650600228486663289456659305 decimal)>

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.gf.DisplayContext at 0x7f8dc24fe630>

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

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

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

In [17]: a * b
Out[17]: 
GF([x^99, x^61, x^75,
    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],
   order=2^100)

# Reset the display mode
In [18]: GF.display()
Out[18]: <galois.gf.DisplayContext at 0x7f8dc209c630>