# 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([33537829449962885674, 16042529009753609664, 10866226205519732295,
14092577331152141395, 16927998116002922971, 36148846129010931589,
2103285211293963847, 4017471091364150894, 28782985382329694155,
2147107529303104376], order=36893488147419103183)

In [6]: b = GF.Random(10); b
Out[6]:
GF([33599818908932537718, 429288030605114487, 9442671844213946301,
20134906419511163876, 14450089583391726852, 11286927895869630241,
12277145855857962698, 22106858698299515242, 30669871520546998723,
4092672160866226338], order=36893488147419103183)

In [7]: a + b
Out[7]:
GF([30244160211476320209, 16471817040358724151, 20308898049733678596,
34227483750663305271, 31378087699394649823, 10542285877461458647,
14380431067151926545, 26124329789663666136, 22559368755457589695,
6239779690169330714], 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._fields._class.DisplayContext at 0x7f5c76d56080>

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._fields._class.DisplayContext at 0x7f5c76d56908>