The integer representation of the polynomial.

Notes¶

Int() and __int__() are inverse operations.

For the polynomial \(f(x) = a_d x^d + a_{d-1} x^{d-1} + \dots + a_1 x + a_0\) over the field \(\mathrm{GF}(p^m)\), the integer representation is \(i = a_d (p^m)^{d} + a_{d-1} (p^m)^{d-1} + \dots + a_1 (p^m) + a_0\) using integer arithmetic, not finite field arithmetic.

Said differently, the polynomial coefficients \(\{a_d, a_{d-1}, \dots, a_1, a_0\}\) are considered as the \(d\) “digits” of a radix-\(p^m\) value. The polynomial’s integer representation is that value in decimal (radix-10).

Examples¶

In [1]: GF = galois.GF(7)

In [2]: f = galois.Poly([3, 0, 5, 2], field=GF); f
Out[2]: Poly(3x^3 + 5x + 2, GF(7))

In [3]: int(f)
Out[3]: 1066

In [4]: int(f) == 3*GF.order**3 + 5*GF.order**1 + 2*GF.order**0
Out[4]: True