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multiplicative inverse from irreducible polynomial

asked 2021-07-04 09:41:40 +0200

Rana gravatar image

Would you please explain how to compute multiplicative inverse from irreducible polynomial like x^4 x 1? Even if you can help me what to study to understand this conversion, that will be a great help.

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Welcome to Ask Sage!

slelievre gravatar imageslelievre ( 2021-07-05 19:20:12 +0200 )edit

Not completely clear to me what you are asking.

slelievre gravatar imageslelievre ( 2021-07-05 19:20:16 +0200 )edit

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answered 2021-07-10 06:17:54 +0200

Max Alekseyev gravatar image

If you ask about computing multiplicative inverse modulo an irreducible polynomial, then this can be done by defining a corresponding quotient ring. For example, the following code inverts $x+1$ modulo $x^4+x+1$ over the field $GF(2)$:

R.<x> = PolynomialRing(GF(2))
K.<a> = R.quotient_ring(x^4+x+1)
print( (a+1)^(-1) )

Here a stands for a zero of $x^4+x+1$, which also may be thought of as the image of $x$ in the quotient ring.

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Asked: 2021-07-04 09:41:40 +0200

Seen: 71 times

Last updated: Jul 10 '21