# Polynomial PowerMod

Is there equivalent of PowerMod for polynomials in Mathematica?

We have Mod[a^e,m]==PowerMod[a,e,m], $$a$$, $$e$$ and $$m$$ all integers.

PowerMod is much more efficient for large $$e$$.

We have also PolynomialMod[p1^e,p2], $$p1$$ polynomial, $$e$$ integer, $$p2$$ integer or polynomial.

For large $$e$$ it is inefficient.

Something like PolynomialPowerMod would be useful. Is there something like it? Or I have to write my own procedure?

There is an undocumented function for this purpose: AlgebraPolynomialPowerModPolynomialPowerMod[]. For example, one could do

AlgebraPolynomialPowerModPolynomialPowerMod[-1 + x + x^2 - x^4 + x^6, 5, x, x^3 + 1]
70 + 79 x + 8 x^2


which gives the same result as

PolynomialMod[(-1 + x + x^2 - x^4 + x^6)^5, x^3 + 1]
70 + 79 x + 8 x^2


(I previously used the function in this answer for computing a modular inverse.)

• (Could a kind soul please evaluate the code snippet I gave, so that other people can see the expected result?) Mar 1, 2019 at 17:17
• I wonder why it is undocumented? Such a useful function. I was trying to write my own but it was not much efficient. Mar 1, 2019 at 17:27
• I cannot open the package :-(. I got this error: Get::noopen: Cannot open AlgebraPolynomialPowerMod. Mar 1, 2019 at 17:31
• It's not documented because it is used directly by PolynomialMod. Mar 1, 2019 at 17:48
• Per J.M.'s request to run the snippet: AlgebraPolynomialPowerModPolynomialPowerMod[-1 + x + x^2 - x^4 + x^6, 5, x, x^3 + 1] gives 70 + 79 x + 8 x^2 as does PolynomialMod[(-1 + x + x^2 - x^4 + x^6)^5, x^3 + 1] Mar 1, 2019 at 23:01

According to the documentation from Wolfram,

AlgebraPolynomialPowerMod

The functionality of PolynomialPowerMod is now available in the kernel function PolynomialRemainder. Modulus is now an option to the kernel functions PolynomialQuotient and PolynomialRemainder.

I believe it is now possible to avoid this undocumented function AlgebraPolynomialPowerModPolynomialPowerMod[] by doing

PolynomialRemainder[(-1 + x + x^2 - x^4 + x^6)^5, x^3 + 1, x]
70 + 79 x + 8 x^2
`