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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?

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2 Answers 2

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

Algebra`PolynomialPowerMod`PolynomialPowerMod[-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.)

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  • $\begingroup$ (Could a kind soul please evaluate the code snippet I gave, so that other people can see the expected result?) $\endgroup$ Mar 1, 2019 at 17:17
  • $\begingroup$ I wonder why it is undocumented? Such a useful function. I was trying to write my own but it was not much efficient. $\endgroup$ Mar 1, 2019 at 17:27
  • $\begingroup$ I cannot open the package :-(. I got this error: Get::noopen: Cannot open AlgebraPolynomialPowerMod. $\endgroup$ Mar 1, 2019 at 17:31
  • $\begingroup$ It's not documented because it is used directly by PolynomialMod. $\endgroup$ Mar 1, 2019 at 17:48
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    $\begingroup$ 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] $\endgroup$
    – Rabbit
    Mar 1, 2019 at 23:01
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According to the documentation from Wolfram,

Algebra`PolynomialPowerMod

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 Algebra`PolynomialPowerMod`PolynomialPowerMod[] by doing

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