shamazmazum / cl-polynomial

Polynomial factorization over finite fields with prime number of elements or over integers

cl-polynomial

cl-polynomial performs factorization of polynomials over integers and finite fields with prime number of elements.

For documentation visit https://shamazmazum.github.io/cl-polynomial/.

Additional reading

Here is some books and links which I found useful

• Geddes, Keith O., Stephen R. Czapor, and George Labahn. Algorithms for computer algebra. Springer Science & Business Media, 1992.
• Lidl, Rudolf, and Harald Niederreiter. Finite fields. No. 20. Cambridge university press, 1997.
• https://www.csd.uwo.ca/~mmorenom/CS874/Lectures/Newton2Hensel.html/node17.html (contains typos!)
• https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#Subresultant_pseudo-remainder_sequence

cl-polynomial's Issues

Quadratic Hensel lifting needs too high precision for polynomials of degree > 123

Mignotte bound $M$ for $x^{123}+1$ is $233944127258145193631070042609340645376$. If we factor in $\mathbb{F} _ 2[x]$ and lift to $\mathbb{Z}_{2^m}[x]$, we need find the smallest $l$ such that $2^{2^l} \ge M$, which is $7$ and $2^{2^7} = 340282366920938463463374607431768211456$. For $x^{124}+1$ the bound is $467888254516290387262140085218681290752$, $l = 8$. $2^{2^8} = 115792089237316195423570985008687907853269984665640564039457584007913129639936$ which seems to be very high and slows down calculations with bignums.

Maybe bring back linear lifting just for this case?

Recombination of factors is too slow

When a polynomial $f$ has only few irreducible factors in $\mathbb{Z}[x]$, but a lot of factors in a finite field $\mathbb{F}_p[x]$ which is used for factorization of $f$, the process of finding the desired factors mod p (which in coded in RECOMBINE in zx.lisp) is too slow.

The worst cases in the form $x^n-1$ are:

• $x^{91}-1$ for $n \le 100$ has 4 factors in $\mathbb{Z}[x]$ ( $Q_1(x)Q_7(x)Q_{13}(x)Q_{91}(x)$ ) and 18 factors in $\mathbb{F}_3[x]$
• $x^{121}-1$ for $n \le 200$ has 3 factors in $\mathbb{Z}[x]$ ( $Q_1(x)Q_{11}(x)Q_{121}(x)$ ) and 25 factors in $\mathbb{F}_3[x]$.

So some another recombination method must be implemented to deal with such "bad" polynomials.