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I wasn't really satisfied with the previous version, but it was
easiest to get Stein's algorithm working on polynomials by doing it
exactly how I already knew to do it for integers. But now I've
improved it in two ways.
The first improvement I got from another implementation: instead of
transforming A into A - kB for some k that makes the constant term
zero, you can scale _both_ inputs, replacing A with mA - kB for some
k,m. The advantage is that you can calculate m and k very easily, by
making each one the constant term of the other polynomial, which means
you don't need to invert something mod q in every step. (Rather like
the projective-coordinates optimisations in elliptic curves, where
instead of inverting in every step you accumulate the product of all
the factors that need to be inverted, and invert the whole product
once at the very end.)
The second improvement is to abandon my cumbersome unwinding loop that
builds up the output coefficients by reversing the steps in the
original gcd-finding loop. Instead, I do the thing you do in normal
Euclid's algorithm: keep track of the coefficients as you go through
the original loop. I had wanted to do this before, but hadn't figured
out how you could deal with dividing a coefficient by x when (unlike
the associated real value) the coefficient isn't a multiple of x. But
the answer is very simple: x is invertible in the ring we're working
in (its inverse mod x^p-x-1 is just x^{p-1}-1), so you _can_ just
divide your coefficient by x, and moreover, very easily!
Together, these changes speed up the NTRU key generation by about a
factor of 1.5. And they remove lots of complicated code as well, so
everybody wins.
This is the README for PuTTY, a free Windows and Unix Telnet and SSH
client.
PuTTY is built using CMake <https://cmake.org/>. To compile in the
simplest way (on any of Linux, Windows or Mac), run these commands in
the source directory:
cmake .
cmake --build .
Documentation (in various formats including Windows Help and Unix
`man' pages) is built from the Halibut (`.but') files in the `doc'
subdirectory using `doc/Makefile'. If you aren't using one of our
source snapshots, you'll need to do this yourself. Halibut can be
found at <https://www.chiark.greenend.org.uk/~sgtatham/halibut/>.
The PuTTY home web site is
https://www.chiark.greenend.org.uk/~sgtatham/putty/
If you want to send bug reports or feature requests, please read the
Feedback section of the web site before doing so. Sending one-line
reports saying `it doesn't work' will waste your time as much as
ours.
See the file LICENCE for the licence conditions.