# A perfect power, and then?

Niels Möller nisse at lysator.liu.se
Sat Oct 27 14:06:30 CEST 2012

Torbjorn Granlund <tg at gmplib.org> writes:

> I realise that "Use a small table to get starting value" might not be
> easy to implement for root since one might need k tables.  Other
> possibilities would be:

I think we discussed some months ago. IIRC, to get a starting value for
a^{1/k}, it should work fine to use a table indexed by low bits of a and
*low bits only* of k.

I think the underlying reason is that

\phi(2^m) = 2^{m-1},

hence

a^n (mod 2^m) = a^{n mod 2^{m-1}} (mod 2^m)

My implementation constructs a 4-bit starting value as

r0 = 1 + (((n << 2) & ((a0 << 1) ^ (a0 << 2))) & 8);

(here, a0 is the low input limb, r0 is the low output limb, and the
iteration computes a^{1/n-1} mod a power of two.

We should be able to get a 8-bit starting value using a table lookup on
at most 13 bits (18 KByte). But maybe it's not worth the effort; a
single iteration getting from 4 bits to 8 shouldn't be terribly
expensive.

BTW, for large n one ought to use n mod the right power of 2 for the
powering in the first few iterations, to avoid doing lots of useless
work in powering.

> (2) iterate single limb code before entering the mpn loop.

One should definitely have an initial single-limb loop. Similar to how
it's doen with binvert and binvert_limb.

Regards,
/Niels

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