The usual way to perform arithmetic in a (small, prime) finite field
is to create the appropriate ring via the pseudo-constructors
NewZZmod (or NewQuotientRing if you prefer) which are
documented in QuotientRing. These functions will automatically
choose a suitable underlying implementation, and you should normally
use them!
If n is a small prime then NewZZmod(n) produces the same result as
NewRingFp(n) (or perhaps NewRingFpDouble(n)). If n is not a
small prime then NewRingFp(n) throws an exception whereas NewZZmod(n)
will produce a working quotient ring. Unless you have a good reason not
to, you should use NewZZmod(n); see QuotientRing.
In some special circumstances, you may wish to choose explicitly the
underlying implementation. CoCoALib offers three distinct
implementations of small prime finite fields: RingFp (described
here), and RingFpLog and RingFpDouble. Of these RingFp is
probably simplest and fastest implementation -- this file describes
how to create a RingFp implementation.
To create a ring of this specific type use one of the pseudo-constructors:
NewRingFp(p) -- p a machine integer or BigInt
NewRingFp(I) -- I an ideal of RingZZ
NewRingFp(p, res) -- p a machine integer, res is either ``GlobalSettings::SymmResidues`` or ``GlobalSettings::NonNegResidues``
These pseudo-constructors are for creating small prime finite fields; they
will fail if the characteristic is not prime or is too large: the error
signalled by throwing a CoCoA::ErrorInfo whose code is
CoCoA::ERR::BadSmallFpChar. You can test whether an argument is
suitable by calling IsGoodForRingFp.
The default convention for printing residues is specified when you create
the GlobalManager; you can also specify explicitly which convention to
use by giving a second argument to the pseudo-ctor NewRingFp. Note
that the internal representation is always least non-negative
regardless of the output convention chosen.
If you seek a means for fast arithmetic in small finite fields consult
the documentation about SmallFpImpl, SmallFpLogImpl, and
SmallFpDoubleImpl. All arithmetic on elements of a RingFp
is actually carried out by a SmallFpImpl object.
The class RingFpImpl is a low-level implementation of (small
prime) finite fields; it is not intended for direct use by casual CoCoA
library users.
The class RingFpImpl is intended to implement small, prime finite
fields. The constructor is more complicated than one might expect,
this is because the RingFpImpl object must store a little extra
information to fulfil its role as a QuotientRingBase. Currently,
the characteristic must be prime (otherwise it wouldn't be a field)
and must also be small enough that its square fits into a
SmallFpElem_t (probably unsigned long, see the file config.H);
if not, an error is signalled.
Extreme efficiency is NOT one of the main features of this version;
contrast this with SmallFpImpl.
The class RingFpImpl derives from QuotientRingBase, which in
turn is derived from RingBase: see QuotientRing and ring
for more details. Note that there is no RingFp class; a
RingFpImpl object can only be accessed as a QuotientRing.
Note the use of "argument checking" static member functions in the ctor:
this is because const data members must be initialized before the main
body of the ctor is entered.
A member typedef RingFpImpl::value_t specifies the type used for
representing the value of an element of a RingFpImpl: this is a
typedef for SmallFpElem_t which is defined in config.H (to facilitate
tuning for different platforms).
The data members are those of a QuotientRingBase (which are used only
for answering queries about a QuotientRing), plus the characteristic
of the field (held as an value_t in myModulus), and an auto-pointer
to a copy of the zero and one elements of the ring.
The zero and one elements of the ring is held in an auto_ptr<> for consistency with the implementation of other rings -- in this simple class it is not really necessary for exception safety.
This implementation is very simplistic: almost every operation is
delegated to the class SmallFpImpl. The implementation class has been
separated so that its inline member functions can be used directly by
some other special case code (e.g. polynomials with SmallFp coeffs).
See SmallFpImpl for details. I note that the residues are
represented internally as the least non-negative value in the residue class
regardless of the user's choice of type of residue.
The largest permitted modulus for a RingFpImpl may depend on the
platform. On a 32-bit machine the modulus must surely be less than
65536 -- refer to SmallFpImpl for details. A 64-bit machine may
allow larger characteristics.
Although it may seem wasteful to use heap memory for the values of
elements in a RingFpImpl, trying to make them "inline" leads to
lots of problems. Originally we had implemented the values as "inline",
and the resulting problems delayed CoCoALib by almost a year.
Why does the class RingFp not exist? Well, my current thoughts
are that since a RingFp would not do anything special which a
QuotientRing cannot do, it seems needless extra complication to
create a "useless" class. In particular, it cannot offer better
run-time performance. If you want to compute quickly modulo a small
prime you must use SmallFpImpl directly.
Probably RingFp, RingFpLog and RingFpDouble could be replaced by
instances of a template class -- the template parameter would be
SmallFpImpl, SmallFpLogImpl or SmallFpDoubleImpl accordingly.
Why do all the member functions blindly forward their calls to the
SmallFpImpl member functions? This means that the error message
for division by zero (say) will refer to SmallFpImpl rather than
RingFpImpl. Does this really matter that much? Obviously the
much same applies to RingFpLogImpl and RingFpDoubleImpl.