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.
In some special circumstances, you may wish to choose explicitly the
underlying implementation. CoCoALib offers three distinct
implementations of small prime finite fields: RingFp, and
RingFpLog (described here) and RingFpDouble. Of these
RingFpLog may be the fastest if your processor has slow division
instructions -- this file describes how to create a RingFpLog
implementation.
To create a ring of this specific type use one of the pseudo-constructors:
NewRingFpLog(p) -- Z ring of integers, p a machine integer or BigInt
NewRingFpLog(I) -- Z ring of integers, I an ideal of Z
NewRingFpLog(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 IsGoodFoRingFpLog.
In the directory examples/ there is a small example program showing
how small finite fields (with known implementation) can be created and
used: ex-RingFp2.C.
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 NewRingFpLog. 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 RingFpLogImpl is a low-level implementation of (small
prime) finite fields; it is not intended for direct use by casual CoCoA
library users. Multiplication and division are effected using discrete
log/exp tables.
The class RingFpLogImpl is intended to represent small, prime finite
fields. The constructor is more complicated than one might expect, this
is because the RingFpLogImpl 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).
Furthermore, the characteristic must also be less than 65536 even on
machines with 64-bit arithmetic -- larger values are prohibited as the
internal tables would become excessively large. Creating a
RingFpLogImpl of characteristic p takes time roughly linear in p;
space consumption is linear in p. An error is signalled if the
characteristic is too large or not prime.
Extreme efficiency is NOT one of the main features of this version.
The class RingFpLogImpl derives from QuotientRingBase, which in turn
is derived from RingBase: see QuotientRing and ring for more
details. Note that there is no RingFpLog class; a RingFpLogImpl
object can only be accessed via 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 RingFpLogImpl::value_t specifies the type used for
representing the value of an element of a RingFpLogImpl: currently
this is a typedef for SmallFpLogElem_t which is defined in config.H.
Essentially all operations are delegated to the class SmallFpLogImpl.
The two classes are separate so that the inline operations of
SmallFpLogImpl can be accessed directly in certain other special case
implementations (e.g. polynomials with coeffs in a SmallFp). See the
documentation on SmallFpLogImpl for details. I note that the
residues are represented as the least non-negative value in the residue
class.
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 a value_t in myModulusValue), 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.
The largest permitted modulus for a RingFpLogImpl may depend on the
platform. On a 32-bit machine the modulus must surely be less than
65536 -- refer to SmallFpLogImpl 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 RingFpLogImpl, trying to make them "inline" leads to
lots of problems -- see RingFp for more details
See also some comments in the "bugs" section of RingFp.txt.
The code is not very smart in the case of characteristic 2.
Run-time performance is disappointing.
I wonder if this code will ever prove useful to anyone.