The usual way to perform arithmetic in a (small, prime) finite field
is to create the appropriate ring via the pseudo-constructors
NewQuotientRing if you prefer) which are
QuotientRing. These functions will automatically
choose a suitable underlying implementation, and you should normally
In some special circumstances, you may wish to choose explicitly the
underlying implementation. CoCoALib offers three distinct
implementations of small prime finite fields:
RingFpDouble (described here). Of these
RingFpDouble may offer the highest limit on the characteristic
(e.g. on 32-bit machines) -- this file describes how to create a
To create a
ring of this specific type use one of the pseudo-constructors:
NewRingFpDouble(p) -- p a machine integer or BigInt NewRingFpDouble(I) -- I an ideal of Z NewRingFpDouble(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
In the directory
examples/ there is a small example program showing
how small finite fields (with known implementation) can be created and
The default convention for printing residues is specified when you create
GlobalManager; you can also specify explicitly which convention to
use by giving a second argument to the pseudo-ctor
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
SmallFpDoubleImpl. All arithmetic on elements of a
is actually carried out by a
RingFpDoubleImpl is a low-level implementation of (small
prime) finite fields; it is not intended for direct use by casual CoCoA
library users. Internally values are represented using
this may permit a higher maximum characteristic on some computers
RingFpDoubleImpl is intended to represent small, prime
finite fields. The constructor is more complicated than one might
expect; this is because the
RingFpDoubleImpl object must store a
little extra information to fulfil its role as a
Currently, the characteristic must be prime (otherwise it wouldn't be a
field). Furthermore, the characteristic p must also be small enough
that all integers up to p*(p-1) can be represented exactly as
RingFpDoubleImpl takes almost constant time (except for the
primality check). An error is signalled (i.e. a
thrown) if the characteristic is too large or not prime.
Extreme efficiency is NOT one of the main features of this version:
RingFpDoubleImpl derives from
which in turn is derived from
ring for more details. Note that there is no
RingFpDoubleImpl object can only be accessed as a
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 specifies the type used internally for representing
the value of an element of a
RingFpDoubleImpl: currently this is just
SmallFpDoubleImpl::value_t which is
Essentially all operations are delegated to the class
The two classes are separate so that the inline operations of
SmallFpDoubleImpl can be accessed directly in certain other special case
implementations (e.g. polynomials with coeffs in a small finite field). See the
SmallFpDoubleImpl for details.
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
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
RingFpImpl may depend on the
platform. If IEEE doubles are used then moduli up to 67108859 are
permitted -- refer to
SmallFpDoubleImpl for details.
Although it may seem wasteful to use heap memory for the values of
elements in a
RingFpDoubleImpl, trying to make them "inline" leads to
lots of problems -- see
RingFp for more details.
Can reduction modulo p be made faster?
Run-time performance is disappointing.
I wonder if this code will ever prove useful to anyone.