© 2005,2010-2011 John Abbott
GNU Free Documentation License, Version 1.2

CoCoALib Documentation Index

User documentation for the class RingFpImpl

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 (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.

NewZZmod or NewRingFp?

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.


Maintainer documentation for the class RingFpImpl

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.

Bugs, shortcomings and other ideas

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.