The class SmallFpDoubleImpl is a very low level implementation class
for fast arithmetic in a small, prime finite field. It is not intended
for use by casual CoCoALib users, who should instead see the documentation
in QuotientRing (in particular the function NewZZmod), or possibly
the documentation in RingFp, RingFpLog, and RingFpDouble.
Compared to SmallFpImpl the main difference is an implementation
detail: values are represented as doubles -- on 32-bit computers this
allows a potentially usefully greater range of characteristics at a
probably minor run-time cost.
All operations on values must be effected by calling member functions
of the SmallFpDoubleImpl class. Here is a brief summary.
SmallFpDoubleImpl::IsGoodCtorArg(p); // true iff ctor SmallFpDoubleImpl(p) will succeed
SmallFpDoubleImpl::ourMaxModulus(); // largest permitted modulus
SmallFpDoubleImpl ModP(p, convention); // create SmallFpDoubleImpl object
long n;
BigInt N;
BigRat q;
SmallFpImpl::value_t a, b, c;
ModP.myModulus(); // value of p (as a long)
ModP.myReduce(n); // reduce mod p
ModP.myReduce(N); // reduce mod p
ModP.myReduce(q); // reduce mod p
ModP.myExport(a); // returns a preimage (of type long) according to symm/non-neg convention.
ModP.myNegate(a); // -a mod p
ModP.myAdd(a, b); // (a+b)%p;
ModP.mySub(a, b); // (a-b)%p;
ModP.myMul(a, b); // (a*b)%p;
ModP.myDiv(a, b); // (a*inv(b))%p; where inv(b) is inverse of b
ModP.myPower(a, n); // (a^n)%p; where ^ means "to the power of"
ModP.myIsZeroAddMul(a,b,c) // a = (a+b*c)%p; result is (a==0)
For myExport the choice between least non-negative and symmetric
residues is determined by the convention specified when constructing
the SmallFpDoubleImpl object. This convention may be either
GlobalSettings::SymmResidues or
GlobalSettings::NonNegResidues.
Most functions are implemented inline, and no sanity checks are
performed (except when CoCoA_DEBUG is enabled). The constructor
does do some checking. The basic idea is to use the extra precision
available in doubles to allow larger prime finite fields than are
permitted when 32-bit integers are used for all arithmetic. If fast
64-bit arithmetic becomes widespread then this class will probably
become obsolete (unless you have a very fast floating point coprocessor?).
SmallFpDoubleImpl::value_t is simply double. Note that the
values are always non-negative integers with maximum value less than
myModulusValue; i.e. each residue class is represented
(internally) by its least non-negative member.
To avoid problems with overflow the constructor checks that all
integers from 0 to p*p-p can be represented exactly. We need to allow
numbers as big as p*p-p so that myIsZeroAddMul can be implemented easily.
It is not strictly necessary that myModulusValue be prime, though division
becomes only a partial map if myModulusValue is composite. I believe it is
safest to insist that myModulusValue be prime.
The implementation is simplistic -- I wanted to dash it off quickly before going on holiday :-)