NAME
    rcmul - REDC multiplication

SYNOPSIS
    rcmul(x, y, m)

TYPES
    x		integer
    y		integer
    m		odd positive integer

    return	integer v, 0 <= v < m.

DESCRIPTION
    Let B be the base calc uses for representing integers internally
    (B = 2^16 for 32-bit machines, 2^32 for 64-bit machines)
    and N the number of words (base-B digits) in the representation
    of m.  Then rcmul(x,y,m) returns the value of B^-N * x * y % m,
    where the inverse implicit in B^-N is modulo m
    and the modulus operator % gives the least non-negative residue.

    The normal use of rcmul() may be said to be that of multiplying modulo m
    values encoded by rcin() and REDC functions, as in:

	    rcin(x * y, m) = rcmul(rcin(x,m), rcin(y,m), m),

    or with only one factor encoded:

	    x * y % m = rcmul(rcin(x,m), y, m).

RUNTIME
    If the value of m in rcmul(x,y,m) is being used for the first time
    in a REDC function, the information required for the REDC
    algorithms is calculated and stored for future use, possibly
    replacing an already stored valued, in a table covering up to 5
    (i.e. MAXREDC) values of m.	 The runtime required for this is about
    two times that required for multiplying two N-word integers.

    Two algorithms are available for evaluating rcmul(x,y,m), the one
    which is usually faster for small N is used when N <
    config("redc2"); the other is usually faster for larger N. If
    config("redc2") is set at about 90 and x and y have both been
    reduced modulo m, the runtime required for rcmul(x,y,m) is at most
    about f times the runtime required for an N-word by N-word
    multiplication, where f increases from about 1.3 for N = 1 to near
    3 for N > 90.  More runtime may be required if x and y have to be
    reduced modulo m.

EXAMPLE
    Using a 64-bit machine with B = 2^32:

    > print rcin(4 * 5, 9), rcmul(rcin(4,9), rcin(5,9), 9), rcout(8, 9);
    8 8 2

LIMITS
    none

LIBRARY
    void zredcmul(REDC *rp, ZVALUE z1, ZVALUE z2, ZVALUE *res)

SEE ALSO
    rcin, rcout, rcsq, rcpow