NAME
    rcout - decode for REDC algorithms

SYNOPSIS
    rcout(x, m)

TYPES
    x		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
    rcout(x,m) returns the value of B^-N * x % m, where the inverse
    implicit in B^-N is modulo m and the modulus operator % gives the
    least non-negative residue.  The functions rcin() and rcout() are
    inverses of each other for all x:

	    rcout(rcin(x,m), m) = rcin(rcout(x,m),m) = x % m.

    The normal use of rcout() may be said to be that of decoding
    values encoded by rcin() and REDC functions, as in:

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

	    x^2 % m = rcout(rcsq(rcin(x,m),m),m),

	    x ^ k % m = rcout(rcpow(rcin(x,m),k,m),m).

RUNTIME
    If the value of m in rcout(x,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 rcout(x, m), the one
    which is usually faster for small N is used when N <
    config("pow2"); the other is usually faster for larger N. If
    config("pow2") is set at about 200, and x has been reduced modulo
    m, the runtime required for rcout(x, m) is at most about f times
    the runtime required for an N-word by N-word multiplication, where
    f increases from about 1 for N = 1 to near 2 for N >
    config("pow2").  More runtime may be required if x has to be
    reduced modulo m.

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

    > for (i = 0; i < 9; i++) print rcout(i,9),:; print;
    0 7 5 3 1 8 6 4 2

LIMITS
    none

LIBRARY
    void zredcdecode(REDC *rp, ZVALUE z1, ZVALUE *res)

SEE ALSO
    rcout, rcmul, rcsq, rcpow
