NAME
    rcin - encode for REDC algorithms

SYNOPSIS
    rcin(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
    rcin(x,m) returns the value of B^N * x % m, where the modulus
    operator % here gives the least nonnegative residue.

    If y = rcin(x,m), x % m may be evaluated by x % m = rcout(y, m).

    The "encoding" method of using rcmul(), rcsq(), and rcpow() for
    evaluating products, squares and powers modulo m correspond to the
    formulae:

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

	    rcin(x^2, m) = rcsq(rcin(x,m), m);

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

    Here k is any nonnegative integer.  Using these formulae may be
    faster than direct evaluation of x * y % m, x^2 % m, x^k % m.
    Some encoding and decoding may be bypassed by formulae like:

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

    If m is a divisor of B^N - h for some integer h, rcin(x,m) may be
    computed by using rcin(x,m) = h * x % m.  In particular, if
    m is a divisor of B^N - 1 and 0 <= x < m, then rcin(x,m) = x.
    For example if B = 2^16 or 2^32, this is so for m = (B^N - 1)/d
    for the divisors d = 3, 5, 15, 17, ...

RUNTIME
    The first time a particular value for m is used in rcin(x, m),
    the information required for the REDC algorithms is
    calculated and stored for future use in a table covering up to
    5 (i.e. MAXREDC) values of m.  The runtime required for this is about
    two that required for multiplying two N-word integers.

    Two algorithms are available for evaluating rcin(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 both been reduced
    modulo m, the runtime required for rcin(x, m) is at most about f
    times the runtime required for an N-word by N-word multiplication,
    where f increases from about 1.3 for N = 1 to near 2 for N > 200.
    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 rcin(x, 9),:; print;
    0 4 8 3 7 2 6 1 5

LIMITS
    none

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

SEE ALSO
   rcout, rcmul, rcsq, rcpow
